1974
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 36
Modeling of Bathymetry-Locked Residual Eddies in Well-Mixed Tidal Channels with
Arbitrary Depth Variations
CHUNYAN LI
Coastal Studies Institute, Department of Oceanography and Coastal Sciences, School of the Coast and Environment,
Louisiana State University, Baton Rouge, Louisiana
(Manuscript received 20 June 2005, in final form 13 February 2006)
ABSTRACT
The concept of the in–out-type exchange flow in estuaries only applies to situations with significant
freshwater discharge and/or elongated channels with relatively simple variations in depth and coastline
along the channel. In waterways with complex bottom topography, the in–out-type exchange flows may be
replaced by residual eddies that are locked to bathymetry. This paper develops an analytic model for such
tidally induced residual eddies. The model allows arbitrary depth variations in both along- and acrosschannel directions. The model produces residual eddies locked to bathymetry features, similar to observations in Wassaw Sound using a ship-mounted ADCP. Analysis indicates that for problems in a “short”
channel with standing wave characteristics, the residual circulation is significantly influenced by advection.
The frictional effect, however, is dominant and the wave propagation effect cannot be uniformly neglected
(i.e., it sometimes can still compete or reinforce the effect of advection). The bathymetry function plays an
important role in the generation of residual eddies. The wind during a half-dozen field observations does
not appear to have significant effect to alter the structure of the flow field. Nor does the tidal range: the
structures of the residual eddies do not change with the spring–neap tidal variation of tidal amplitude and
remain robust in location. The persistent nature of these residual eddies makes it useful to map them in a
specific area in a way similar to a coastal current. Although variabilities are anticipated in response to wind
and coastal low-frequency sea level changes, the residual eddies will have significant implications to the
flushing of a tidal channel, material transport, and the ecosystem dynamics.
1. Introduction
Residual circulations in estuaries are often described
as exchange flows with inward flows at some locations
and outward flows at other locations. With significant
freshwater discharge, the estuarine flow is usually top
out and bottom in (Hansen and Rattray 1965). When
the cross-channel depth gradient has to be considered,
the exchange flow develops a tendency to become
channel in and shoal out (e.g., Hamrick 1979; Wong
1994; Valle-Levinson et al. 2003). When tidally induced
nonlinear effect is dominant over the density-driven
flow, the exchange flow is shoal in and channel out for
a progressive tidal wave (Li and O’Donnell 1997) but
Corresponding author address: Chunyan Li, Coastal Studies Institute, Department of Oceanography and Coastal Sciences,
School of the Coast and Environment, Louisiana State University,
3rd floor, Howe-Russell Geoscience Complex, Baton Rouge, LA
70803.
E-mail: [email protected]
© 2006 American Meteorological Society
JPO2955
shoal out and channel in for a standing tidal wave (Li
and O’Donnell 2005). Therefore, the density-driven
and tidally driven flows can either compete with each
other (Li et al. 1998) or reinforce each other, depending
on the tidal wave characteristics.
These conceptual pictures, though useful in understanding the physical processes, are based on simplified
conditions. When the complexity of the bathymetry and
coastlines have to be considered, these theories do not
necessarily apply. Intuitively, if the river discharge is
large enough, the above mentioned exchange flow of
Hansen and Rattray (1965), Hamrick (1979), and Wong
(1994) will most likely prove to be qualitatively correct
unless the river flow dominates in which case there is
only outward flow of river water. In a long estuary
where tidal propagation is important, the tidally induced flows will also prove to be qualitatively correct if
the density-driven flow is relatively small. With a strong
density-driven flow, the tidally induced flow will be superimposed on the former. If, on the other hand, the
river discharge is small and the tide is a standing wave,
OCTOBER 2006
the above theories need to be modified where bathymetry and coastlines are complicated. This is because
tidal wave–induced flow reaches its minimum under
standing wave conditions (Li and O’Donnell 2005) and
thus the strong bathymetric gradients and complicated
coastlines will likely to interact with the flow field (advective nonlinear processes) and dominate over the
baroclinic pressure gradient and tidal propagation related nonlinear processes. This interaction between advective nonlinearity and complex bathymetry has not
been well understood.
Observations, laboratory experiments, and modeling
have indeed shown flows more complicated than a
simple in–out style such as the transient and residual
eddies caused by headlands (e.g. Pingree and Maddock
1977a,b,1979; Pingree 1978; Maddock and Pingree
1978; Fang and Yang 1985; Geyer and Signell 1990;
Signell and Geyer 1991; Geyer 1993; Wolanski et al.
1996; Chant and Wilson 1997) and islands (e.g. Wolanski et al. 1984a; Pattiaratchi et al. 1987; Pawlak et al.
2003; Alaee et al. 2004). However, lateral boundary
features (coastline curvature, headland, and island) are
not the only factors that generate transient and residual
eddies. In this paper, we will focus on the effect of
bottom topography and use an analytic model to study
bathymetry-locked residual eddies in a tidal channel.
The model study is motivated by findings from multiple
ship-based observations of flow field in a tidal channel
in Wassaw Sound, on the coast of northeastern Georgia. The model allows arbitrary depth variations both
along and across the channel, allowing us to study the
effect of complex bottom features. In the following sections, the observational findings are introduced first.
The model is then developed and validated. We will
then present the model results and discuss the main
findings from the model.
2. Observations
The study site is at the western part of the Wassaw
Sound (Fig. 1). The Sound has an area of about 5 ⫻ 5
km2. It is surrounded by intertidal salt marsh and connected to numerous tidal creeks with no freshwater input at the head. There is a very limited connection
between the Sound and the Savannah River through a
narrow and shallow channel of the Wilmington River
north of the Sound. The flow field is generally in the
same direction top to bottom during either flood or
ebb.
In the present study, six surveys are conducted in
2001 and 2003 (Table 1), using a vessel-based ADCP.
Each of these surveys spans 11–12 h in time. The observations are conducted during different tidal phases
covering neap tides (25 September 2001), spring tides
LI
1975
(15 May 2003), and different stages in between (all the
other surveys).
In each survey, the research vessel (25-ft R/V Gannet) ran along a predefined route repeatedly from early
morning to 0.5–1 h after sunset. The repeated surveys
are mostly continuous except when a short detour is
made into adjacent creeks to collect data for other purposes. The two surveys conducted on 20 and 25 September 2001 use a “stretched hourglass” ship route
(Figs. 2a,b) that covers part of the main channel in the
Wassaw Sound and the Wilmington River. The
stretched hourglass consists of two sharp triangles with
the two most acute angles facing each other and connected by two straight lines. The total length for this
route is 8.8 km. The stretched hourglass covers mostly
the deep channels north and south of the cape at
31.94°N.
The survey on 15 May 2003 uses a “zigzagged” ship
route (Fig. 2c) aimed at concentrating on a smaller segment of the main channel. The total length of this route
is 5.7 km. The last three surveys on 26 June, 13 and 24
October 2003 use a radiator-shaped route (Figs. 2d,e,f)
covering the same area as the 15 May survey. The radiator ship route can cover the channels by crossing
them and cover the banks extensively along the channel. The total length of this route is 6.3 km, except for
the last survey (Fig. 2f) when the cross-channel line
near the Romerly Marsh Creek is extended farther into
the mouth of the creek. The ADCP is configured such
that the velocity data are recorded every 0.5 m along
the vertical and temporal averages are applied with 5-s
intervals for the fourth and fifth surveys but 15 s for the
rest of the surveys (Table 1).
By applying a harmonic-statistical analysis (Li et al.
