MAPPING TIDAL CURRENTS AND RESIDUAL CURRENTS BY USE OF
COASTAL ACOUSTIC TOMOGRAPHY
Xiao-Hua Zhu, State Key Laboratory of Satellite Ocean Environment Dynamics, Second
Institute of Oceanography, State Oceanic Administration, China
Ze-Nan Zhu, State Key Laboratory of Satellite Ocean Environment Dynamics, Second
Institute of Oceanography, State Oceanic Administration, China
Xinyu Guo Center for Marine Environmental Study, Ehime University, 2-5 Bunkyo-cho,
Matsuyama 790-8577, Japanxhzhu@sio.org.cn
A coastal acoustic tomography (CAT) experiment for mapping the tidal currents in the
Zhitouyang Bay was successfully carried out with seven acoustic stations during July 12 to
13, 2009. The horizontal distributions of tidal current in the tomography domain arecalculated by the inverse analysis in which the travel time differences for sound travelingreciprocally are used as data. Spatial mean amplitude ratios M 2 : M 4 : M 6 are 1.00 : 0.15 :0.11. The shallow-water equations are used to analyze the generation mechanisms of M 4
and M 6. In the deep area, velocity amplitudes of M 4 measured by CAT agree well withthose of M 4 predicted by the advection terms in the shallow water equations, indicatingthat M 4 in the deep area where water depths are larger than 60 m is predominantly
generated by the advection terms. M 6 measured by CAT and M 6 predicted by the nonlinear
quadratic bottom friction terms agree well in the area where water depths are less than 20
m, indicating that friction mechanisms are predominant for generating M 6 in the shallow
area . Dynamic analysis of the residual currents using the tidally averaged momentum
equation shows that spatial mean values of the horizontal pressure gradient due to residualsea level and of the advection of residual currents together contribute about 75% of thespatial 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.
Key words: tidal current structures, residual currents, M 4 and M 6 generation mechanisms,
coastal acoustic tomography
I. INTRODUCTION
The continuous monitoring of tidal currents in bays, inland seas, and straits over a large
area is difficult, especially in the coastal seas of China, where fishing and shipping pressures arehigh. Furthermore, in order to examine generation mechanisms of the nonlinear tidal currents,
the 2-dimensional continuous current field also should be mapped accurately to allow calculationof the advection terms.
To obtain observations of rapidly varying tidal current structures, bottom-mounted or
subsurface moored current meter arrays are needed as conventional tools. However, heavy
shipping traffic and fishery activities make it difficult to deploy such mooring arrays in almost
all coastal regions in China, even if the antitrawl bottom-mounted acoustic Doppler current
profiler (ADCP) systems are applicable. Coastal Acoustic Tomography (CAT) can achieve thiseasily by deploying only a few instruments on the periphery of the observation region [1]-[2].

II. SITE AND METHOD
An experiment with seven CAT systems was performed on July 12–13, 2009, in a neap
tidal period at Zhitouyang Bay (hereinafter ZTYB) near Zhoushan Island around the mouth ofHangzhou Bay, China (Fig. 1).

