Continental Shelf Research 143 (2017) 219–227

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Continental Shelf Research
journal homepage: www.elsevier.com/locate/csr

Research papers

Coastal tomographic mapping of nonlinear tidal currents and residual
currents

MARK⁎

Ze-Nan Zhua,b, Xiao-Hua Zhub,a, , Xinyu Guob,caOcean College, Zhejiang University, Zhoushan 316021, China
State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012,
ChinacCenter for Marine Environmental Study, Ehime University, 2-5 Bunkyo-cho, Matsuyama 790-8577, Japanb

A R T I C L E I N F O

A BS T RAC T

Keywords:
Nonlinear tidal currents
Residual currents
M4 and M6 generation mechanisms
Coastal acoustic tomography

Depth-averaged current data, which were obtained by coastal acoustic tomography (CAT) July 12–13, 2009 in
Zhitouyang Bay on the western side of the East China Sea, are used to estimate the semidiurnal tidal current(M2) as well as its ﬁrst two overtide currents (M4 and M6). Spatial mean amplitude ratios M2:M4:M6 in the bay
are 1.00:0.15:0.11. The shallow-water equations are used to analyze the generation mechanisms of M4 and M6.
In the deep area, where water depths are larger than 60 m, M4 velocity amplitudes measured by CAT agree wellwith those predicted by the advection terms in the shallow water equations, indicating that M4 in the deep area
is predominantly generated by the advection terms. M6 measured by CAT and M6 predicted by the nonlinear
quadratic bottom friction terms agree well in the area where water depths are less than 20 m, indicating thatfriction mechanisms are predominant for generating M6 in the shallow area. In addition, dynamic analysis of
the residual currents using the tidally averaged momentum equation shows that spatial mean values of thehorizontal pressure gradient due to residual sea level and of the advection of residual currents together
contribute about 75% of the spatial mean values of the advection by the tidal currents, indicating that residual
currents in this bay are induced mainly by the nonlinear eﬀects of tidal currents. This is the ﬁrst ever nonlinear
tidal current study by CAT.

1. Introduction
The overtides M4 and M6, are common shallow-water constituentsin coastal regions where the M2 tide is dominant. They are generated
when the tide propagates into a shallow-water area. Sea level observational data (e.g., Aubrey and Speer (1985), Speer and Aubrey (1985))
have shown obvious distortion from the combination of the M2 tidewith its overtides. But there are few observational data of M4 and M6
overtides in tidal current data over a large area.
Sea level observations (Aubrey and Speer, 1985; Parker, 1991) andtidal current observations (Blanton et al., 2002; Parker, 1991) haveshown that presence of the M4 tide is a general feature of many
estuaries or coastal regions. Then theoretical analyses (Speer andAubrey, 1985; Friedrichs and Aubrey, 1988; Parker, 1991) andnumerical models (Pingree and Maddock, 1978; Hench and Luettich,
2003; Sheng and Wang, 2004) have indicated that the continuity andadvection terms are the main contributors to generation of the M4 tide.
Sheng and Wang (2004) demonstrated that the advection terms aremost important for generating the M4 tide in Lunenburg Bay, Nova
Scotia, while Pingree and Maddock (1978) pointed out that continuity
terms played a more important role in the English Channel. Therefore
the M4 generation mechanisms are complicated and can be diﬀerent in
diﬀerent regions.
Earlier studies of the M6 tide are mainly based on theoreticalderivation, with Fourier expansion of quadratic bottom friction (Fang,
1987; Parker, 1991; Le Provost, 1991). They point out that quadraticbottom friction is partly nonlinear, and the nonlinear part plays animportant role in generating the M6 tide. Observational data (Blanton
et al., 2002) and numerical models (Sheng and Wang, 2004) have also
been used to discuss the M6 tide generation mechanisms, and conﬁrm
that quadratic bottom friction plays an important role in generating theM6 tide.
Previous M4 and M6 tide analyses based on observational data
mainly used sea level data rather than tidal current data (e.g., Aubrey
and Speer (1985), Parker (1991), Dong and Su (1991a)). Idealized
experiments (e.g., Pingree and Maddock (1980) and Hench et al.

⁎
Corresponding author at: State Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012,
China.
E-mail address: xhzhu@sio.org.cn (X.-H. Zhu).

http://dx.doi.org/10.1016/j.csr.2016.06.014
Received 16 November 2015; Received in revised form 12 June 2016; Accepted 24 June 2016
Available online 25 June 20160278-4343/ © 2016 Published by Elsevier Ltd.

Continental Shelf Research 143 (2017) 219–227

Z.-N. Zhu et al.

2

Residual current speed (m/s)(2002)) and numerical models (e.g., Dong and Su (1991b), Walters andWerner (1991)) were widely used to understand the dynamics ofnonlinear tides. The dynamical mechanisms of nonlinear tides havebeen well documented by these studies.
In previous studies, the observational current data were mainly
point measurements from current meters (e.g., Blanton et al. (2002)
and Parker (1991)), which do not readily enable mapping of the tidal
current structure over a large area. In order to examine M4 generation
mechanisms, the 2-dimensional continuous current ﬁeld should bemapped accurately to allow calculation of the advection terms.
However, with conventional measurement techniques it is diﬃcult to
deploy a large number of current meters within an observation region
to obtain the needed 2-dimensional continuous current ﬁeld. Coastal
Acoustic Tomography (CAT) can achieve this easily by deploying only a
few instruments on the periphery of the observation region (Yamaoka
et al., 2002). Thus, we can analyze the generation mechanisms of M4
and M6 using the current ﬁeld mapped with the CAT.
In order to study the dynamics of overtides M4 and M6 using CAT
data, we chose for our research region a shallow bay, Zhitouyang Bay(hereinafter ZTYB) having a mean water depth of about 25 m andcomplex topography, which is beneﬁcial for generating nonlinear tides.
Zhu et al. (2013) had already determined the horizontal distribution of
tidal currents and residual currents in ZTYB by using CAT (Fig. 1).
They pointed out that the M2 tide was the predominant tidal
constituent, and the M2 velocity amplitudes reached 1.0 m s−1. The
residual currents were largest along the deep channel, with a spatial
mean speed of 0.2 m s−1. Thus, relatively large overtides of the M2 tidewere expected. But they did not further discuss the M2 overtides andthe dynamics of the residual current. The correlation coeﬃcient
between the M2 tidal current amplitude and the residual current speed
is 0.52 (Fig. 2), indicating that the residual current may be induced by
the tidal currents. Moreover, these data of Zhu et al. (2013) are ideal
for analyzing the M2 overtides and the relative importance of diﬀerent
generation mechanisms of these overtides.
In this paper, we use the CAT data to estimate the M4 and M6 tidal
current ellipses. Then we calculate the M4 tidal current ellipses
obtained from the nonlinear advection terms in the shallow waterequations (hereinafter, M4_cal) and the M6 tidal current ellipses
obtained from the nonlinear components of the quadratic bottom
friction terms (hereinafter, M6_cal) to demonstrate the M4 and M6
overtide generation mechanisms. We also discuss the residual current
dynamical mechanisms in the observation region by using the tidally
averaged momentum equation.