2000) to the velocity field, the residual flow field is
obtained as shown in Fig. 3. The results shown in this
figure included the nonlinear effect of the finite surface
elevation, that is, the mean velocity is calculated by
averaging through the entire water column (bottom to
surface), rather than an average from the bottom to the
mean surface, which is often called the Eulerian mean
velocity (which is only part of the Eulerian mean velocity omitting the finite surface elevation effect). This
residual flow field from the six surveys is quite consistent which suggests that they are “locked” to the
bathymetry, despite the variable wind conditions. It can
be seen that these residual eddies are small in size (on
kilometer scales). The letters A, B, C, and D marked in
Fig. 4 show deep waters. A more detailed description
about the method and results of these surveys can be
found in Li et al. (2006).
If the observations had been conducted only on a
single cross-channel transect, the in–out exchange flow
1976
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 36
FIG. 1. The study area in Wassaw Sound.
would have been identifiable but different transects
would have given different exchange flow patterns because the residual flow field has strong two-dimensional
variabilities. Also, strong lateral velocity would be seen.
This is different from a simple in–out estuarine circulation in which the cross-channel flow is usually smaller
than the along-channel flow. In the observed two-
dimensional flow field, the in–out-type exchange flow is
no longer consistent any more. Instead, the residual
flow within the channel is in the form of eddies. The
fact that these eddies are locked to the bathymetry
prompt us to consider a modeling experiment to determine the effect of bathymetry and identify the causes.
In this paper, an analytic model is developed to allow
OCTOBER 2006
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TABLE 1. Wassaw Sound surveys (Li et al. 2006).
Date
20
25
15
26
13
24
Sep 2001
Sep 2001
May 2003
Jun 2003
Oct 2003
Oct 2003
Moon phase
Ship route
Repetition
Avg interval
First quarter in 4 days
1 day after first quarter
Full moon
New moon in 3 days
3 days after full moon
1 day before new moon
Stretched hourglass
Stretched hourglass
Zigzags
Radiator
Radiator
Radiator
11
9.5
12
12
10
9
15 s
15 s
15 s
5s
5s
15 s
almost arbitrary bathymetric functions in both alongand cross-channel directions. Complicated bathymetry
functions with large bottom slopes are used. Large bottom slopes can potentially cause problems in numerical
models. In addition, because of the small scales of the
residual eddies, small grid size would be needed and
short time steps would be required if a numerical model
is used. The analytic model is thoroughly validated by
examining the momentum balance and mass conservations. By using this analytic model, we can evaluate the
contributions to momentum balance by different
mechanisms relatively easily. This unique model allows
us to take a close look at the effect of complicated
bathymetry in producing tidally induced residual eddies. The effect of the coastline curvatures, though may
also be very important, is neglected in this study because of the significant increase in complexity, which
prevents an analytic solution.
3. Model development
To examine the effect of complicated bathymetry,
a model that includes the depth function variable in
FIG. 2. The ship tracks along which the vessel-based ADCP observations were conducted on (a) 20 Sep 2001, (b) 25 Sep 2001, (c) 15
May 2003, (d) 26 June 2003, (e) 13 Oct 2003, and (f) 24 Oct 2003. The grids over the ship tracks are used to bin the data for
harmonic-statistic analysis.
1978
JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 36
FIG. 3. Residual velocity surveys conducted on 20 and 25 Sep 2001 and 15 May, 26 Jun, and 13 and 24 Oct 2003 (Li et al. 2006).
both along- and cross-channel directions is needed. The
effect of coastline, though also important in many
occasions, will not be considered to limit the complexity of the problem. Our assumption here is that the
bathymetry effect alone will be able to produce tidally
induced residual eddies in a tidal channel. The
combined effect of the coastline and bathymetry
will complicate the problem in mathematics and should
be investigated separately.
In this work, we are looking at problems in tidal
channels with significant depth variations, the influence
of depth variations on the tidal propagation will have to
be included. Therefore, we will have two major parameters: one is the nonlinear parameter ␧ ⫽ ␨0/h0, in which
␨0 is the tidal amplitude and h0 is the mean depth;
and the second is the depth ratio (hmax ⫺ hmin)/h0, in
which the symbols for maximum and minimum depths
are self-explanatory and h0 is the spatially averaged
water depth. A double perturbation scheme will be
used to solve the problem. The idea is to obtain solutions by separating the first- and second-order equations in the parameter ␧, which measures the importance of the nonlinearity. Though the lowest-order
equations in ␧ are linear, the variation in depth produces variable coefficients in the bottom friction and
continuity. This difficulty is addressed using a second
expansion in the amplitude of the bottom depth variation ␥ ⬃ (hmax ⫺ hmin)/h0. The second-order timeaveraged problem can be solved without a second expansion in ␥ as demonstrated below.
a. Analytic model with arbitrary depth function
The geometry of the model is taken to be a rectangular tidal channel. The depth-averaged, shallow-water
momentum and continuity equations are used:
⭸u
⭸␨ CDu公u2 ⫹ ␷2
⭸u
⭸u
⫹u
⫹␷
⫽ ⫺g ⫺
,
⭸t
⭸x
⭸y
⭸x
h⫹␨
⭸␷
⭸␨ CD␷公u2 ⫹ ␷2
⭸␷
⭸␷
⫹ u ⫹ ␷ ⫽ ⫺g ⫺
,
⭸t
⭸x
⭸y
⭸y
h⫹␨
and
⭸␨ ⭸共h ⫹ ␨兲u ⭸共h ⫹ ␨兲␷
⫹
⫹
⫽ 0,
⭸t
⭸x
⭸y
共1兲
where u, ␷, ␨, h, x, y, t, CD, and g are longitudinal velocity, lateral velocity, elevation, water depth, longitudinal coordinate, lateral coordinate, time, drag coefficient, and the gravitational acceleration, respectively.
OCTOBER 2006
1979
LI
FIG. 4. The study area that shows the bathymetry and observed residual eddies (black lines
with arrows) from the multiple surveys (Fig. 3). The letters A, B, C, and D indicate the
locations of four depth depressions where water depth is on the order of 15 m. The dots along
the channel show the ship track of two of the surveys.
The longitudinal velocity at the head and the lateral
velocity at the side boundaries are zero. The boundary
conditions for the model can be expressed as
i␴t
u | x⫽L ⫽ 0, ␷ | y⫽0,D ⫽ 0, and ␨ | x⫽0 ⫽ Re共ae
兲,
共2兲
where the constants D, ␴, and a are the channel width,
tidal angular frequency, and amplitude at the mouth,
respectively, i ⫽ 公⫺1, and Re( ) indicates the real part
of the complex function inside the parentheses.
As shown in Li and O’Donnell (1997), a Fourier decomposition of the quadratic friction leads to the following equations, correct to the second order:
⭸u
⭸␨
⭸u
u ␤
⭸u
⫹u
⫹␷
⫽ ⫺g ⫺ ␤ ⫹ 2 u␨,
⭸t
⭸x
⭸y
⭸x
h h
␤⫽
8CDU0
,
3␲
共4兲
where CD and U0 are the bottom drag coefficient and
the magnitude of the longitudinal velocity, respectively.
In (3) the quadratic friction has been decomposed into
a linear part and a higher-order term. The advantage of
(3) over the original shallow-water equations is that the
perturbation method can be readily applied for an analytical solution.
The depth is taken to be a function of x and y:
h ⫽ h0共1 ⫹ ␥h1兲,
共5兲
where h0 and ␥ are constants, and h1 is a specified dimensionless function of x and y. To solve the problem
analytically, we assume that ␥ is a number smaller than
O (1). The depth range is
⭸␷
⭸␨
␷
⭸␷
␤
⭸␷
⫹ u ⫹ ␷ ⫽ ⫺g ⫺ ␤ ⫹ 2 ␷␨,
⭸t
⭸x
⭸y
⭸y
h h
␦h ⫽ ␥h0␦h1.
and
⭸␨ ⭸共h ⫹ ␨兲u ⭸共h ⫹ ␨兲␷
⫹
⫹
⫽0
⭸t
⭸x
⭸y
in which
共3兲
共6兲
We further require that the range of h1 or ␦h1 is of O (1).