Fig. 1 Location maps of the experimental site. The positions of the CAT stations (C1–C7) are
shown with solid circles. The thick solid lines connecting the CAT stations indicate the soundtransmission lines. The interval of bathymetric contours is 20 m.
The seven CAT systems were set up at seven stations numbered C1–C7 using fishingboats anchored on the periphery of the bay. The transducer was suspended down to 5–10 m
depths from the boat by rope, while the major components of the system, such as electronic
housing, batteries, and Global Positioning System (GPS) antennas were put onboard the boat.
For the geographical configuration of acoustic stations, the total number of transmission
lines of 21 results in an average horizontal resolution of 1.8 km [3]. Grid size for the data display
in the inverse analysis is set to be 1.0×1.0 km, which is slightly smaller than the horizontal
resolution.
The inverse method developed by Park and Kaneko [3] is applied here to reconstruct the
tidal current distributions by averaging any vertical dependency. The travel-time differences
obtained from the reciprocal sound transmission for the CAT station pairs are used as data in the
inverse analysis. The inverse domain is taken to be 22×22 km , which is twice the tomographydomain, to suppress the artificial periodicity of inverse results (Fig. 1). The equation for the
inverse method is as follows:
𝒚𝒚 = 𝑬𝑬𝑬𝑬 + 𝒆𝒆
(1)where 𝒚𝒚 is a 21-element column vector, each column denotes the travel-time difference data ofeach station pair; 𝒙𝒙 is a vector of the 20-element row vector of the unknown coefficients used in
the Fourier function expansion of the stream function used to estimate the current field; 𝑬𝑬 is a
21×20 matrix, determined by the locations of the 7 CAT stations; 𝒆𝒆 is the error vector. The
optimum solution of (1) is obtained by minimizing the objective function (𝑱𝑱) with the tapered
least squares method. The objective function is given by:
𝑱𝑱 = (𝒚𝒚 − 𝑬𝑬𝑬𝑬)𝑇𝑇 (𝒚𝒚 − 𝑬𝑬𝑬𝑬) + 𝛼𝛼 2 𝒙𝒙𝑇𝑇 𝒙𝒙(2),where the weighting factor α is determined by the L-curve method [4].
After obtaining the current field at each spatial grid point for a period of 27 hours byusing the inverse method, we then compute the M 2 , M 4 and M 6 harmonic constants at each grid
point by using harmonic tidal analysis [5].
𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

To obtain the M 4 and M 6 tidal currents generated by the advection terms (𝑢𝑢 𝜕𝜕𝜕𝜕 + 𝑣𝑣 𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝜕𝜕𝜕𝜕

𝐶𝐶

𝐶𝐶

𝑑𝑑
𝑑𝑑
and 𝑢𝑢 𝜕𝜕𝜕𝜕 + 𝑣𝑣 𝜕𝜕𝜕𝜕) and quadratic bottom friction terms (ℎ+𝜂𝜂𝑢𝑢√𝑢𝑢2 + 𝑣𝑣 2 and ℎ+𝜂𝜂𝑣𝑣√𝑢𝑢2 + 𝑣𝑣 2 ), we

firstly substitute the tidal current forms of M 2 into the advection terms and quadratic bottom
friction terms of the shallow water equations, respectively. Then, we obtain the terms having the
M 4 and M 6 frequencies as the predicted M 4 and M 6 tidal currents (hereinafter M 4_cal and M 6_cal ).
Comparing the predicted M 4 and M 6 currents with those observed directly by CAT, we canquantify the contribution of advection terms and quadratic bottom friction terms for generating
M 4 and M 6 [6].
III. RESULTS AND DISCUSSION
A. Horizontal distribution of tidal current
The inversion results provide synoptic snapshots of horizontal tidal current distributions
at hourly intervals. The hourly mean tidal current structures from 18:00:00 July 12, 2009 to
17:00:00 July 13, 2009, are shown in Fig. 2. The strong eastward tidal current, with a maximum
current of 2.05 m/s, entered the tomography site from the Luotou channel at the western part ofthe bay and separated into two branches flowing into the Laoshumen channel and the Qingzimen
and Fodu channels during the ebb tide (Fig. 2a–2c). During the flood tide, the strong westward
tidal current, with a maximum current of 1.03 m/s, merged with the currents from the
Laoshumen, Qingzimen, and Fodu channels and went back to the Luotou channel during theflood tide (Fig. 2f–2h). This tidal current variation was repeated at the half-day cycle.

