Continental Shelf Research 108 (2015) 65–75

Contents lists available at ScienceDirect

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

Measurement of tidal and residual currents and volume transport
through the Qiongzhou Strait using coastal acoustic tomography
Xiao-Hua Zhu a,b,n, Ze-Nan Zhu b,a, Xinyu Guo a,c, Yun-Long Ma a, Xiaopeng Fan a,
Menghong Dong a, Chuanzheng Zhang aaState Key Laboratory of Satellite Ocean Environment Dynamics, Second Institute of Oceanography, State Oceanic Administration, Hangzhou 310012, China
Ocean College, Zhejiang University, Hangzhou 310058, ChinacCenter for Marine Environmental Study, Ehime University, 2-5 Bunkyo-cho, Matsuyama 790-8577, Japanb

art ic l e i nf o

a b s t r a c t

Article history:
Received 6 February 2015
Received in revised form
10 August 2015
Accepted 18 August 2015
Available online 20 August 2015

Quantifying the tidal current and volume transport through the Qiongzhou Strait (QS) is vital to understanding the circulation in the northern South China Sea. To measure the tidal current in the strait, a15-day coastal acoustic tomography (CAT) experiment was carried out at four acoustic stations in March
2013. The horizontal distributions of the tidal currents were calculated by inverse analysis of CAT data.
The diurnal tidal current constituents were found to dominate: the ratio of the amplitudes O1, K1, M2, S2,
and MSF was 1.00:0.60:0.47:0.21:0.11. The residual currents were found to ﬂow westward in the northern
QS and turn southward in the southern QS. The residual current velocities were larger in the northernarea than in the southern area, with a maximum westward velocity of 12.4 cm s  1 in the northern QS.
Volume transport estimated using the CAT data varied between  0.710 Sv and 0.859 Sv, with residual
current transport of  0.044 Sv, where negative values indicate westward. We conducted a dynamic
analysis of the observations made during the study, which suggested that tidal rectiﬁcation and sea level
difference between the two entrances of the QS are important in maintaining the residual current
through the strait. This is the ﬁrst estimation, from synchronous measurements, of major tidal current
constituents, residual currents, and volume transport in this strait.
& 2015 Elsevier Ltd. All rights reserved.

Keywords:
Tidal current
Residual current
Volume transport
Coastal acoustic tomography
Qiongzhou Strait
South China Sea

1. Introduction
The Qiongzhou Strait (QS) is a channel and key point of waterexchange between the northern South China Sea and the Beibu
Gulf (Fig. 1a). Quantifying the tidal current and volume transport
through the QS is vital to understanding the coastal circulationand mass balance in the northern South China Sea and the cyclonic
circulation in the Beibu Gulf (Wang et al., 2010; Wu et al., 2008).
Analysis performed by Shi et al. (2002) of current meter data
and tide gauge data from 1963 to 1999 revealed a westward residual current through the QS throughout the year. They alsoroughly estimated the corresponding volume transport: approximately 0.2–0.4 Sv in winter and spring, and approximately 0.1–
0.2 Sv in summer and autumn. These results agree with the more
recent current meter mooring data collected at several stations
and at different times (Chen et al., 2006, 2007; Yan et al., 2008),
repeat shipboard ADCP data (Zhu et al., 2014), drifting bottle datanCorresponding author at: State Key Laboratory of Satellite Ocean Environment
Dynamics, Second Institute of Oceanography, State Oceanic Administration,
Hangzhou 310012, China. Fax: þ 86 571 88839374.
E-mail address: xhzhu@sio.org.cn (X.-H. Zhu).

http://dx.doi.org/10.1016/j.csr.2015.08.016
0278-4343/& 2015 Elsevier Ltd. All rights reserved.

(Yang et al., 2003; Bao et al., 2005), and numerical studies (Chenet al., 2009). However, synchronous data measurements to directlyprobe the spatial structure of the tidal and residual currents andthe resulting volume transport through the strait have not yet
been performed.
To observe tidal current structures and volume transport
through the QS, bottom-mounted acoustic Doppler current proﬁlers (ADCP), arrays of subsurface moored current meters, or repeat shipboard ADCP surveys would normally be utilized. However, in the QS, the heavy shipping trafﬁc and ﬁshery activity
prohibit the deployment of mooring arrays. The QS is studded
with ﬁshing nets, which also makes shipboard ADCP surveys difﬁcult at night. Thus, in situ point current data are insufﬁcient todetermine the complete structure of tidal and residual currents in
the QS.
To measure the current in the QS, we carried out a 15-day experiment using coastal acoustic tomography (CAT) systems (Kaneko et al., 2005; Yamaguchi et al., 2005) that do not disturb
shipping trafﬁc, ﬁsheries, or marine aquaculture. To the best of ourknowledge, this study is the ﬁrst synchronous measurement that
separates the major tidal current constituents and residual current
in the QS.