Therefore, ␥ is a nondimensional measure of the varia-
1980
JOURNAL OF PHYSICAL OCEANOGRAPHY
tion of the depth function. In other words, ␥ is roughly
proportional to (hmax ⫺ hmin)/h0.
The variables are made dimensionless using the following transformations:
y
u
h
x
␷
t̂ ⫽ ␴t, x̂ ⫽ , ŷ ⫽ , û ⫽ , ␷ˆ ⫽ , ĥ ⫽ ,
L
D
U
V
h0
冑
␨
␨˜ ⫽ , U ⫽ a
a
a
g
, V ⫽ ␧␴D, ␧ ⫽ ,
h0
h0
⭸û
⭸t̂
⭸␷ˆ
⭸t̂
⭸␨ˆ
⭸t̂
⫹
L⫽
VOLUME 36
T公gh0
2␲
␭
⫽
, and T ⫽ ,
2␲
2␲
␴
共7兲
where ␴, L, D, U, V, a, and T are the angular frequency
of semidiurnal tide, the tidal length scale, the width of
the estuary, the scale of the longitudinal velocity, the
scale of the lateral velocity, the tidal amplitude at the
mouth, and the period of the tide, respectively.
Equation (3) can then be written as
⫹
V ⭸û
ga ⭸␨ˆ
U ⭸û
␤ û
␤a û␨ˆ
û ⫹
⫺
␷ˆ ⫽ ⫺
⫹ 2 2,
␴L ⭸x̂ ␴D ⭸ŷ
␴UL ⭸x̂ ␴h0 ĥ ␴h0 ĥ
⫹
V ⭸␷ˆ
ga ⭸␨ˆ
U ⭸␷ˆ
␤ ␷ˆ
␤a ␷ˆ ␨ˆ
û ⫹
⫺
␷ˆ ⫽ ⫺
⫹ 2 2,
␴L ⭸x̂ ␴D ⭸ŷ
␴VD ⭸ŷ ␴h0 ĥ ␴h0 ĥ
and
h0U ⭸共ĥ ⫹ ␧␨ˆ 兲û h0V ⭸共ĥ ⫹ ␧␨ˆ 兲␷ˆ
⫹
⫽ 0.
␴aL
⭸x̂
␴aD
⭸ŷ
共8兲
Now we assume that the parameter ␧ is smaller than 1:
It can be shown that
V
ga
h0U
h0V
U
⫽ ␧,
⫽ ␧,
⫽ 1,
⫽ 1, and
⫽ 1.
␴L
␴D
␴UL
␴aL
␴aD
共9兲
␧⫽
ga
⫽H⫽
␴VD
冉公 冊
gh0
␴D
2
.
共11兲
⭸u
⭸␨
u
u␨
⭸u
⭸u
⫹ ␧u ⫹ ␧␷ ⫽ − ⫺ D ⫹ ␧D 2 ,
⭸t
⭸x
⭸y
⭸x
h
h
⭸␷
⭸␨
⭸␷
␷
␷␨
⭸␷
⫹ ␧u ⫹ ␧␷ ⫽ ⫺H ⫺ D ⫹ ␧D 2 ,
⭸t
⭸x
⭸y
⭸y
h
h
and
共12兲
where the dimensionless depth is
h ⫽ 1 ⫹ ␥h1.
␷ ⫽ ␷0 ⫹ ␧␷1 ⫹ ␧2␷2 ⫹ · · · , and
␨ ⫽ ␨0 ⫹ ␧␨1 ⫹ ␧2␨2 ⫹ · · · ,
共13兲
共15兲
in which
ui ⫽ ui共x, y, t兲,
␷i ⫽ ␷i共x, y, t兲, and
␨i ⫽ ␨i共x, y, t兲 共i ⫽ 0, 1, 2, . . .兲.
Omitting the carets, (8) is therefore equivalent to
⭸␨ ⭸共h ⫹ ␧␨兲u ⭸共h ⫹ ␧␨兲␷
⫹
⫹
⫽ 0,
⭸t
⭸x
⭸y
u ⫽ u0 ⫹ ␧u1 ⫹ ␧2u2 ⫹ · · · ,
共10兲
and the coefficient of the across-channel pressure gradient using
共14兲
A power series expansion for the velocity and elevation
in terms of ␧ can be obtained
To reduce the complexity of the notation, we now
write the coefficient of bottom friction using
␤
⫽D
␴h0
␨0
⬍ 1.
h0
共16兲
Since ␧ is smaller than 1, the convergence of (15) is
generally guaranteed for a nonsingular problem. The
smaller ␧ is, the faster the convergence of the solution
will be. Therefore it is desirable that this parameter be
much smaller than unity, for a faster convergence, or
smaller error once the number of terms is fixed. When
the number of terms is given, the smaller this perturbation parameter is, the smaller the errors. On the
other hand, for practical problems, one is most likely
interested in problems with larger ␧ values, which will
require more terms to achieve the same accuracy of the
approximation.
By substituting (15) into (12), the O (␧0) equations,
and the O (␧1) equations are obtained, respectively, as
OCTOBER 2006
1981
LI
latter does not. The expansion in ␥ is used to obtain
an approximate solution to the O (␧0) problem and
this is used to force the O (␧1) problem, which is the
lowest order at which a residual circulation can be resolved.
For a depth function given by (13), since
⭸␨0
u0
⭸u0
⫽⫺
⫺D ,
⭸t
⭸x
h
⭸␷0
␷0
⭸␨0
⫽ ⫺H
⫺D ,
⭸t
⭸y
h
and
⭸␨0 ⭸hu0 ⭸h␷0
⫹
⫹
⫽ 0,
⭸t
⭸x
⭸y
共17兲
1
1
⫽
⫽
h 1 ⫹ ␥h1
and
⭸␨1
u1
u0␨0
⭸u0
⭸u0
⭸u1
⫹ u0
⫹ ␷0
⫽⫺
⫺D ⫹D 2 ,
⭸t
⭸x
⭸y
⭸x
h
h
⭸␨1
⭸␷0
⭸␷0
␷1
␷0␨0
⭸␷1
⫹ u0
⫹ ␷0
⫽ ⫺H
⫺D ⫹D 2 ,
⭸t
⭸x
⭸y
⭸y
h
h
⬁
兺 共⫺1兲 共␥h 兲 ⫽ 1 ⫺ ␥h
i
i
1
1
⫹ ␥2h21 ⫹ · · · ,
i⫽0
共22兲
the problem can be restated as a series of equations—
for ␧0␥0:
⭸␨00
⭸u00
⫽⫺
⫺ Du00,
⭸t
⭸x
and
⭸␨1 ⭸hu1 ⭸␨0u0 ⭸h␷1 ⭸␨0␷0
⫹
⫹
⫹
⫹
⫽ 0.