Fig. 2 Vector plots of the hourly mean current reconstructed from 18:00:00 July 12, 2009 to
17:00:00 July 13, 2009, by the inverse analysis. The tidal phases are indicated by a dot on thesea surface height (SSH) plot at the upper right of each panel.
B. Harmonic analysis of tidal currents
The harmonic analysis showed that the M 2 tide was the predominant tidal constituent.
The semimajor axes of the tidal ellipse were directed to the east–west at the western half of thesite and were separated into two directions at the eastern half (Fig. 3a). The mean semimajor andsemiminor axes of the tidal ellipses were 1.00 and 0.07 m/s, respectively. The vector plot of theresidual current is shown in Fig. 3b. The residual current was strong near the Luotou channel andbecame weak in the eastern part of the bay. The residual current flowed out as the eastward
current from the Luotou channel and then turned southward along the 40-m bathymetric contourin the middle of the site.

Fig. 3 The distribution of (a) M 2 tidal current ellipses, and (b) residual currents in ZhitouyangBay. 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 spatially-averaged amplitude ratios M 2 : M 4 : M 6 are 1.00 : 0.15 : 0.11. The
relatively large M 4 and M 6 overtides were observed in ZTYB. Then, we will discuss thegeneration mechanisms of M 4 and M 6 by using the ZTYB CAT data.
The horizontal distribution of the M 4 tidal ellipses measured by CAT and those of M 4_cal
are shown in Fig. 4. The M 4 currents (Fig. 4a) are relatively large near the Luotou and Qinzimen
channels with a maximum value 0.27 m s-1, corresponding well to the distribution of the M 2 tidal
current (Fig.3a). The area averaged semimajor and semiminor axis lengths of the M 4 tidal
ellipses are 0.15 and 0.04 m/s, respectively (Table 1). The spatial mean ellipticity of the M 4 tide
is 0.27, which is larger than that of M 2 (0.07). The M 4_cal currents (Fig. 4b) are also relatively
large near the Luotou and Qinzimen channels with a maximum value 0.32 m/s near Luotou
channel. The M 4_cal ellipses have the same area averaged semimajor and semiminor axis lengths
as the observed M 4 values (Table 1).
The M 4 semimajor axis direction is mainly northeast-southwest in the west part of theobservation region, and becomes northwest-southeast in the east part (Fig. 4a). The M 4_cal
semimajor axis direction is mainly northeast-southwest in the west part of the region, andbecomes north-south in the east part (Fig. 4b).
We compare the velocity amplitudes of M 4 and M 4_cal to confirm the contribution of
advection terms in generating the M 4 tide in different areas of the observational region. We
consider the M 4_ and M 4_cal ellipses to be similar at those points (red ellipses in Fig. 4) for which
the differences of both velocity amplitude and phase between M 4_ and M 4_cal are less than one
STD of the differences between the M 4_ and M 4_cal values, i.e., 0.07m s-1 and 110º, respectively.
We find that the tidal current ellipses are most similar in the deep area where water depths arelarger than 60 m. The distribution of M 2 ellipses (Fig.3a) shows that relatively large M 2 currents
appear in the deep area, which means the values of the advection terms are also larger in the deeparea. Thus the advection terms make a primary contribution to M 4 generation in the deep area.
Moreover the tidal current ellipses (black ellipses in Fig. 4) are not generally similar in shallow
waters, that is to say the M 4 tidal currents in the shallow area are not mainly caused by theadvection terms. There are other key factors for generating M 4 in the shallow area, such as thecontinuity terms or the friction mechanism [7].