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X.-H. Zhu et al. / Continental Shelf Research 108 (2015) 65–75

25oN

CHINA

Guangdong

22oNoGulf of Beibu

Qiongz19oN

0hou Str

ait

ai
na
n

20oN

H

−20

−20

21oN

a
Se

−4000

10oN
5oN

ina
Ch
uth
So

15oN

VIETNAM

0

Luzon

20 N

Borneoo018oN o108 E 109oE 110oE 111oE 112oE

104oE108oE112oE116oE120oE

Inverse domain
Leizhou Peninsula
15’

5 km

Tomography domain
C4

C3

90

10

30

70
50

50

10’

70

90

30
10

C1
C2

5’

Hainan Island
20oN

5’

10’

110oE
15.00’

20’

25’

Fig. 1. Qiongzhou Strait and measurement sites. (a) Bathymetry of the South China Sea; (b) Location of the Qiongzhou Strait and Hainan Island, and (c) Bathymetry of the
Qiongzhou Strait. The solid circles (C1–C4) indicate the positions of the CAT stations. The blue dashed lines indicate the sound transmission lines, the red lines indicate the
shipboard ADCP tracks. The ‘★’ indicates the position of the Haikou tidal gauge station. The interval of bathymetric contours is 20 m. (For interpretation of the references tocolor in this ﬁgure legend, the reader is referred to the web version of this article.)

2. Site and methods
Four CAT systems measured the currents in the QS (Fig. 1b) over a

15-day experiment period (March 17–April 1, 2013), which covered
one fortnightly spring/neap tidal cycle (Fig. 2a). The CAT systems
were set up using ﬁshing ships anchored on both sides of the QS, at

↔
↔
↔
↔
↔
↔

Fig. 2. Synchronized CAT and ADCP measurements. (a) The tidal level anomaly (black lines) observed at the Haikou tidal gauge station and (b) time periods of successfulacquisition of reciprocal sound transmission (gray cross) between each station pair and the time schedule of the shipboard ADCP survey (black horizontal bar). The cruise
number is indicated above the bars.

X.-H. Zhu et al. / Continental Shelf Research 108 (2015) 65–75

four stations numbered C1–C4, spanning an 11.5 km  16.5 km area.
The broadband transducers were suspended at a depth of about 7 musing a rope. During the CAT observation period, 26 along-line tracksof shipboard ADCP (RDI Workhorse 300-kHz) were performed
across the QS (Figs. 1c and 2b).
An 11th order M sequence phase-modulated the transmitted
5 kHz sound used in this experiment, increasing the signal-to-noise
ratio (SNR) of received signals by 20 × log 211 − 1 = 33.1 dB. A Qvalue of 3 cycles per digit set the frequency band-width to 5 kHz/
3¼ 1.7 kHz. The time resolution of multi-ray arrivals, deﬁned as
one-digit length of the M sequence, was 0.6 ms. One-period (2048
digits at 1.23 s) M sequences were transmitted every 5 min, simultaneously from all four stations. The clocks of the CAT systems
at the four stations were synchronized by GPS with a very hightiming accuracy of about 7500 ns (Yamaguchi et al., 2005). Yamaguchi et al. (2005) gives detailed methods for various M sequence transmissions and resolution of acoustic signals from multiple stations. For current velocity measurements, positioning error
due to anchored ship motion is negligible and does not affect themeasured travel time difference of reciprocally transmitted sounds,
which requires precise clock accuracy rather than the precise positioning accuracy (Munk et al., 1995).
Reciprocal sound transmissions were successfully carried out
along the six lines spanning the four CAT stations. The successful
sound transmission acquisition rates varied from 78% (between C1and C4) to 97% (between C3 and C4), with a mean rate of 91%(Fig. 2b). The primary cause of data dropout in the QS was likely
due to shipping wake generated bubbles or noises. The differential
travel-time data were smoothed by a one-hour running mean, andthen used in the inverse method to reconstruct the tidal currents.
The shipboard ADCP observations were performed in the daytime and synchronized with the CAT schedule inside the tomography domain (Figs. 1c and 2b). The ADCP was attached to theside of a wooden ﬁshing ship using a stainless steel frame with its
transducers maintained at a depth of 1 m. The ADCP sampled dataat one ping per 2 s, with a bin number and bin length of 60 and
2 m, respectively. Under these conditions, the raw velocity error(standard deviation) for a single ping was less than 7.0 cm s  1(Teledyne RD Instruments, 2013). The ﬁrst ADCP bin depth and theblank distance from the bottom were approximately 3 m and 4 m,
respectively. The velocity of the top 3 m and the bottom 4 m could
not be measured directly and were determined by extrapolation,
following the method proposed by Gordon (1989). We correctedcompass biases, which were mainly due to the magnetic angle(Kaneko et al., 1992). After compass correction, velocity proﬁles
obtained using bottom-tracking mode were averaged over a 100 mhorizontal range (approximately 15 proﬁles mean) to produce a
standard data set with a spatial interval of 100 m and an estimatederror of 7.0/ 15 ¼ 1.8 cm s  1. During the shipboard ADCP surveys,
one conductivity, temperature, depth (CTD) cast was executed at
the center of the tomography area to acquire parameters for use in
the sound speed proﬁle calculations for the ray simulation.
The inverse method (Park and Kaneko, 2001) reconstructed thehorizontal distribution of depth averaged tidal currents from thedifferential travel-time data obtained from the reciprocal soundtransmissions for all the CAT station pairs. The equation system
relates the travel-time difference vector y to the unknown variable
vector x as follows:y = Ex + e ,(1)