r=0.52

1.5

1

0.5

0

0

0.2

0.4

0.6

M2 tidal current ellipse semimajor axis length (m/s)
Fig. 2. Correlation between M2 tidal current amplitude and residual current speed. The
thin solid line indicates the regression line derived from the least-squares method. The
two gray broken lines indicate the standard deviation range.

2. Data set
The data used in this study were obtained from the CAT observation
experiment in ZTYB (Fig. 1). The 3-min interval travel-time diﬀerence
raw data were measured for about 27 h during July 12–13, 2009 (Zhu
et al., 2013). With 27 h observation, we can extract information ofsemidiurnal tides as represented by M2 tidal constituent and of diurnal
tides as represented by K1 tidal constituent as well as that of overtidesof M4 and M6 tidal constituents.
In order to obtain the horizontal distribution of the depth averaged
current ﬁeld from the diﬀerential travel-time data measured by CAT,
the inverse method (Park and Kaneko, 2001) was used. The equation
for the inverse method is as follows:(1)

y = Ex + e,

where y is a 21-element column vector, each column denotes thetravel-time diﬀerence data of each station pair; x is a vector of the 20element row vector of the unknown coeﬃcients used in the Fourier
function expansion of the stream function used to estimate the current
ﬁeld; E is a 21×20 matrix, determined by the locations of the 7 CAT
stations; e is the error vector. The optimum solution of Eq. (1) is
obtained by minimizing the objective function ( J ) with the tapered

15

100 (m)

Ch

ina

Se

a

ZhoushanIslandst

80

Ea

Distance (km)10

LuotouChannelLuotouChannelLaoshumen
ChannelLaoshumen
Channel60
40

5Fodu Channel

2

0

0

4

m/s

5

20

Fodu Channel

1m/s

Qingzimen Channel

10

15

0

Qingzimen Channel

5

10

0

15

Distance (km)Fig. 1. The distribution of (a) M2 tidal current ellipses, and (b) residual currents in Zhitouyang Bay, observed by Zhu et al. (2013). The gray solid circles are the positions of the CAT
stations (C1–C7). The area enclosed by the dashed lines indicates the CAT observational region. The ﬁgure in the upper right of panel (a) shows the East China Sea with a red circle
denoting the location of the study. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).

220

Continental Shelf Research 143 (2017) 219–227

Z.-N. Zhu et al.

⎧ uσ = Uc(x, y )cos(σt ) + Us(x, y )sin(σt )⎨⎩ vσ = Vc(x, y )cos(σt ) + Vs(x , y )sin(σt )least squares method. The objective function is given by:

J = (y − Ex)T (y − Ex) + α 2xT x,(2)

Here, Uc(x, y ), Us(x, y ), Vc(x, y ), Vs(x, y ) are harmonic constants of theM2 tidal current at each spatial point, σ is the M2 angular frequency.
The M4 tide is mainly caused by the nonlinear advection terms of
the M2 tidal current (Parker, 1991). Substituting Eq. (6) into Eq. (5),we ﬁnd that the nonlinear advection terms can be written as follows:

where the weighting factor α is determined by the L-curve method(Hansen and O’Leary, 1993).
Grid size for the data display in the inverse analysis is set to be
1.0×1.0 km, which is the same as that used in Zhu et al. (2013).
For a traditional harmonic tidal theory (Godin, 1972), an observed
current velocity time series v(t ) can be written in the following form:⎧ A0 + Bc cos(2σt ) + Bssin(2σt )⎨⎩ C0 + Dc cos(2σt ) + Dssin(2σt )mv(t ) = v0(t )+ ∑ (Ai sinσti + Bi cosσti )

(3)

i =1

A0 =
=

⎪
⎪

(4)

where, A M4 , BM4 , A M6 , BM6 are harmonic constants of M4 and M6,
respectively. σ M4, σ M6 (i.e., 1.01 and 1.52 h−1) are the angular frequencies of M4 and M6. After obtaining the current ﬁeld at each spatial grid
point for a period of 27 h by using the inverse method, we thencompute the M4 and M6 harmonic constants at each grid point by using
harmonic tidal analysis (Pawlowicz et al., 2002).
The minimum observation time for separating the M4 and M6 is
2π=12.4 h , when the time interval of data is one hour (Godin,σ −σM61972). Our data length is about 27 h which is longer than 12.4 h and is
suﬃcient to separate the M4 and M6 constituents. On the other hand, toresolve well the M4 and M6 overtides, we used 15-min running meansof CAT data, instead of the hourly running means used in Zhu et al.
(2013). This 15-minute time interval in our data allows us to have atleast 16 data points in the M6 period and more than that in the M4
period. The root mean square diﬀerences (RMSDs) of the diﬀerential
travel-time data between the 15-minute running mean data and thehourly running mean data range from 0.05 to 0.30 ms, and thecorresponding RMSDs of velocity range from 0.01 to 0.06 m s−1.
Thus, the 15-min running mean data are credible in this study.