Fig. 4 Tidal current ellipses of (a) M 4 , and (b) M 4_cal . Red lines indicate similar ellipses for M 4
and M 4_cal . The area enclosed by the dashed lines indicates the CAT observational region.
The horizontal distribution of the M 6 tidal ellipses measured by CAT and those of M 6_cal
are shown in Fig. 5. The maximum value of the M 6 semimajor axes is 0.20 m/s, appearing near
Luotou channel. The spatial mean semimajor and semiminor axis lengths of the M 6 current
ellipses are 0.11 and 0.03 m/s (Table 1). The M 6 ellipticity is relatively small in the north part ofthe observational region, and becomes larger near Fodu channel. The spatial mean value of
ellipticity is 0.27. The M 6_cal currents (Fig. 5b) are relatively larger near Luotou channel with amaximum value 0.10 m/s. The spatial mean semimajor and semiminor axis lengths of the M 6_cal
tidal current ellipses are 0.07 and 0.01 m s-1, respectively (Table 1). The spatial mean ellipticityof the M 6_cal tide is 0.09.
The orientations of the M 6 and M 6_cal tidal current ellipses correspond well throughout
the observational region: both are mainly directed east-west in the north part of the observational
region, and turn north-south along the 40 m bathymetric contour.
We further show the relationship of M 6_ and M 6_cal velocity amplitudes to confirm thecontribution of the quadratic bottom friction terms to the generation of M 6 in different areas ofthe observational region. We consider the M 6 and M 6_cal ellipses to be similar at those points (redellipses in Fig. 5) for which the difference of velocity amplitude between M 6_ and M 6_cal is lessthan the STD of the differences between the M 6_ and M 6_cal values (i.e., 0.05m s-1). Where thetidal current ellipses are similar (red ellipses in Fig. 5) this similarity indicates M 6 mainly
generated by the quadratic bottom friction terms; this occurs principally in the east part of theobservational region where water depth is less than 20 m. These results indicate that thequadratic bottom friction terms play a primary role for generating M 6 in shallow waters, while
M 6 in the deep area is also affected by such factors such as the other nonlinear terms in the
momentum equations [8].

Fig. 5 Tidal current ellipses of (a) M 6 , and (b) M 6_cal . Red lines indicate similar ellipses for M 6
and M 6_cal . The area enclosed by the dashed lines indicates the CAT observational region.
Table 1. Spatial mean parameters of tidal current ellipses of M 4 , M 4_cal , M 6 , M 6_cal and their
STDs
M4

M 4_cal

M6

M 6_calSemimajor
length±STDs (m/s)

axis 0.15±0.05

0.15±0.07

0.11±0.04

0.07±0.02

Semiminor
length±STDs (m/s)

axis 0.04±0.03

0.04±0.03

0.03±0.02

0.01±0.00

C. Dynamic mechanisms of residual currents
To comprehend the main dynamic processes of the residual currents in ZTYB, we�⃗ = 𝑢𝑢𝚤𝚤⃗ + 𝑣𝑣𝚥𝚥⃗) observed by CAT into two parts: the tidal
decompose the depth-averaged currents (𝑢𝑢′′′�⃗ = 𝑈𝑈𝚤𝚤⃗ + 𝑉𝑉𝚥𝚥⃗). The momentum equation,
currents (u�⃗ = 𝑢𝑢 𝚤𝚤⃗ + 𝑣𝑣 𝚥𝚥⃗) and the residual currents (𝑈𝑈
integrated from the sea bottom to the sea surface and averaged over one tidal cycle [6], [8]-[9]can be written as
����⃗
𝜏𝜏�
𝜏𝜏�
𝑤𝑤
������
�⃗ × 𝑈𝑈�⃗𝑈𝑈�⃗� + 𝐴𝐴𝑚𝑚 ∇ℎ ∙ �∇ℎ 𝑈𝑈
�⃗� + �����⃗�⃗∇ℎ ∙ �𝑢𝑢�⃗ ′ 𝑢𝑢�⃗ ′ � = −𝑔𝑔∇ℎ 𝜂𝜂̅ − ∇ℎ �𝑈𝑈− 𝑏𝑏 − 𝑓𝑓𝑘𝑘
(3)
𝜌𝜌ℎ