where y is a 6th order column vector, corresponding to the total
number of station pairs; x is a vector of the 20 unknown coefﬁcients used in the Fourier function expansion of the stream function to estimate the current ﬁeld; E is the 6  20 transform matrix,
determined by the locations of the 4 CAT stations; e is the matrix

67

of observation errors. The maximum and minimum wavelengths(wavenumbers) in the Fourier expansion are 33 km ( 6π km  1) and
33

11 km ( 2π km  1), respectively. The objective function, with the33

tapered least squares method, is

J = (y − Ex )T (y − Ex ) + α 2xT x,(2)

where α is the weighting factor, which can be determined by the
L-curve method. The expected solution is determined byminimizing J .
The inverse domain is 33 km  33 km (Fig. 1c), and the grid sizefor the data display is 2.8 km  2.8 km, which is slightly smaller
than the mean horizontal resolution of
A /N ¼4.4 km, whereA¼115.6 km2 is the area of the tomography domain, and N ¼6 is
the ray number. Here we used a small grid size to interpolate thecurrent vector from the stream function to show the ﬁne structureof the measured current distribution. Apparently, the interpolation
did not change the mean horizontal resolution of data.

3. Results
3.1. Ray simulation
The CTD cast at the center of the tomography area measured
the water temperature to decrease from the surface to a depth ofabout 15 m and remain nearly constant from 15 to 80 m (Fig. 3a). A
shallow water thermocline formed at a depth of approximately
10 m. As a result, a shallow water sound channel formed at a depthof around 7 m (Fig. 3a).
Ray patterns were obtained for two station pairs, C1–C3 andC2–C4 (Fig. 3b and c), by ray simulations. Ray simulations were
performed using range-dependent bathymetry and a range-independent sound speed proﬁle determined from the CTD dataobtained during the CAT experiment (Fig. 3a). All rays were conﬁned to the upper 30 m. The travelling sound formed direct rays
along the shallow water sound channel axis or made three- or
four-time subsurface bends below the direct rays. The values of thetravel-time mean 7 standard deviation for these rays are(11986.212 70.044) ms for C1–C3 and (11064.161 70.052) ms for
C2–C4. The standard deviations of travel-time varied within anarrow range shorter than time resolution (0.6 ms). This indicatesthat the multi-path signals arrived at nearly the same time, making them indistinguishable by our instruments. All arrival rays
within a one-digit length constructed a broad arrival peak, which
determines the travel-time.
3.2. Along-strait current velocity and transport
The reciprocal sound transmission method (Munk et al., 1995)
determines the section-averaged current velocity between two
acoustic stations to be

um ≈

2
cm
Δτ ,
2L

(3)

where cm and um are the sound speed and current velocity averaged along the ray path, respectively; Δτ is the differential reciprocal travel time; and L the station-to-station distance.
Hourly mean section-averaged current velocity um along eachtransect (Fig. 4a–f) was estimated from Δτ using Eq. (1). V¯ADCPdenotes the shipboard ADCP velocities averaged over the entire
section. um for each transect agrees well with V¯ADCP ; the rootmean-squares differences (RMSD) between the two variables are
0.041 m s  1, 0.036 m s  1, 0.029 m s  1, and 0.030 m s  1 for station pairs C1–C3, C1–C4, C2–C3, and C2–C4, respectively.
um for station pairs C1–C2, C1–C3, C2–C4, and C3–C4 change