∂V
∂V ⎞
∂V
1 ⎛ ∂Vc+ Vc c + Us s + Vs s ⎟;⎜Uc
2 ⎝ ∂x
∂y
∂x
∂y ⎠
∂U
∂U ⎞
∂U
1 ⎛ ∂Uc+ Vc c + Us s + Vs s ⎟; Bs⎜Uc
2 ⎝ ∂x
∂y
∂x
∂y ⎠

=

dU
∂U
∂U ⎞
1 ⎛ ∂Us+ Us c + Vc s + Vs c ⎟;⎜Uc
2 ⎝ ∂x
∂x
∂y
∂y ⎠

Dc =

∂V ⎞
∂V
∂V
1 ⎛ ∂Vc+ Vc c + Us s + Vs s ⎟; Ds
⎜Uc
∂y ⎠
∂x
2 ⎝ ∂x
∂y

=

∂V
∂V ⎞
∂V
1 ⎛ ∂Vs+ Us c + Vc s + Vs c ⎟ .
⎜Uc
2 ⎝ ∂x∂x∂y∂y ⎠Then we can obtain the currents from the advection terms (i.e.,
M4_cal) by integrating the M4 angular frequency terms in Eq. (7) withrespect to time, as follows:B0⎧⎪ u 2σ = 2σ sin(2σt − αu )⎨D0
⎪ v2σ = sin(2σt − αv )
⎩
2σ

(8)

Here,

⎛D ⎞
⎛B ⎞
Dc 2 + Ds 2 ; αu = arctan⎜ Bs ⎟; αv = arctan⎜ Ds ⎟. The
⎝ c⎠
⎝ c⎠
velocities in Eq. (8), having the M4 angular frequency, are considered
to be the predicted M4 tidal currents. Then, because the predicted M4
currents are derived from the advection terms, if the predicted M4 aresimilar to the M4 currents measured by CAT, we can conﬁrm that theadvection terms make a primary contribution to the generation of M4
tidal currents, otherwise the M4 tidal currents may also be generated byother factors or propagate to the observation site from adjacentregions.
On the other hand, the overtide that has three times the M2 angularfrequency (i.e., 3σ ) is mainly caused by the friction terms (Fang, 1987;
Parker, 1991; Le Provost, 1991). The quadratic bottom friction termsof the linear tide can be written in the following form:
B0 =3. Theoretical derivation of M4 and M6
After the tide propagates into a shallow-water region, the nonlinear
terms in the momentum equations become non-negligible; the nonlinear advection term and the nonlinear friction term are the important
terms in the momentum equations (Fang, 1987). To obtain the M4 andM6 tidal currents generated by the advection and quadratic bottom
friction terms, the derivation of these terms in the two-dimensional
shallow-water equations has been done. In our theoretical derivation,
the two-dimensional shallow-water equations (Le Provost, 1991) canbe written as follows:

Bc 2 + Bs 2 ; D0 =⎧⎪ τmaj = Cd⎪⎨⎪
⎪ τmin = Cd⎩

(5)

Here, x, y are the horizontal spatial coordinates (positive denoting
eastward and northward respectively), t is time, u, v are the horizontal
velocity components; g is gravitational acceleration (=9.8 m s−2); η issurface elevation; Cd is bottom drag coeﬃcient (=0.0025); h is thewater depth; f is Coriolis parameter (=7.27×10−5 s−1). Thus, the
∂u
∂u
∂v
∂v
nonlinear advection terms are u ∂x +v ∂y and u ∂x +v ∂y ; the nonlinear
C

∂U ⎞
∂U
∂U
1 ⎛ ∂Uc+ Vc c + Us s + Vs s ⎟; C0
⎜Uc
∂y ⎠
∂x
2 ⎝ ∂x
∂y

Bc =

M4

⎧ ∂u
Cd
∂η
∂u
∂u
2
2
⎪ ∂t + u ∂x + v ∂y − fv=−g ∂x − h + η u u + v⎨⎪ ∂v + u ∂v + v ∂v + fu=−g ∂η − Cd v u 2 + v 2
∂x
∂y
∂y
h+η
⎩ ∂t

(7)

Here,

where i denotes tidal constituents, Ai , Bi are harmonic constants, σi isthe angular frequency, t denotes the time, v0(t ) is the residual current.
Following this style, the tidal currents of M4 and M6 can be written as:⎧ vM (t ) = A M sinσ M t + BM cosσ M t
4
4
4
4
⎨ 4
⎩ vM6(t ) = A M6 sinσ M6t + BM6cosσ M6t(6)umaj2 + umin 2 umaj/ humaj2 + umin 2 u min / h(9)Here, τmaj and τmin are the bottom friction along the major andminor axis directions of the linear tide, respectively; umaj and umin are
the velocities along the major and minor axis directions of the linear
tide, respectively; h is the water depth. Considering umaj ≫ umin (Fig. 1),
Eq. (9) can be written asC⎧ τmaj = Cd umaj umaj/ h⎨⎩ τmin = Cd umaj umin / hfriction terms are h +d η u u 2 + v 2 and h +d η v u 2 + v 2 . We then derive thetwo-dimensional shallow-water equations in the following steps.
The tidal velocity components of M2 (i.e., uσ and vσ ) can be writtenas⎪⎪Eq. (6) can also be written in the following form:
221

(10)

Continental Shelf Research 143 (2017) 219–227

Z.-N. Zhu et al.

⎧ umaj = Umaj(x, y )cosσt⎨⎩ umin = Umin(x, y )sinσt

tively. Here we should note that, due to the duration of the observation
data, we use M2 as representative of the semidiurnal tidal constituents.

(11)Here, Umaj and Umin are the amplitudes along the major and minor
axis directions of the linear tide, respectively.
When we consider only one linear tidal constituent, the Fourier
expansion of Eq. (10) can be written as follows:

4.1. Results for M4 tidal currents
The horizontal distribution of the M4 tidal ellipses measured by
CAT and those of M4_cal are shown in Fig. 3. The M4 currents (Fig. 3a)are relatively large near the Luotou and Qinzimen channels with amaximum value 0.27 m s−1, corresponding well to the distribution of
the M2 tidal current (Fig. 1). The area averaged semimajor andsemiminor axis lengths of the M4 tidal ellipses are 0.15 and0.04 m s−1, respectively (Table 1). The spatial mean ellipticity of theM4 tide is 0.27, which is larger than that of M2 (0.07). The M4_cal
currents (Fig. 3b) are also relatively larger near the Luotou andQinzimen channels with a maximum value 0.32 m s−1 near Luotou
channel. The M4_cal ellipses have the same area averaged semimajor
and semiminor axis lengths as the observed M4 values (Table 1).
The M4 semimajor axis direction is mainly northeast-southwest in
the west part of the observation region, and becomes northwestsoutheast in the east part (Fig. 3a). The M4_cal semimajor axis direction
is mainly northeast-southwest in the west part of the region, andbecomes north-south in the east part (Fig. 3b).
Comparing the results for the tidal ellipses of M4 (Fig. 3a) andM4_cal (Fig. 3b), the spatial distributions agree better in the west part ofthe observational region (red ellipses in Fig. 3) where the directions of
the semimajor axes of both sets of ellipses are mainly northeastsouthwest. The area averaged semimajor and semiminor axis lengths
are the same for the two sets. The diﬀerence between their maximum
speeds (0.27 m s−1 and 0.32 m s−1, respectively) is 0.05 m s−1, which isless than the standard deviation (STD) of the amplitude diﬀerences
between M4 and M4_cal (0.07 m s−1). From such close maximum speedsof M4 and M4_cal, we therefore conﬁrm that the advection terms play animportant role in generating M4 currents in ZTYB.
Furthermore, we compare the velocity amplitudes of M4 and M4_cal
to conﬁrm the contribution of advection terms in generating the M4tide in diﬀerent areas of the observational region. The relationshipbetween the M4 and M4_cal velocity amplitudes and phase is shown in
Fig. 4. In this ﬁgure we see that not all the data are gathered around the“perfect agreement” diagonal line. We consider the M4 and M4_cal
ellipses to be similar at those points (red dots in Fig. 4) for which thediﬀerences of both velocity amplitude and phase between M4 andM4_cal are less than one STD of the diﬀerences between the M4 andM4_cal values, i.e., 0.07 m s−1 and 110°, respectively. The corresponding ellipses are shown in red in Fig. 3; we ﬁnd that the tidal current
ellipses are most similar in the deep area where water depths are larger
than 60 m. The distribution of M2 ellipses (Fig. 1a) shows that8⎧ τ = U (x , y ) 2 ∑(−1)m +1 (2m −1)(2m +1)(2m +3)π cos(2m+1)σtmaj
m =0,1,2,3,…
⎪
⎪ maj
⎨
⎞
⎛
⎞
⎛
8
⎪ τmin = Umaj⎜x, y⎟Umin⎜x, y⎟ ∑m =0,1,2,3,… (−1)msin(2m+1)σt⎪(2m −1)(2m +3)π
⎠
⎝
⎠
⎝
⎩

(12)
The predicted M6 tidal currents (i.e., M6_cal) are obtained by setting
m=1 and integrating Eq. (12) with respect to time as follows:
8C⎧
u3σ = Umaj(x, y )2 45hπd cos3σt⎪⎨⎪ v3σ =−Umaj(x , y )Umin(x , y ) 4Cd sin3σt
⎩
15hπ

(13)

The velocities in Eq. (13) having the M6 angular frequency are
considered to be the predicted M6 tide. From the similarity between themeasured M6 and predicted M6 currents, we can conﬁrm the contribution of quadratic bottom friction to the generation of the M6 tide. Here
we should note that the phases in Eq. (11) are ignored to makederivation of the Fourier expansion easier.
In this theoretical derivation, we ﬁrstly substitute the tidal current
forms of M2 (Eqs. (6) and (11)) into the advection terms and quadraticbottom friction terms of the shallow water equations (Eq. (5)),
respectively. Then, we obtain the terms having the M4 and M6
frequencies (Eqs. (8) and (13)) as the predicted M4 and M6 tidal
currents. Comparing the predicted M4 and M6 currents (i.e., Eqs. (8)and (13)) with those observed directly by CAT, we can quantify thecontribution of advection terms and quadratic bottom friction terms
for generating M4 and M6.
4. Results and discussion
We estimate the tidal constituents and their errors by a tidal
harmonic analysis using the T_TIDE program (Pawlowicz et al.,2002). For tidal currents in ZTYB, M2 is the strongest constituent.
The M2 tidal current ellipses calculated using the 15-min running meandata have no signiﬁcant diﬀerences from the hourly running mean
results of Zhu et al. (2013) (Fig. 1). The spatial mean semimajor andsemiminor axis lengths of the M2 tidal ellipses are 0.97 and 0.07 m s−1,
respectively. We estimate the spatial mean errors of M2, M4 and M6
velocity amplitudes as 0.06 m s−1, 0.05 m s−1 and 0.05 m s−1, respec-

15

ZhoushanIslandZhoushanIsland100 (m)Distance (km)80

10LuotouChannelLuotouChannelLaoshumen
ChannelLaoshumen
Channel60
40

5Fodu Channel
0.5
m/s
0
1

0

5

Qingzimen Channel

10

15

20

Fodu Channel
0.5
m/s
0
1

0

5

Qingzimen Channel

10

0

15

Distance (km)Fig. 3. Tidal current ellipses of (a) M4, and (b) M4_cal. Red lines indicate similar ellipses for M4 and M4_cal. The area enclosed by the dashed lines indicates the CAT observational region.
(For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)222

Continental Shelf Research 143 (2017) 219–227

Z.-N. Zhu et al.

Table 1
Spatial mean parameters of tidal current ellipses of M4 and M4_cal and their STDs.

M4
M4_calSemimajor axis length ± STDs (m s−1)

Semiminor axis length ± STDs (m s−1)0.15 ± 0.05
0.15 ± 0.07

0.04 ± 0.03
0.04 ± 0.03

those of M6. These results conﬁrm that the quadratic bottom friction
terms play an important role in generating M6 currents in ZTYB.
We further show the relationship of M6 and M6_cal velocity
amplitudes (Fig. 6) to conﬁrm the contribution of the quadratic bottom
friction terms to the generation of M6 in diﬀerent areas of theobservational region. We consider the M6 and M6_cal ellipses to be
similar at those points (red dots in Fig. 6) for which the diﬀerence ofvelocity amplitude between M6 and M6_cal is less than the STD of thediﬀerences between the M6 and M6_cal values (i.e., 0.05 m s−1). Where
the tidal current ellipses are similar (red ellipses in Fig. 5) thissimilarity indicates M6 mainly generated by the quadratic bottom
friction terms; this occurs principally in the east part of the observational region where water depth is less than 20 m. These results
indicate that the quadratic bottom friction terms play a primary rolefor generating M6 in shallow waters, while M6 in the deep area is also
aﬀected by such factors such as the other nonlinear terms in the
momentum equations (Sheng and Wang, 2004).