𝜌𝜌ℎ

�⃗ are unit vectors in the eastward, northward, upward directions, respectively; ∇ℎ =here, 𝚤𝚤⃗, 𝚥𝚥⃗, 𝑘𝑘𝜕𝜕𝜕𝜕𝚤𝚤⃗ 𝜕𝜕𝜕𝜕 + 𝚥𝚥⃗ 𝜕𝜕𝜕𝜕 ; 𝐴𝐴𝑚𝑚 is the horizontal eddy viscosity coefficient, which is set as 90 m2/s [10].
Overbars denote a temporal average over one tidal cycle.
The term on the left-hand side of (3) is known as the tidally averaged advection of thetidal currents [8]. It is balanced by the six terms on the right-hand side of (3). To quantify therole of each term in (3) in generating the residual current in ZTYB, the maximum and meanvalues of these terms are calculated during one M 2 tidal period (Table 2). The horizontal
distributions of the six terms (except −𝑔𝑔∇ℎ 𝜂𝜂̅ ) in (3) are shown in Fig. 6a-6f.
������
Fig. 6a shows that the tidally averaged advection term of the tidal currents (�∇ℎ ∙ �𝑢𝑢�⃗ ′ 𝑢𝑢�⃗ ′ ��)-42
is large where the residual current is strong, the maximum value being 3.36×10 m/s near
Luotou channel. The mean value in the observation region is about 1.25×10-4 m/s2 (Table 2). The�⃗𝑈𝑈�⃗�) (Fig. 6b) is similar to that of
spatial distribution of the residual current advection term (∇ℎ �𝑈𝑈�������∇ℎ ∙ �𝑢𝑢�⃗ ′ 𝑢𝑢�⃗ ′ ��, with larger values near Luotou channel, and its value becomes about 2×10-5 m/s2
near Fodu channel. The maximum and mean values in the observation region are 1.76×10-4 m/s2������
�⃗𝑈𝑈�⃗ �� takes up 37.9% of �∇ℎ ∙ �𝑢𝑢
and 4.74×10-5 m/s2, respectively. The mean value of �∇ℎ �𝑈𝑈�⃗ ′ 𝑢𝑢�⃗ ′ ��.

�⃗�) (Fig. 6c) is
The spatial distribution of the time mean horizontal mixing term (𝐴𝐴𝑚𝑚 ∇ℎ ∙ �∇ℎ 𝑈𝑈
�⃗�� is only about
different from that of the first two terms, and the mean value of �𝐴𝐴𝑚𝑚 ∇ℎ ∙ �∇ℎ 𝑈𝑈������
5×10-7 m s-2, which takes up 0.4% of �∇ℎ ∙ �𝑢𝑢�⃗ ′ 𝑢𝑢�⃗ ′ ��. The spatial pattern of the time mean bottom���𝜏𝜏������
friction term (𝜌𝜌ℎ𝑏𝑏 ) (Fig. 6d) is similar to that of �∇ℎ ∙ �𝑢𝑢�⃗ ′ 𝑢𝑢�⃗ ′ ��, with mean value about 8.1×10-6
����������⃗
𝜏𝜏𝑤𝑤������
�⃗ ′ 𝑢𝑢�⃗ ′ ��. The time mean wind stress term (�𝜌𝜌ℎ�) (Fig. 6f) is
m/s2, which is about 6.5% of �∇ℎ ∙ �𝑢𝑢������
5.18×10-6 m/s2, which is 4.2% of �∇ℎ ∙ �𝑢𝑢�⃗ ′ 𝑢𝑢�⃗ ′ ��. The mean value of the time mean Coriolis term������
�⃗ × 𝑈𝑈�⃗) (Fig. 6e) is 14.6% of �∇ℎ ∙ �𝑢𝑢(−𝑓𝑓𝑘𝑘�⃗ ′ 𝑢𝑢�⃗ ′ �� (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 (−𝑔𝑔∇ℎ 𝜂𝜂̅ ) by using (3). The results show that the mean value of |𝑔𝑔∇ℎ 𝜂𝜂̅ | in this������
region is about 4.55×10-5 m/s2, which is about 36.4% of �∇ℎ ∙ �𝑢𝑢�⃗ ′ 𝑢𝑢�⃗ ′ ��.
Therefore, the momentum equation (3) of the residual currents in ZTYB is mainly������
balanced by the tidally averaged advection of the tidal currents (∇ℎ ∙ �𝑢𝑢�⃗ ′ 𝑢𝑢�⃗ ′ �), the averaged
�⃗𝑈𝑈�⃗�). Inhorizontal pressure gradient (−𝑔𝑔∇ℎ 𝜂𝜂̅ ) and the advection of the residual currents (∇ℎ �𝑈𝑈�����⃗
����⃗
𝜏𝜏�
𝜏𝜏�
𝑤𝑤
�⃗ × 𝑈𝑈�⃗) play relatively), bottom friction ( 𝑏𝑏 ) and Coriolis force (−𝑓𝑓𝑘𝑘
this region, wind stress (𝜌𝜌ℎ𝜌𝜌ℎ�⃗�) is negligible.
less important roles, and horizontal mixing ((𝐴𝐴𝑚𝑚 ∇ℎ ∙ �∇ℎ 𝑈𝑈������
�⃗𝑈𝑈
�⃗�� ,
Fig. 6 Contour plot (in units of 10-6 m s-2) for terms in (3): (a) �𝛻𝛻ℎ ∙ �𝑢𝑢�⃗ ′ 𝑢𝑢�⃗ ′ �� , (b) �𝛻𝛻ℎ �𝑈𝑈
𝜏𝜏𝑏𝑏
𝜏𝜏𝑤𝑤
�⃗ × 𝑈𝑈�⃗�� , (d)� ����⃗�, (f)� ����(c) �𝐴𝐴𝑚𝑚 𝛻𝛻ℎ ∙ �𝛻𝛻ℎ 𝑈𝑈
�, (e) �𝑓𝑓𝑘𝑘
�.
𝜌𝜌ℎ
𝜌𝜌ℎ