⭸t
⭸x
⭸x
⭸y
⭸y
The O (␧0) problem is forced by sea level at the open
end of the channel (x ⫽ 0), which is specified as
␨0|x⫽0 ⫽ e jt,
共19兲
and no normal flow at the closed end and sides is enforced by the boundary conditions
and ␷0|y⫽0,1 ⫽ 0.
u0|x⫽1 ⫽ 0
共20兲
Since (17) is linear, the main difficulty in solving the
O (␧0) problem results from the variable coefficient in
the friction and continuity divergence terms. To solve
the O (␧0) problem, (17), the functions are expanded in
terms of the bottom slope parameter ␥ (⬎0) with the
condition that ␥ is smaller than 1. The expansions are
then
u0 ⫽ u00 ⫹ ␥u01 ⫹ ␥2u02 ⫹ · · · ,
共21兲
Note that only the O (␧ ) variables are expanded in ␥,
and that the O (␧0) and O (␧0␥0) are different. The
former includes the effect of the bathymetry and the
0
and
⭸␨00 ⭸u00 ⭸␷00
⫹
⫹
⫽ 0,
⭸t
⭸x
⭸y
共23兲
for ␧0␥1:
⭸␨01
⭸u01
⫽⫺
⫺ D共u01 ⫺ u00h1兲,
⭸t
⭸x
⭸␨01
⭸␷01
⫽⫺
⫺ D共␷01 ⫺ ␷00h1兲,
⭸t
⭸y
and
⭸␨01 ⭸u01 ⭸␷01 ⭸h1u00 ⭸h1␷00
⫹
⫹
⫹
⫹
⫽ 0,
⭸t
⭸x
⭸y
⭸x
⭸y
共24兲
and for ␧0␥i:
⭸␨0i
⭸u0i
⫽⫺
⫺ D u0i ⫹ F̃xi,
⭸t
⭸x
⭸␨0i
⭸␷0i
⫽⫺
⫺ D ␷0i ⫹ F̃yi,
⭸t
⭸y
␷0 ⫽ ␷00 ⫹ ␥␷01 ⫹ ␥2␷02 ⫹ · · · , and
␨0 ⫽ ␨00 ⫹ ␥␨01 ⫹ ␥2␨02 ⫹ · · · .
⭸␨00
⭸␷00
⫽ ⫺H
⫺ D␷00,
⭸t
⭸y
共18兲
⭸␨0i ⭸u0i ⭸␷0i
⫹
⫹
⫽ F̃zi
⭸t
⭸x
⭸y
and
共25兲
—where the error resulting from the truncation of the
expansion is O (␧0, ␥ i⫹1) and
i
i
F̃xi ⫽ ⫺D 关⫺u0,i⫺1h1 ⫹ u0,i⫺2h21 ⫹ · · · ⫹ 共⫺1兲mu0,i⫺mhm
1 ⫹ · · · ⫹ 共⫺1 兲 u0,0h1兴,
i
i
F̃yi ⫽ ⫺D 关⫺␷0,i⫺1h1 ⫹ ␷0,i⫺2h21 ⫹ · · · ⫹ 共⫺1兲m␷0,i⫺mhm
1 ⫹ · · · ⫹ 共⫺1 兲 ␷0,0h1兴,
F̃zi ⫽ ⫺
⭸h1u0,i⫺1 ⭸h1␷0,i⫺1
⫺
.
⭸x
⭸y
and
共26兲
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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 36
The largest neglected term in the O (␧0␥M) solution is
O (␧0␥M⫹1). Since the error for u0, ␷0, ␨0 is O (␧1), consistency requires that
are linear and the tidal forcing frequency is ␴, the solution for u0i, ␷0i and ␨0i can be expressed as
␥M⫹1 ⬃ ␧1,
␷0i ⫽ Re共Vi e jt兲, and
u0i ⫽ Re共Ui e jt兲,
共27兲
␨0i ⫽ Re共Ai e jt兲,
which yields
M⫽
冋
册
2 loge␧
⫺ 1 ⫹ 1,
loge␥
共28兲
where the brackets indicate the largest integer not bigger than the number inside. For instance, for ␧ ⫽ 0.1,
␥ ⫽ 0.5, M ⫽ [2.3] ⫹ 1 ⫽ 3. Therefore, three terms for
u0 are enough in this example.
Since all the systems of the O (␧0␥ i) equations above
共29兲
where Re( ) indicates the real part of the complex function inside the braces. Denoting
F̃xi ⫽ Re共Fxi e jt兲,
F̃yi ⫽ Re共Fyi e jt兲,
and
F̃zi ⫽ Re共Fzi e jt兲,
共30兲
then
i
i
Fxi ⫽ ⫺D 关⫺Ui⫺1h1 ⫹ Ui⫺2h21 ⫹ · · · ⫹ 共⫺1兲mUi⫺mhm
1 ⫹ · · · ⫹ 共⫺1 兲 U0h1兴,
i
i
Fyi ⫽ ⫺D 关⫺Vi⫺1h1 ⫹ Vi⫺2h21 ⫹ · · · ⫹ 共⫺1兲mVi⫺mhm
1 ⫹ · · · ⫹ 共⫺1 兲 V0h1兴,
Fzi ⫽ ⫺
冉
冊
⭸h1Ui⫺1 ⭸h1Vi⫺1
⫺
.
⭸x
⭸y
共31兲
Substituting (29) into (25) and with some straightforward algebra, the following equations are obtained
⭸2Ai
⭸x2
⫹H
⭸2Ai
⭸y2
⫹ ␣2Ai ⫽ Fai,
Ui ⫽ ⫺
Fxi
1 ⭸Ai
⫹
,
j ⫹ D ⭸x
j⫹D
Vi ⫽ ⫺
Fyi
H ⭸Ai
⫹
,
j ⫹ D ⭸y
j⫹D
and
共32兲
for Ai (i ⫽ 1, 2, . . .) is given in the appendix. The
solutions for Ui and Vi are derived from the last two
equations in (32).
Now that the O (␧0) equations have been solved for
u0, ␷0 and ␨0 with an error O (␧1␥M⫹1), assuming M ⫹ 1
terms are used for each variable for the approximation,
the solution of the O (␧1) problem can be addressed.
Since we are dominantly interested in the mean circulation, the equations in (18) are averaged to obtain
in which
␣2 ⫽ 1 ⫺ jD and Fai ⫽
⭸Fxi ⭸Fyi
⫹
⫺ 共 j ⫹ D兲Fzi.
⭸x
⭸y
共33兲
The boundary conditions (19) lead to
Ui|x⫽1 ⫽ 0,
Fyi|y⫽0,1 ⫽ 0,
冏
⭸Ai
⭸y
⫽ 0,
and
Vi|y⫽0,1 ⫽ 0,
⭸␨1
⭸u0
⭸u0
u1
u0␨0
⫹ ␷0
⫽⫺
⫺D ⫹D 2 ,
⭸x
⭸y
⭸x
h
h
u0
⭸␨1
⭸␷0
⭸␷0
␷1
␷0␨0
⫹ ␷0
⫽ ⫺H
⫺D ⫹D 2 ,
⭸x
⭸y
⭸y
h
h
and
⭸hu1 ⭸␨0u0 ⭸h␷1 ⭸␨0␷0
⫹
⫹
⫹
⫽0
⭸x
⭸x
⭸y
⭸y
A0|x⫽0 ⫽ 1,
共35兲
with O (␧2) error, and the boundary conditions are
and
i ⫽ 1, 2, . . . .
u0
共34兲
y⫽0,1
It is obvious that the O (␧0␥i) solution, (u00, ␷00, ␨00), is
the first-order approximation of Li and O’Donnell
(1997), therefore, A0, U0, and V0 are already solved.
After Ak, Uk, and Vk (k ⫽ 0, 1, 2, . . . , i ⫺ 1) are solved,
Ai, Ui, and Vi can then be derived from (32). Thus, ␨0,
u0 and ␷0 can be found using (21) and (29). The solution
u1|x⫽1 ⫽ 0
and ␷1|y⫽0,1 ⫽ 0.