relatively large M2 currents appear in the deep area, which means thevalues of the advection terms are also larger in the deep area. Thus theadvection terms make a primary contribution to M4 generation in the
deep area. Moreover the tidal current ellipses (black ellipses in Fig. 3)
are not generally similar in shallow waters, that is to say the M4 tidal
currents in the shallow area are not mainly caused by the advection
terms. There are other key factors for generating M4 in the shallow
area, such as the continuity terms or the friction mechanism (Parker,
1991).
4.2. Results of M6 tidal currents

4.3. Asymmetric distortion of M4 and M6
From the results above, the spatially-averaged amplitude ratios
M2:M4:M6 are 1.00:0.15:0.11. The relatively large M4 and M6 overtides
generate strong asymmetric distortion in ZTYB. In this section, we
discuss the characteristics of this asymmetric distortion by using the
ZTYB CAT data.
Asymmetric distortion is mainly caused by the interaction of M2
and M4 (Aubrey and Speer, 1985; Speer and Aubrey, 1985; Friedrichs
and Aubrey, 1988). The ratio of the velocity amplitudes of M4 to M2(hereinafter M4/M2) is calculated to show the asymmetric distortion
resulting from M4 currents in the observational region (Fig. 7a). M4/M2
ranges from 0.02 to 0.39 in the observational region. It is relatively
small ( < 0.2) and varies little in the north and near Fodu channel,
while it abruptly increases to its highest value (0.39) near Qinzimen
channel, indicating that the asymmetry changes dramatically near

0.4

360

M4 tidal current ellipse phase (°)

M4 tidal current ellipse semimajor axis length (m/s)

The horizontal distribution of the M6 tidal ellipses measured by
CAT and those of M6_cal are shown in Fig. 5. The maximum value of theM6 semimajor axes is 0.20 m s−1, appearing near Luotou channel. The
spatial mean semimajor and semiminor axis lengths of the M6 current
ellipses are 0.11 and 0.03 m s−1 (Table 2). The M6 ellipticity isrelatively small in the north part of the observational region, andbecomes larger near Fodu channel. The spatial mean value of ellipticity
is 0.27. The M6_cal currents (Fig. 5b) are relatively larger near Luotou
channel with a maximum value 0.10 m s−1. The spatial mean semimajor and semiminor axis lengths of the M6_cal tidal current ellipses
are 0.07 and 0.01 m s−1, respectively (Table 2). The spatial mean
ellipticity of the M6_cal tide is 0.09.
The orientations of the M6 and M6_cal tidal current ellipses
correspond well throughout the observational region: both are mainly
directed east-west in the north part of the observational region, andturn north-south along the 40 m bathymetric contour.
Comparing the tidal current ellipses of M6 (Fig. 5a) and M6_cal(Fig. 5b), we see that the patterns agree well; the directions of the
major axes of them both turn north-south along the deep area in the
middle of the observational region. The M6 and M6_cal currents areboth relatively large near Luotou channel, and become small near
Qinzimen channel, but the M6_cal currents are slightly smaller than

0.3

0.2

0.1

300
240
180
120
60

0

0

0.1

0.2

0.3

0

0.4

0

M4_cal tidal current ellipse semimajor axis length (m/s)

60

120

180

240

300

360

M4_cal tidal current ellipse phase (°)

Fig. 4. Scatter plot of M4 and M4_cal (a) velocity amplitudes, (b) phases. In each panel the diagonal blue solid line denotes perfect agreement, and the blue dashed lines denote thestandard deviation (STD) range of the velocity amplitudes from the diagonal line. The red dots (black crosses) indicate points at which the diﬀerences of both velocity amplitude andphase between M4 and M4_cal are less (larger) than the STD of the diﬀerences between M4 and M4_cal (i.e., 0.07 m s−1 and 110°). (For interpretation of the references to color in thisﬁgure legend, the reader is referred to the web version of this article.)223

Continental Shelf Research 143 (2017) 219–227

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15

ZhoushanIslandZhoushanIsland100 (m)Distance (km)80

10LuotouChannelLuotouChannelLaoshumen
ChannelLaoshumen
Channel60
40

5Fodu Channel
0.5
m/s
(a) 0
1

0

Qingzimen Channel

5

10

15

20

Fodu Channel
0.5
m/s
(b) 0
1

0

5

Qingzimen Channel

10

0

15

Distance (km)Fig. 5. Tidal current ellipses of (a) M6, and (b) M6_cal. Red lines indicate similar ellipses for M6 and M6_cal. The area enclosed by the dashed lines indicates the CAT observational region.
(For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article).
Table 2
Spatial mean parameters of tidal current ellipses of M6 and M6_cal and their STDs.

M6 tidal current ellipse semimajor axis length (m/s)

M6
M6_calSemimajor axis length ± STDs (m s−1)

Semiminor axis length ± STDs (m s−1)0.11 ± 0.04
0.07 ± 0.02

0.03 ± 0.02
0.01 ± 0.00

0.2

M4 and M4_cal current ellipses (Fig. 3) are not similar and the M4_cal
tidal currents (Fig. 3b) are relatively small, indicating that the advection terms contribute less in this area; furthermore, there are no abruptchanges in M6/M2 (Fig. 7b). Thus the continuity terms should bedominant in generating M4 here, and the M2 energy should mainlytransfer to M4 (Parker, 1991; Blanton et al., 2002).
M6 also plays a minor role in the tidal asymmetry (Blanton et al.,2002). The ratio of the velocity amplitudes of M6 to M2 (hereinafter
M6/M2) is also calculated to show the asymmetric distortion resulting
from M6 in the observational region (Fig. 7b). M6/M2 is less than 0.2
and changes gradually throughout the observational region. It isrelatively small near Fodu channel and in the north part of the region,
as in the case of M4/M2, while M6/M2 is only about 0.05 near Qinzimen
channel. Highest M6/M2 is observed in middle of the CAT region with avalue of 0.17. The relative velocity phase of 3 M2-M6 (Fig. 7d) is about
90° in the middle and south part of the observational region. It
becomes larger than 90° or less than −90° in the west and east partof the observational region. However, the values of M6/M2 (Fig. 7b) arevery small (about 0.1), so the asymmetric distortion caused by M6 isweak (Blanton et al., 2002). Compared with the results of M4/M2, M6/
M2 is relatively small in most of the ZTYB region, indicating that M6contributes less to the asymmetric distortion.