Table 2. Maximum and average values (in units of 10-6 m/s2) of dynamic terms in the tidally
averaged momentum equation, and percentages of each term relative to the forcing term.
𝜏𝜏��𝜏𝜏���
𝑤𝑤������
�⃗ × 𝑈𝑈�⃗𝑈𝑈�⃗�� �∇ℎ ∙ �𝐴𝐴𝑚𝑚 ∇ℎ 𝑈𝑈
�⃗�� � 𝑏𝑏 � �𝑓𝑓𝑘𝑘�⃗�� ��∇ℎ ∙ �𝑢𝑢�⃗ ′ 𝑢𝑢�⃗ ′ �� �∇ℎ �𝑈𝑈𝜌𝜌ℎ𝜌𝜌ℎMaximum
value

336

176

0.08

75

48.7

11.0

Percentage,% 100

52.4

0.02

22.3

14.5

3.3

Average
value

125

47.4

0.5

8.1

18.3

5.2

Percentage,% 100

37.9

0.4

6.5

14.6

4.2

IV. Summary
In this study, we use CAT data to reconstruct the horizontal distributions of tidal currents
and residual currents in ZTYB. Furthermore, we also demonstrate that the overtides M 4 and M 6
are mainly generated by the nonlinear advection and quadratic bottom friction terms of M 2 ,respectively. All these results indicate that the M 4 and M 6 tidal currents directly measured by the
CAT are credible. To the best of our knowledge, this study is the first nonlinear tidal current
measurement by CAT.
We also analyze the main dynamic processes responsible for the residual currents byusing 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 the bottom friction and Coriolis
force terms contribute less.
The results show the advantages of the CAT system for accurately 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.
V. ACKNOWLEDGEMENT
This study is supported by the National Natural Science Foundation of China (41276095,
41476020, 41576001, 41321004 and 41576031); the Scientific Research Fund of SIO under
grant JT1402; the projects of the State Key Laboratory of Satellite Ocean Environment
Dynamics, SIO (SOEDZZ1501); the National Programme on Global Change and Air-Sea
Interaction (GASI-IPOVAI-01-02).

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