共36兲
The dimensional total (vertically integrated) residual
transport (an Eulerian quantity), integrating the velocity vertically from the bottom to the surface, expressed
in components, is
Tu ⫽ ud共hd ⫹ ␨d兲 and
T␷ ⫽ ␷d共hd ⫹ ␨d兲,
共37兲
OCTOBER 2006
1983
LI
Since the tidally averaged O (␧0) solution is zero for a
weakly nonlinear problem, u0 ⫽ 0 and ␷0 ⫽ 0, the above
equation reduces to
where the superscript “d” indicates a dimensional
quantity to avoid confusion. Dividing the above equations by the mean depth hd, we obtain a quantity with a
unit of velocity:
udT
␷dT
⫽
⫽
Tu
hd
T␷
d
h
⫽u
冉 冊
冉 冊
d
⫽␷
d
1⫹
1⫹
␨d
and
hd
␨d
d
h
冉
冉
共38兲
.
udT
␨u
⫽u⫹␧
U
h
␷T ⫽
␷dT
␨␷
⫽␷⫹␧ .
V
h
冋
冋
and
冉
冉
冊
冊
␷0␨0 h
⭸␷0
⭸␷0
⫺
⫹ ␷0
␷T ⫽ ␧ 2
u0
h
D
⭸x
⭸y
⫺
⫺
in which uT1, uT2, and uT3 are the three components for
the longitudinal residual velocity identified in Li and
O’Donnell (1997):
␨0u0 1
␧,
uT1 ⫽ 2
h
冉
冊
uT2 ⫽ ⫺
h
⭸u0
⭸u0
⫹ ␷0
u0
␧,
D
⭸x
⭸y
uT3 ⫽ ⫺
h ⭸␨1
␧.
D ⭸x
and
冉
␨0␷0
⫹ O 共␧2兲.
h
冉
冉
u0␨0 h
⭸u0
⭸u0
⫺
⫹ ␷0
u0
h
D
⭸x
⭸y
␷1 ⫽
␷0␨0 h
⭸␷0
⭸␷0
⫺
⫹ ␷0
u0
h
D
⭸x
⭸y
冊
h
⭸␷0
⭸␷0
⫹ ␷0
u0
␧,
D
⭸x
⭸y
␷T3 ⫽ ⫺
h ⭸␨1
H
␧.
D ⭸y
共40兲
⫺
⫺
h ⭸␨1
D ⭸x
and
H h ⭸␨1
.
D ⭸x
共41兲
册
册
h ⭸␨1
⫹ O 共␧2兲 ⫽ uT1 ⫹ uT2 ⫹ uT3 and
D ⭸x
H h ⭸␨1
⫹ O 共␧2兲 ⫽ ␷T1 ⫹ ␷T2 ⫹ ␷T3,
D ⭸y
共42兲
has to be solved first. This is accomplished by 1) an
assumption that the mean along-channel pressure gradient is a function of x and thus uniform across the
channel as argued in Li and O’Donnell (1997), and 2)
an integration across the channel to the third equation
in (35). By a total mass conservation requirement under
the condition that the averaged water volume in the
channel remains constant, that is,
冕
1
huT dy ⫽ 0.
共45兲
0
The resulting mean pressure gradient is
⭸␨1
⫽
⭸x
␷T2 ⫽ ⫺
冊
冊
u1 ⫽
共43兲
Similarly ␷T1, ␷T2, and ␷T3 are the three components
for the lateral residual velocity defined by
␨0␷0
␧,
␷T1 ⫽ 2
h
␷T ⫽ ␧ ␷1 ⫹
Therefore, the residual transport velocity components are
共39兲
u0␨0 h
⭸u0
⭸u0
⫺
⫹ ␷0
uT ⫽ ␧ 2
u0
h
D
⭸x
⭸y
␨0u0
⫹ O 共␧2兲 and
h
From the momentum equation, (35), the O (␧1) mean
velocity components can be expressed as
In other words, when the quantity (udT, ␷dT) is multiplied
by the mean depth, it is the total transport in the entire
water column of unit width [ud(␨d ⫹ hd), ␷d(␨d ⫹ hd)].
Note that this velocity is an Eulerian quantity. The dimensionless form of this velocity can be expressed as
uT ⫽
冊
冊
uT ⫽ ␧ u1 ⫹
冕
D
1
2
h dy
冕冋
1
2␨0u0 ⫺
0
冉
h2
⭸u0
⭸u0
⫹ ␷0
u0
D
⭸x
⭸y
冊册
dy.
0
共46兲
and
共44兲
To obtain the residual transport velocity using (42),
the O (␧1) solution for the residual pressure gradient,
The longitudinal component of the residual velocity can
then be obtained via the first equation in (35) or (42).
We cannot use the second equation of (35) or (42) to
solve the lateral component of the residual velocity because we do not know the lateral pressure gradient.
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JOURNAL OF PHYSICAL OCEANOGRAPHY
Applying the continuity equation [i.e., the third equation in (35)], the lateral component of the residual velocity can be solved straightforwardly as
␷T ⫽ ⫺
1 ⭸
h ⭸x
冉冕
y
冊
huT dy .
0
共47兲
The idea of the solution is to use the boundary condition (no net flux across the channel) or the integrated
mass conservation constraint to solve the mean pressure gradient and then solve the longitudinal flow from
the momentum equation and then the lateral flow using
the continuity equation. The lateral momentum equation cannot be solved directly without knowledge of the
lateral pressure gradient (Li and O’Donnell 1997,
2005).
b. Validation of the perturbation solution
We now apply the above model to some depth functions. We will first verify the convergence of the perturbation expansion in (21) and check the momentum
balance and continuity in (17). To this end, the balance
of all major equations at each computational stage is
checked. Here we only show the errors of the equations
for the complex amplitude in (32) and those of the
momentum and continuity equations in (17). An almost
linear (exponential) depth function, as defined in Li
and O’Donnell (1997), is used,
h ⫽ aeby,
共48兲
where a and b are constants related to the minimum
and maximum depth (hmin and hmax) as defined by
a ⫽ hmin
and
b ⫽ ln共hminⲐhmax兲ⲐD.
共49兲
In the present study, we choose the maximum and
minimum depths to be 10 and 5 m, respectively. The
width of the channel is 2 km. Other depth functions
used in Li and Valle-Levinson (1999) have also been
used to check the errors of the model and the conclusion is essentially the same. For brevity, we only discuss
the simplest case without losing its generality. Equation
(32) is for the ith term of the expansion; Eq. (17) is for
the sum of all terms of the expansion. They are therefore representative and sufficient for the test of convergence of the first-order solution. Figure 5a shows the
error of the first equation in (32). The left panel shows
the maximum dimensional error of the equation across
the channel (the value of the difference between the
left-hand side and the right-hand side of the equation)
and the right panel shows the dimensional magnitude of
the right-hand side. The maximum error is smaller than
1%. Note that these are the maximum relative errors
across the channel and therefore are functions of the
VOLUME 36
along-channel position (the horizontal axis). The maximum errors appear to be along the lateral boundaries.
In the interior, the errors are much smaller. Figure 5b
shows the convergence of the perturbation expansion in
(21). Shown in the figure are the dimensional differences of adjacent terms for ␨, u, and ␷, that is, |Ai␥ i ⫺
Ai⫺1␥ i⫺1|, |Ui␥ i ⫺ Ui⫺1␥ i⫺1|, and |Vi␥ i⫺Vi⫺1␥ i⫺1| (i ⫽ 1,
2, . . . , 10). At the fourth term, the error for ␨ is already
almost zero. The convergence of u and ␷ are also shown
to be satisfied in the other two panels. Figure 5c shows
the maximum relative error of the continuity equation
and the magnitude of the terms in the continuity equation along the horizontal axis (the value on the x axis
indicates the position of the grid point of the calculations). The magnitude of the terms is 10⫺4 m s⫺1. The
maximum relative error is about 2% on the boundaries
(solid line in the left panel) and the mean relative error
averaged across the channel is less than 0.8% (dashed
line in the left panel). In summary, the solution satisfies
both the convergence of the perturbation expansion
and the consistency of the equations.