0.15

0.1

0.05

0

0

0.05

0.1

0.15

0.2

M6_cal tidal current ellipse semimajor axis length (m/s)
Fig. 6. Scatter plot of M6 and M6_cal velocity amplitudes. In each panel the diagonal blue
solid line denotes perfect agreement, and the blue dashed lines denote the standard
deviation (STD) range of the velocity amplitudes from the diagonal line. The red dots
(black crosses) indicate the points at which the diﬀerences of velocity amplitudes
between M6 and M6_cal are less (larger) than the STD of the diﬀerences between M6
and M6_cal (i.e., 0.05 m s−1). (For interpretation of the references to color in this ﬁgurelegend, the reader is referred to the web version of this article.)

4.4. Dynamic mechanisms of residual currents
The depth-averaged residual currents are shown in Fig. 1b. The
residual currents are strong at the inﬂow from Luotou channel andweak in the eastern part of the observation region. The residual current
vectors are eastward in the north, then turn southward near Fodu
channel. To comprehend the main dynamic processes of the residual
currents in ZTYB, we decompose the depth-averaged currents→⎯u⎯ =ui ⃗+vj ) observed by CAT into two parts: the tidal currents
(→
→
→
⎯
→
⎯
→
→
( u ′ = u′ i + v′j ) and the residual currents (U =Ui ⃗+Vj ). The momentum equation, integrated from the sea bottom to the sea surface andaveraged over one tidal cycle can be written (Sheng and Wang, 2004;
Zhu et al., 2015) as

Qinzimen channel. The type of asymmetric distortion (ﬂood- or ebbdominant) is determined by the relative phase of 2M2-M4 (Speer andAubrey, 1985). The relative velocity phase of 2M2-M4 (Fig. 7c) changes
gradually from about −90° to 90°, from the west part to the east part ofthe observational region, indicating that the type of asymmetricdistortion is ﬂood-dominant in this area. But the type becomes ebbdominant (2M2-M4 relative velocity phase larger than 90°) in the
middle area near Qinzimen channel (Friedrichs and Aubrey, 1988).
Regardless of type, a larger M4/M2 ratio denotes more distortion of thetide (Friedrichs and Aubrey, 1988), so there are relatively strong ﬂoodand ebb-dominant distortions near Qinzimen channel, with weak
distortions in the rest of the region. Near Qinzimen channel, the tidal

→
⎯τ
→
⎯τ
→
⎯ →
⎯→
→
⎯
⎯
w
′u⎯ ′)=−g∇ η − ∇ (→⎯∇h ∙(→u⎯ →− b − fk × U
hh UU ) + Am ∇h ∙(∇h U ) +
ρh
ρh
224

(14)

Continental Shelf Research 143 (2017) 219–227

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15

ZhoushanIsland0.4

ZhoushanIsland0.3

10LuotouChannelLuotouChannelLaoshumen
ChannelLaoshumen
Channel0.2

Distance (km)50.1
Fodu Channel
0.5
m/s
0
1

Fodu Channel
0.5
m/s
0
1

Qingzimen Channel

15

Qingzimen Channel

ZhoushanIsland180

ZhoushanIsland90

10LuotouChannelLuotouChannelLaoshumen
ChannelLaoshumen
Channel05−90Fodu Channel

Fodu Channel
Qingzimen Channel

0

5

10

Qingzimen Channel

15

0

5

10

−180

15

Distance (km)Fig. 7. The distribution of (a) the ratio of the velocity amplitudes of M4 to M2, (b) the ratio of the velocity amplitudes of M6 to M2, and the velocity phase in degrees of (c) 2M2-M4, (d)3M2-M6. The black (white) lines in (c) and (d) indicate the 90° (−90°) contour lines.

⎯→⎯ →Here, i ⃗, j , k are unit vectors in the eastward, northward, upward→⎯ ∂∂directions, respectively; ∇h = i ⃗ ∂x + j ∂y ; Am is the horizontal eddy
viscosity coeﬃcient, which is set as 90 m2 s−1 (Cáceres et al., 2003).
Overbars denote a temporal average over one tidal cycle.
The term on the left-hand side of Eq. (14) is known as the tidally
averaged advection of the tidal currents (Sheng and Wang, 2004). It is
the forcing term for the residual ﬂow and is part of the “tidal stress”(Nihoul and Ronday, 1975). It is balanced by the six terms on the righthand side of Eq. (14). They are the pressure gradient of residual sea→⎯⎯→
level (−g∇h η ); advection of the residual currents (∇h (UU )); horizontal⎯→→⎯
mixing of residual currents ( Am ∇h ∙(∇h U )); time mean wind stress ( τw );
ρh
→
⎯τ
→
⎯ →
⎯
b
time mean bottom stress ( ); Coriolis term (−fk × U ). The wind stressρh→⎯⎯τ = C ρ ⎯→u ⎯→u ; here, ⎯→u = u i ⃗ + v j , with u andwas calculated as →
w

w a

w

w

w

w

w

Table 3
Maximum and average values (in units of 10−6 m s−2) of dynamic terms in the tidally
averaged momentum equation, and percentages of each term relative to the forcing term.

Maximum value
Percentage,%
Average value
Percentage,%—∇h ∙(→
u ′→
u ′)

→→
∇h (U U )→∇h ∙(Am ∇h U )

336
100
125
100

176
52.4
47.4
37.9

0.08
0.02
0.5
0.4

τb
ρh

75
22.3
8.1
6.5

→ →
fk × U

48.7
14.5
18.3
14.6

τw
ρh

11.0
3.3
5.2
4.2

(Table 3). The spatial distribution of the residual current advection→⎯⎯→′u⎯ ′) , with larger valuesu⎯ →term (∇h (UU )) (Fig. 8b) is similar to that of ∇h ⋅(→near Luotou channel, and its value becomes about 2×10−5 m s−2 near
Fodu channel. The maximum and mean values in the observation
region are 1.76×10−4 m s−2 and 4.74×10−5 m s−2, respectively. The→⎯⎯→′u⎯ ′) . The spatialu⎯ →mean value of ∇h (UU ) takes up 37.9% of ∇h ⋅(→→⎯distribution of the time mean horizontal mixing term ( Am ∇h ∙(∇h U ))(Fig. 8c) is diﬀerent from that of the ﬁrst two terms, and the mean→⎯value of Am ∇h ∙(∇h U ) is only about 5×10−7 m s−2, which takes up 0.4%
′u⎯ ′) . The spatial pattern of the time mean bottom friction termu⎯ →of ∇h ∙(→τb′u⎯ ′) , with mean value aboutu⎯ →( ) (Fig. 8d) is similar to that of ∇ ∙(→w