To further demonstrate the validity of the model, we
now compare results from the present work with that
solved with a different analytic method (Li and ValleLevinson 1999). This earlier method does not use the
perturbation expansion but assumes that the lateral
variation of the surface elevation can be neglected for
the first-order tide. The present method does not have
this requirement and it uses a more rigorous perturbation method. On the other hand, the earlier model is
verified by examination of momentum and mass conservations and has an error estimate of ␦a/a (Li and
Valle-Levinson 1999), in which a and ␦a are the tidal
amplitude and the across-channel difference of the tidal
amplitude, respectively. The error was estimated to be
on the order of 2%–3% for a model of 2 km in width.
It is not obvious that the two models should give almost
identical results. The comparison therefore provides
further confidence that the present model produces reliable results. By comparing the results from these two
methods, we can cross-examine these two models. The
resulting tidal velocity and elevation between the two
models are almost identical in magnitude with a maximum difference on the order of 1%. For brevity, we
ignore the details of comparisons between the magnitudes of the velocities or the elevations, which are almost identical between the two models. Instead, we
compare the results of phase differences between u and
␨ and ␷ and ␨ for the two models. The phase of tide and
tidal velocity are generally sensitive to depth variations.
Figure 6 shows the comparison of the results from these
models with exactly the same conditions. The small dif-
OCTOBER 2006
LI
FIG. 5. Validation of the perturbation method. (a) (left) The error of the approximate
solution of the first equation in (32) and (right) the magnitude of the right-hand side of the first
equation in (32), all as functions of the along-channel distance. (b) The magnitude of difference
of adjacent terms for the first 10 terms of the elevation (m) and the two velocity components
(m s⫺1) in the perturbation expansions. (c) (left) The relative error of the continuity equation
[the third equation of (17)] and (right) the magnitude of the terms in the continuity, all as
functions of the channel distance.
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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 36
although the two methods provide almost identical results, the perturbation method can be applied to problems with both longitudinal and transverse depth variations but is limited to only moderate ratios (⬃0.5) of
depth between shoal and channel to ensure convergence of the solution as required by the perturbation
expansion.
4. Results and discussion
a. Residual eddies
FIG. 6. Validation of the perturbation method (continued). (a),
(b), (c) The phase difference between the along-channel velocity
and the elevation, phase difference between the two velocity components, and the phase difference between the cross-channel velocity and the elevation, respectively, for a simplified model (Li
and Valle-Levinson 1999). (d), (e), (f) Same as (a)–(c), but for the
present model.
ferences between the results can provide a reliable
evaluation of the methods.
As can be seen, the two methods produce almost
identical phase differences (Fig. 6). Particularly, Figs
6a,d show the best agreement. The phase difference
between ␷ and ␨ from the perturbation (present) solution shows small alterations of the contours close to the
lateral boundaries (Fig. 6f) but otherwise it is the same
as the Li and Valle-Levinson (1999) solution (Fig. 6c).
The phase difference between ␷ and u from the Li and
Valle-Levinson (1999) solution shows straight-line contours (Fig. 6b), while the perturbation (present) solution shows small wiggles at the mouth (Fig. 6e). In addition, there is about a 1° difference between the two
methods (cf. Figs. 6b,e). It is important to note that
We now apply the analytic solution to a problem in a
short tidal channel with different depth functions. Here
“short” is relative to one-quarter of the wavelength.
The reason for choosing a short channel is because we
are interested in problems with standing wave characteristics in which the phase difference between tidal
elevation and velocity is about 90°. This is the case in
Wassaw Sound and is true for most intercoastal waterways, inlets, lagoons, and tidal creeks. The length and
width of the model domain are 9.3 and 3 km, respectively. We have experimented with different lengths
and widths and no structural changes are seen within
the ranges of interest (width ⬃km, length ⬃10 km). It is
therefore sufficient to just discuss some representative
examples (Fig. 7). Figures 7a,b have three, while Fig. 7c
has only two, depressions of depth. For the first two
examples, the maximum depth of the depressions is
about 16 m while the minimum depth is 8 m. These
depth functions are used to represent the main features
of the depth distribution along the lower portion of the
main channel of the Wassaw Sound (Fig. 4). The difference between Figs. 7a and 7b is that the former has
two depressions near the mouth aligned in the same
direction as the channel, while the two near-mouth depressions of the latter are not aligned in the exact direction of the channel. The small variations of the depth
functions are used to evaluate the sensitivity of the residual eddies. The third example, in which there are
only two depth depressions, has a maximum depth of
12 m.
Figure 8 shows the results of the subtidal (mean) flow
field from the analytic solution. For the first model (Fig.
8a), the landward flow occurs mostly in the deep water
in the two depth depressions. The net oceanward flow
in the shallow water is also quite consistent along the
channel. This is similar to the case within a short channel (shorter than a quarter of the wavelength) with only
cross-channel depth variations (Li and O’Donnell
2005). The present model, however, generates quite
distinct residual eddies which are apparently associated
with the bathymetry. For the second model (Fig. 8b),
OCTOBER 2006
LI
FIG. 7. Depth distributions used in the model. (a), (b), (c) Three different depth functions that have both alongand cross-channel depth variations. These depth functions have depressions similar to those seen in Wassaw Sound
(Fig. 1).
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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 36
FIG. 8. Depth-averaged net flow field in the model with three different depth functions as shown in Figs. 5a–c.
the flow field has some similarities to the first one but
more residual eddies are seen and the eddies are located at different positions along the channel. The largest eddy that is seen near the entrance in the first model
(between the two aligned depth depressions) is now
moved toward the interior of the channel (between the
deepest and shallowest depth depressions) and its size
is significantly smaller. The net flow through the largest
depression now has both landward and oceanward flow
components, different from the first example. This suggests that these residual eddies are quite sensitive to the
bottom topography. In channels where bottom sedi-
mentation significantly changes the topography, the residual eddies may change accordingly. In both Figs.
8a,b, the main residual eddy is clockwise, similar to the
one in Wassaw Sound (Figs. 3 and 4).
For the third model with only two depth depressions
(Fig. 8c), the flow field is again different such that the
residual flow shows some along-channel convergence at
the two depth depressions. Using the flow field within
the depression near the mouth as an example, the net
flow in the deep water is landward on the left-hand side
(closer to the mouth) but oceanward on the right-hand
side (closer to the head). The cross-channel distribution
OCTOBER 2006
1989
LI
FIG. 9. Percentage of contributions to net flow from different mechanisms. Percentage of contributions to net flow from (a) the tidal
wave propagation (uT1), (b) advection (uT2), and (c) net pressure gradient (uT3). (d) A comparison between the contribution from the
first mechanism (uT1) and the combined contribution from the other two mechanisms (uT2 and uT3).
of the net flows on the shallow water outside of the
depression changes accordingly.
For these models, if several cross-channel transects
were used to measure the flow field, each transect
would give rise to different conclusions about the net
transport. The residual eddies are apparently associated with the depth distributions and the mean flow
field is no longer a simple in–out style as in typical flows
in estuaries with significant river discharges. The results
also demonstrate that even without coastline curvatures
including headlands, residual eddies can still be generated, in this case, by bottom topography. The size of the
eddies are on the order of the size of the bathymetry
features.
b. Different contributions to net flow
As (42)–(44) show, there are three contributions to
the net flow. The first (uT1) is from tidal wave propagation and nonlinear bottom friction, the second (uT2)
from advection, and the third (uT3) from the mean pressure gradient, as explained in Lee and O’Donnell
(1997). To evaluate the magnitude of contributions of
these terms to the total net flow, we define the percentage of contribution by uTi (i ⫽ 1, 2, 3) as
Ci ⫽
| uTi |
3
兺 |u
.
共50兲
Tj |
j⫽1
Using the first depth function (Fig. 7a) as an example,
the contributions from different mechanisms are shown
in Fig. 9. Specifically, Figs. 9a,b,c are the spatial distributions of the percentage of contribution from the first,
second, and third terms, respectively, as defined in (50).
In the calculation, the x and y directions have 63 and 41
points, respectively. This yields a total of 2583 points of
calculation on the two-dimensional model domain. The
maximum contributions to the total net flow at an in-
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JOURNAL OF PHYSICAL OCEANOGRAPHY
dividual point from the first, second, and third terms,
are 40.9%, 70%, and 98%, respectively. The spatial average of the contributions from the first, second, and
third terms are 7.3%, 45.0%, and 47.6%, respectively.
For the first term, uT1, it is found that about 76.4% of
these points have contributions to the total net flow
below 0% and 10% (the majority of the points have
small uT1 values). About 20.2% of the points have contributions between 10% and 30% and only 3.4% of
these points have contributions greater than 30%.
For the second term, uT2, it is found that 0.3% of
these points have contributions to the total net flow
below 10%; and 2.8% of the points have contributions
between 10% and 30%. About 64.5% of the points
have contributions between 30% and 50% and 32.4%
of the points have contributions greater than 50%.
For the third term, uT3, it is found that there is no
place within the model having contributions less than
10%; while only 1.6% of the points have contributions
between 10% and 30%. About 60.9% of the points
have contributions to the total net flow between 30%
and 50%; and 36.5% of the points have contributions
between 50% and 70%. About only 1% of the points
have contributions above 70%.
For comparison, the contribution from the first term,
and that of the sum of the second and third terms are
shown in Fig. 9d. From these statistics, we may conclude that in general the first term has little contribution while the second and third terms have comparable
contributions to the total net flow. This, however, is
true with the understanding that at certain points, especially around the depth variations, the contributions
from the first term can vary significantly and therefore
the first term cannot be neglected in the momentum
equation (the maximum contribution is 40% at certain
points). Note that Li and O’Donnell (2005) have discussed the effect of channel length on the residual circulation: a nondimensional length ratio ␦, the ratio between the channel length, and a quarter of the tidal
wavelength, determines if the wave propagation effect
(the first term uT1) is important and if the mean flow
regime falls into the long-channel (␦ ⬎ 1) or shortchannel (␦ ⬍ 1) categories. In the present case, the
channels are short and the first term of the residual
velocity is generally small. This, however, does not imply that the frictional effect is negligible since the primary (or first order) momentum balance is through the
frictional force.
c. Notes on the methodology
Eddies in the flow and residual flow fields can be
generated by different mechanisms. In general, eddies
have been observed around headland or capes (e.g. Pin-
VOLUME 36
gree and Maddock 1977a,b, 1979; Pingree 1978; Maddock and Pingree 1978; Fang and Yang 1985; Signell
and Geyer 1991; Geyer 1993; Wolanski et al. 1996;
Chant and Wilson 1997), behind islands and coral reefs
(e.g. Wolanski et al. 1984a,b; Pattiaratchi et al. 1987;
Pawlak et al. 2003), and in the open ocean. In elongated
tidal channels, the existence of eddies and residual eddies are not well known and studied. The current analytic model only includes the effect of a complex
bathymetry without considering the effect of the coastline curvature including headlands or capes. It is therefore not a replication of the geometry of the Wassaw
Sound where persistent residual eddies have been identified. The model nevertheless demonstrates that the
depth depressions can cause significant alteration of the
flow field to produce residual eddies such that the net
flow is no longer simply in–out as observed in many
estuaries with significant freshwater discharges.
The coastline curvature in the study area does exist,
though not as dramatic as many other places (e.g., farther upstream of the Wilmington River, where there is
an almost 180° turn). The model has completely neglected the curvature of the coastline. On the other
hand, the neglect of the curvature demonstrates that
residual eddies can be generated without the existence
of curvature. When both bottom topography and coastline curvature are complex, the two will likely combine
to generate residual eddies and we expect that the residual eddies would become more pronounced at least
in some cases in which the effect of the coastline and
that of the bathymetry reinforce each other. This remains to be further investigated.
The effect of wind is not included in the model either.
This is based on the fact that wind speed during the
surveys is generally small. More specifically, the daily
average wind speed for the six surveys is 4.8, 5.5, 4.0,
3.6, 5.3, and 4.1 m s⫺1, respectively, in the chronicle
order. Wind direction during the day of the first survey
(20 September 2001) gradually changes clockwise from
the northeast to the southwest, almost reversing the
direction within 24 h. Wind direction during the day of
the second survey (25 September 2001) varies from the
south to the northwest. The day of the third survey (15
May 2003) has a wind direction gradually varying from
the southeast to the northwest, reversing the direction
within 24 h. The day of the forth survey (26 June 2003)
has a wind direction varying from the south to the west,
and then to the south again. Wind direction during the
fifth survey (13 October 2003) is mostly from the northeast. Wind direction during the day of the last survey
(24 October 2003) varies from the south to the southwest and then to the northeast.
The residual eddies do not seem to be dependent on
OCTOBER 2006
the tidal range either. From Table 1 we can see that the
six surveys were conducted during different tidal phases
within the fortnightly cycle and yet the residual eddies
remain robust. The variable wind directions and tidal
ranges, and the fact that the residual eddies are quite
persistent indicate that wind and tidal range do not
have a major impact on the formation of the residual
eddies during these surveys. The persistency of the
large picture of the residual flow patterns also indicates
that the density-driven flow, if any, must also be small
relative to that of the tide. In an estuary where the
density-driven flow can be significant, it usually is a
function of the freshwater discharge, and one should
expect to have a variable effect in different seasons.
Last, with regard to the mathematical method, we
have solved the residual circulation in a channel with
depth functions variable both along and cross the channel. The depth function needs to have a continuous
spatial derivative and the parameter ␥[⬃(hmax ⫺ hmin)/
h0] must be smaller than 1 (preferably 0.5 or smaller) to
ensure the convergence of the power series expansion.
While in reality, these conditions are not always met.
For instance, the water depth at the banks in Wassaw
Sound can be as shallow as less than 1 m, in which case
the analytic model is not applicable. Therefore, the
model can only represent the main features along the
channel, ignoring possible effect in the very shallow
waters. The possible effect of the shallow waters, which
can be exposed at low tides, cannot be assessed with
this model. We expect that, with a more dramatic
change in depth range, the residual eddies may be more
pronounced because of stronger derivatives of bathymetry and flow and the stronger nonlinear effect in the
shallow water.
5. Summary
Motivated by observations of persistent residual eddies in a tidal channel with significant depth variations
in both along- and cross-channel directions, we have
developed an analytic model allowing arbitrary depth
functions to reproduce residual eddies generated by
bathymetry in the absence of coastline curvatures (particularly headlands). The model uses a “double perturbation” procedure to solve the flow fields at tidal and
zero frequencies, respectively. The tidal model is validated by checking momentum balance, mass conservation, convergence of the perturbation expansions, and
by comparison with another model. The residual circulation is checked for momentum balance and mass conservation. The model results show that residual eddies
can be generated by bottom topography in a tidal channel. The eddies can have the scale comparable to those
1991
LI
of the bathymetry features. These eddies are associated
with the depth functions and are sensitive to the depth
variations. In a short channel (shorter than a quarter of
the tidal wavelength in which tide is a standing wave
such that the velocity and elevation have a 90° phase
difference), the main contributions to the residual eddies are the advection and the net pressure gradient.