vw denoting the eastward and northward wind velocity components
from reanalysis of data with six hour intervals (http://apps.ecmwf.int/
datasets/data/interim-full-daily/); Cw denotes the wind-speed dragCw = 10−3(0.63 + 0.066 u w 2 + vw 2 );coeﬃcient
given
by
−3
ρa = 1.29 kg m is air density (Smith and Banke, 1975). Here, weshould note that, considering the spatial resolution of wind data (0.
75°x0.75°), we use wind data from a single point (122.25°E, 30.0°N) to
represent the whole observational region.
To quantify the role of each term in Eq. (14) in generating theresidual current in ZTYB, the maximum and mean values of theseterms are calculated during one M2 tidal period (Table 3). The
horizontal distributions of the six terms (except −g∇h η ) in Eq. (14)are shown in Fig. 8a–f.
Fig. 8a shows that the tidally averaged advection term of the tidal′u⎯ ′) ) is large where the residual current is strong, theu⎯ →currents ( ∇h ∙(→maximum value being 3.36×10−4 m s−2 near Luotou channel. The
mean value in the observation region is about 1.25×10−4 m s−2hρh′u⎯ ′) . The time mean windu⎯ →8.1×10−6 m s−2, which is about 6.5% of ∇h ∙(→⎯→τ

stress term ( ρhw ) (Fig. 8f) is 5.18×10−6 m s−2, which is 4.2% of⎯ →⎯′u⎯ ′) . The mean value of the time mean Coriolis term (−→fk × U )∇h ∙(→u⎯ →→⎯→⎯(Fig. 8e) is 14.6% of ∇h ∙(u ′u ′) (Table 1). The CAT does not measure sea
surface level in the observation region, therefore we can only estimate
the time mean horizontal pressure gradient term (−g∇h η ) by using Eq.
(14). The results show that the mean value of g∇h η in this region is225

Continental Shelf Research 143 (2017) 219–227

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15

ZhoushanIslandZhoushanIsland0.5

10LuotouChannel10150LuotouChannel50

LuotouChannel

0.1

Laoshumen
ChannelLaoshumen
Channel6

Laoshumen
Channel

3

2

00

0

10
0

0.
1

200

Fodu Channel

Fodu Channel

Fodu Channel

Qingzimen Channel

Qingzimen Channel

Qingzimen Channel

ZhoushanIslandZhoushanIslandZhoushanIsland

15
10

6

5

150

LuotouChannel

6
3

40

9

15

60

25

15
Fodu Channel

Fodu Channel

Qingzimen Channel

5

Laoshumen
Channel20

Fodu Channel

0

LuotouChannelLaoshumen
Channel25

80

5Laoshumen
Channel4080

5LuotouChannel35
30

Distance (km)ZhoushanIsland10

15

Qingzimen Channel

0

5

10

Distance (km)15

Qingzimen Channel0→⎯→′u⎯ ′) , (b) ∇ (→⎯u⎯ →Fig. 8. Contour plot (in units of 10-6 m s−2) for terms in Eq. (14): (a) ∇h ∙(→h UU ) , (c) Am ∇h ∙(∇h U ) , (d)′u⎯ ′) .
u⎯ →about 4.55×10−5 m s−2, which is about 36.4% of ∇h ∙(→Therefore, the momentum equation (Eq. 14) of the residual
currents in ZTYB is mainly balanced by the tidally averaged advectionof the tidal currents, the averaged horizontal pressure gradient and theadvection of the residual currents. In this region, wind stress, bottom
friction and Coriolis force play relatively less important roles, andhorizontal mixing is negligible.

5
→
⎯ →
⎯
τb
, (e) fk × U , (f)
ρh

10

15

τw
.
ρh

where depths are less than 20 m, indicating that the M6 currents aremainly caused by the quadratic bottom friction terms in shallow
waters, but the friction mechanisms have less impact in the deep area.
All these results indicate that the M4 and M6 tidal currents directly
measured by the CAT are credible. To the best of our knowledge, thisstudy is the ﬁrst nonlinear tidal current measurement by CAT.
Furthermore, we also demonstrate that the overtides M4 and M6 aremainly generated by the nonlinear advection and quadratic bottom
friction terms of M2, respectively.
Following previous studies (Sheng and Wang, 2004; Zhu et al.,2015) we also analyze the main dynamic processes responsible for the
residual currents by using the averaged horizontal momentum equation. The predominant terms in balancing the momentum equation of
the residual currents are advection of the tidal currents, horizontalpressure gradient, and advection of the residual currents, while thebottom friction and Coriolis force terms contribute less.
The analyses of M4 and M6 and the residual currents that we havecarried out in this study are based on simultaneous current measurements by CAT. The results show the advantages of the CAT system foraccurately mapping horizontal variations of tidal currents and residual
currents. Our study suggests that CAT measurements have advantages
for current monitoring over a large area and understanding dynamic
mechanisms in coastal regions.

5. Summary
In this study, we use observational data to reconstruct thehorizontal distributions of M4 and M6 tidal ellipses and residual
currents in ZTYB. Furthermore, we also discuss the dynamic mechanisms of the nonlinear tides and residual currents.
The mean semimajor and semiminor axis lengths of the M4 current
ellipses measured by CAT are 0.15 and 0.05 m s−1, respectively. Also,
substituting the measured tidal currents of the M2 semidiurnal tidal
constituent into the advection terms of the two-dimensional shallowwater equations we calculated the predicted M4 (i.e., M4_cal). The mean
semimajor and semiminor axis lengths of the M4_cal ellipses are 0.15
and 0.05 m s−1, respectively. Comparing the velocity amplitudes of M4
and M4_cal, spatial points with good agreement mainly appear near the
Luotou and Qinzimen channels, indicating that M4 currents are mainly
caused by the advection terms in the deep area, but are also generated
by other factors in the observational regions where the values of theadvection terms are small.
The mean semimajor and semiminor axis lengths of the M6 current
ellipses are 0.11 and 0.03 m s−1. Substituting the M2 tidal currents into
the quadratic bottom friction terms of the two-dimensional shallowwater equations, we calculated the predicted M6 (i.e., M6_cal). The
mean semimajor and semiminor axis lengths of the M6_cal ellipses are
0.07 and 0.01 m s−1, respectively, slightly smaller than the values for
M6. Comparing the velocity amplitudes of M6 and M6_cal, the tidal
current ellipses of M6 and M6_cal are in good agreement in the area

Acknowledgments
This study is supported by the National Natural Science Foundation ofChina (41276095, 41476020, 41576001, 41321004 and 41576031); theScientiﬁc Research Fund of SIO under Grant JT1402; the projects of theState Key Laboratory of Satellite Ocean Environment Dynamics, SIO(SOEDZZ1501); the Open Fund of State Key Laboratory of Satellite Ocean
Environment Dynamics, SIO (SOED1511), the National Programme on
Global Change and Air-Sea Interaction (GASI-IPOVAI-01-02).
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Continental Shelf Research 143 (2017) 219–227

References

Park, J.-H., Kaneko, A., 2001. Computer simulation of the coastal acoustic tomography
by a two-dimensional vortex model. J. Oceanogr. 57, 593–602.
Parker, B.B., 1991. The relative importance of the various nonlinear mechanisms in awide range of tidal interactions (review). In: Parker, B.B. (Ed.), Tidal
Hydrodynamics. Wiley New York, USA, 237–268.
Pawlowicz, R., Beardsley, B., Lentz, S., 2002. Classical tidal harmonic analysis including
error estimates in MATLAB using T_TIDE. Comput. Geosci. 28, 929–937.
Pingree, R.D., Maddock, L., 1978. The M4 tide in the english channel derived from a nonlinear numerical model of the M2 tide. Deep Sea Res. 25, 53–63.
Pingree, R.D., Maddock, L., 1980. Tidally induced residual ﬂows around an island due toboth frictional and rotational eﬀects. J. Geophys. Res. 63, 533–546.
Sheng, J., Wang, L., 2004. Numerical study of tidal circulation and nonlinear dynamics in
Lunenburg Bay, Nova Scotia. J. Geophys. Res. 109, C10018. http://dx.doi.org/
10.1029/2004JC002404.
Smith, S.D., Banke, E.G., 1975. Variation of the sea surface drag coeﬃcient with windspeed. Q. J. R. Meteorol. Soc. 101, 665–673.
Speer, P.E., Aubrey, D.G., 1985. A study of non-linear tidal propagation in shallow inlet/
estuarine systems Part II: theory. Estuar., Coast. Shelf Sci. 21, 207–224.
Walters, R.A., Werner, F.E., 1991. Nonlinear generation of overtides compound tides,
and residuals. In: Parker, B.B. (Ed.), Tidal Hydrodynamics. Wiley New York, USA,
269–295.
Yamaoka, H., Kaneko, A., Park, J.-H., Zheng, H., Gohda, N., Takano, T., Zhu, X.-H.,
Takasugi, Y., 2002. Coastal acoustic tomography system and its ﬁeld application.
IEEE J. Ocean. Eng. 27 (2), 283–295.
Zhu, X.-H., Kaneko, A., Wu, Q., Zhang, C., Taniguchi, N., Gohda, N., 2013. Mapping tidal
current structures in Zhitouyang Bay, China, using coastal acoustic tomography.
IEEE J. Ocean. Eng. 38 (2), 285–296.
Zhu, X.-H., Zhu, Z.-N., Guo, X., Ma, Y.-L., Fan, X., Dong, M., Zhang, C., 2015.
Measurement of tidal and residual currents and volume transport through the
Qiongzhou Strait using coastal acoustic tomography. Cont. Shelf Res. 108, 65–75.

Aubrey, D.G., Speer, P.E., 1985. A study of nonlinear tidal propagation in shallow inlet/
estuarine systems, Part I: observations. Estuar., Coast. Shelf Sci. 21, 185–205.
Blanton, J.O., Lin, G., Elston, S.A., 2002. Tidal current asymmetry in shallow estuaries
and tidal creeks. Cont. Shelf Res. 22, 1731–1743.
Cáceres, M., Valle-Levinson, A., Atkinson, L., 2003. Observations of cross-channel
structure of ﬂow in an energetic tidal channel. J. Geophys. Res. 108 (C4), 3114.
http://dx.doi.org/10.1029/2001JC000968.
Dong, L.X., Su, J.L., 1991a. Tide response and wave distortion in Xiangshan Bay, I,
observation and analysis. Acta Ocean. Sin. 21 (2),
1–10, (in Chinese with English abstract).
Dong, L.X., Su, J.L., 1991b. Tide response and wave distortion in Xiangshan Bay, II,numerical modelling study in the Xiangshan Bay. Acta Ocean. Sin. 21 (1),
1–8, (in Chinese with English abstract).
Fang, G.H., 1987. Nonlinear eﬀects of tidal friction. Acta Ocean. Sin. 6 (Supp I),
S105–S122.
Friedrichs, G.T., Aubrey, D.G., 1988. Non-linear tidal distortion in shallow well-mixed
estuaries: a synthesis. Estuar., Coast. Shelf Sci. 27, 521–545.
Godin, G., 1972. The Analysis of Tides. University of Toronto Press, Toronto.
Hansen, P.C., O’Leary, D.P., 1993. The use of the L-cure in the regularization of discrete
ill-posed problems. SIAM J. Sci. Comput. 14 (6), 1487–1503.
Hench, J.L., Luettich, R.A., 2003. Transient tidal circulation and momentum balances at
a shallow inlet. J. Phys. Oceanogr. 33, 913–932.
Hench, J.L., Blanton, B.O., Luettich, R.A., 2002. Lateral dynamic analysis andclassiﬁcation of barotropic tidal inlets. Cont. Shelf Res. 22, 2516–2631.
Le Provost, C., 1991. Generation of overtides and compound tides (review). In: Parker,
B.B. (Ed.), Tidal Hydrodynamics. Wiley New York, USA, 269–295.
Nihoul, J.C.J., Ronday, F.C., 1975. The inﬂuence of the “tidal stress” on the residual
circulation-application to the Southern Bight of the North Sea. Tellus 5, 484–489.

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