The tidal wave effect is minimum. On average, there
are about 7.3%, 45%, and 47.6% of the contributions to
the net flow from the tidal wave propagation, advection, and net pressure gradient, respectively. Despite
the average small contribution, the tidal wave propagation in the present examples can still contribute to the
net flow to as large as 10%–40% within 23.6% of the
domain. Therefore, all the mechanisms should be kept
in the model to ensure an accurate representation
throughout the entire model domain. The significance
of the results can be seen when considering the fact that
the bottom topography in natural tidal channels and
estuaries are usually complex, which may generate residual eddies. These eddies are probably more common
than we anticipate. The existence of residual eddies will
further complicate the flushing process of estuaries and
coastal waterways such that recirculation will be common and residence times can be longer than the cases in
which simple in–out residual flows dominate the flushing process.
Acknowledgments. This work was supported by the
University of Connecticut, Georgia Sea Grants
(RR746-007/7512067, R/HAB-12-PD, R/HAB-18-PD,
and RR746-011/7876867), South Carolina Sea Grant
(NOAA NA 960PO113), and Georgia DNR (RR 100279-9262764). The author acknowledges the support of
and discussion with J. O’Donnell of the Department of
Marine Science, University of Connecticut. The author
also thanks Captains H. Carter and R. Thomas for their
assistance in the field to collect the flow data during
multiple surveys. Several students from different institutions participated in the field work. They include S.
Armstrong, S. Traynum, J. Jurisa, and A. Bode from
the Honors College of the University of South Carolina
and B. Coleman from Savannah State University. Two
anonymous reviews that proved to be useful in improving the presentation of the results are appreciated.
APPENDIX
Solution for the Tide
The first-order solution of section 2a is derived in this
appendix. The elevation, and both components of the
tidal velocity can be expressed as
1992
JOURNAL OF PHYSICAL OCEANOGRAPHY
u0i ⫽ Re 共Uie jt兲,
Ui ⫽ ⫺
1 ⭸Ai
Fxi
⫹
,
j ⫹ D ⭸x
j⫹D
Vi ⫽ ⫺
Fyi
H ⭸Ai
⫹
,
j ⫹ D ⭸y
j⫹D
␷0i ⫽ Re共Vie jt 兲, and
␨0i ⫽ Re共Aie jt兲,
共A1兲
where Re ( ) indicates the real part of the complex function inside the braces.
The equations for Ai, Ui and Vi are
⭸2Ai
⭸x2
⫹H
⭸2Ai
and
共A2兲
in which
␣2 ⫽ 1 ⫺ jD
Fai ⫽
and
⭸Fxi ⭸Fyi
⫹
⫺ 共 j ⫹ D兲Fzi,
⭸x
⭸y
共A3兲
⫹ ␣2Ai ⫽ Fai,
⭸y2
VOLUME 36
with
i
i
Fxi ⫽ ⫺D 关⫺Ui⫺1h1 ⫹ Ui⫺2h21 ⫹ · · · ⫹ 共⫺1兲mUi⫺mhm
1 ⫹ · · · ⫹ 共⫺1 兲 U0h1兴,
i
i
Fyi ⫽ ⫺D 关⫺Vi⫺1h1 ⫹ Vi⫺2h21 ⫹ · · · ⫹ 共⫺1兲mVi⫺mhm
1 ⫹ · · · ⫹ 共⫺1 兲 V0h1兴,
Fzi ⫽ ⫺
冉
冊
⭸h1Ui⫺1 ⭸h1Vi⫺1
⫹
.
⭸x
⭸y
共A4兲
To solve Ai for any i ⫽ 1, 2, . . . , divide Ai into two
parts
The boundary conditions for Ai are
冏
⭸Ai
⭸x
⫽ 0 and
x⫽1
冏
⭸Ai
⭸y
⫽ 0, i ⫽ 1, 2, . . . .
Ai ⫽ Ãi共x, y兲 ⫹ Bi共y兲
y⫽0,1
共A5兲
⭸2Ãi
⭸x2
⫹H
⭸2Ãi
⭸y2
⭸Ãi
⭸x
冏
⫽ 0,
⫹ ␣2Bi ⫽ gi and
冏
dBi
dy
and
冏
⫽ 0,
共A8兲
dy2
y⫽0,1
F̃ai ⫽ Fai ⫺ Fai|x⫽0 and
gi ⫽ Fai|x⫽0.
共A9兲
By doing this, the inhomogeneous term in (A7) is
zero at x ⫽ 0, a condition which allows the use of Fourier series expansion containing only sine functions,
thus simplifying the procedure of solving the solution.
共A7兲
⫺ ␻n2 ai,n ⫽ f̃i,n and
dai,n
dy
冏
f̃i,n ⫽
2
H
⫽ 0, n ⫽ 1, 2, 3, . . . ,
共A11兲
F̃ai sin共␭nx兲 dx
共A12兲
y⫽0,1
where
␻n2 ⫽
冕
1
and
0
␭n2 ⫺ ␣2
,
H
共A13兲
which gives rise to the following solution:
a. The solution for Ãi
Using Fourier series expansion, Ã i can be expressed as
⬁
2n ⫺ 1
ai,n sin共␭nx兲, ␭n ⫽
␲,
Ãi ⫽
2
n⫽1
⫽ 0, i ⫽ 1, 2, . . .
y⫽0,1
d2ai,n
where
兺
⭸Ãi
⭸y
which satisfies both boundary conditions at x ⫽ 0 and
x ⫽ 1.
Substituting (A10) into (A7) yields
2
dy2
in which Ãi and Bi satisfy the following equations:
x⫽1
and
d Bi
共A6兲
⫹ ␣2Ãi ⫽ F̃ai,
Ãi|x⫽0 ⫽ 0,
H
and
共A10兲
ai,n ⫽ ci,n共e␻ny ⫹ e⫺␻ny兲
⫹ e⫺␻ny
冕冋 冕
y
0
e2␻ny⬘
y⬘
0
册
e⫺␻ny⬙f̃i,n共y⬙兲 dy⬙ dy,
共A14兲
OCTOBER 2006
LI
in which
ci,n ⫽
1
⫺␻n
␻n共e
Ki,n ⫽
冕
冕
y
⫺ e␻n兲
关⫺␻ne⫺␻nKi,n共1兲 ⫹ e␻nJi,n共1兲兴,
e2␻ny⬘Ji,n共y⬘兲 dy⬘,
and
0
Ji,n ⫽
y
e⫺␻ny⬘f̃i,n共y⬘兲 dy⬘.
共A15兲
0
b. The solution for Bi
Similarly, the solution for Bi can be obtained as
Bi ⫽ di共e␻y ⫹ e⫺␻y兲
⫹ e⫺␻y
冕冋 冕
y
e2␻y⬘
0
y⬘
册
e⫺␻y⬙gi共y⬙兲 dy⬙ dy
0
共A16兲
in which
di ⫽
Gi ⫽
1
⫺␻
␻共e
冕
冕
y
⫺ e␻兲
关⫺␻e⫺␻Gi共1兲 ⫹ e␻Hi共1兲兴,
e2␻y⬘Hi共y⬘兲 dy⬘,
and
0
Hi ⫽
y
e⫺␻y⬘gi共y⬘兲 dy⬘.
共A17兲
0
c. Summary: Procedures of computing the
first-order solution
The procedures of computing the first-order solution
are the following:
1) Given all parameters, including M (number of terms
in the expansion in terms of the depth ratio ␥),
2) find A0 and U0 (V0 ⫽ 0) [complex amplitude for
order (␧1, ␥0) solution].
3) For i ⫽ 1 to M, do loop to find Ai, Ui and Vi.
a. Find Fxi, Fyi, Fzi and Fai, using known Aj, Uj and
Vj ( j ⫽ 1, 2, . . . , i ⫺ 1).
b. Find Bi and Ãi.
c. Find Ai ⫽ Ãi ⫹ Bi.
d. Find Ui and Vi.
4) Find the first-order solution.
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