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X.-H. Zhu et al. / Continental Shelf Research 108 (2015) 65–75

↔

±

↔

±
°

Fig. 3. (a) Water temperature and salinity proﬁles obtained in the centre of the QS, and their corresponding sound speed proﬁles (SSP). Ray patterns between (b) C1–C3 and(c) C2–C4 station pairs, simulated by the ray-tracing method using the range-averaged SSP proﬁle. The (mean7 standard deviation) of the travel times are shown at thebottom of each panel.

Fig. 4. Velocities along the sound transmission lines. (a)–(f) Time series of the section-averaged velocities along the sound transmission lines (C1–C2, C1–C3, C1–C4, C2–C3,
C2–C4, and C3–C4), respectively; (g) the measured volume transport (QCAT, black line) and the value predicted based on harmonic constants (blue line), through the
Qiongzhou Strait; (h) time series of the volume transport due to residual current (black line) and spatially mean residual current (red line) obtained by the inverse method.
Shaded horizontal bars indicate mean volume transports in each diurnal tide period, separated by blue broken lines. Red rectangles in (b)–(e) and (g) indicate the sectionaveraged ADCP velocity and the volume transport given by ADCP velocity. Positive (negative) value indicates the eastward (westward) velocity or volume transport. (Forinterpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)with tidal phase. Diurnal tidal variation dominates the um fromMarch 17 to 24, and semidiurnal tidal variation dominates fromMarch 28 to 29 (Fig. 4a–f). Temporal variations of um are consistent
with sea level anomalies (Fig. 2a). Station pair C3–C4 um reaches amaximum of about 7 1 m s  1 in the direction nearly parallel tothe strait. The um of station pairs C1–C4 and C2–C3 are small

because these two sections are nearly perpendicular to the strait.
The um between the station pairs based on travel-time difference data are the mean velocities within the upper 30 m of thestrait, because the acoustic rays only distributed in that upper
layer (see Fig. 3b and c). Consequently, volume transport acrossthe whole section cannot be estimated using um and the areas of

X.-H. Zhu et al. / Continental Shelf Research 108 (2015) 65–75

Δτ

69

Δτ

Δτ

Δτ

Δτ
Fig. 5. The relationship between the differential travel time and the volume transport along the strait estimated from the ADCP data for station pairs. (a) C1–C3, (b) C2–C4,
and (c) the mean of C1–C3 and C2–C4. The solid line indicates the regression line derived from the least squares method. The two gray broken lines indicate the range ofstandard deviation between the volume transport estimated from the ADCP data and the volume transport derived from differential travel time.

the vertical sections. On the other hand, volume transport along
the strait can be estimated from the ADCP data (QADCP) collectedduring the 26 cruises. However, the QADCP data are available only
during the time of the shipboard surveys. To acquire a time seriesof volume transport through the QS that allows for harmonic
analysis, we established an empirical relationship between QADCPand Δτ , and applied this relation to all Δτ to obtain a Δτ -based
volume transport through the QS (QCAT).
QADCP and Δτ are linearly related (Fig. 5) for station pairs C1–C3(Fig. 5a), C2–C4 (Fig. 5b), and the mean of station pairs C1–C3 andC2–C4 (Fig. 5c), with correlation coefﬁcients r2 ¼ 0.988, 0.981 and
0.990, and standard deviations of the differences between QADCPand Δτ regression values of 0.051 Sv, 0.064 Sv, and 0.046 Sv, respectively. Therefore, the empirical formulae that relate Δτ tovolume transport are

Q CAT = 0.110 × Δτ C1− C3 − 0.075,(4)

Q CAT = 0.104 × Δτ C2− C4 + 0.038,(5)

Q CAT = 0.100 × Δτmean − 0.016,(6)

where QCAT is the volume transport estimated from a Δτ ; andΔτ C1− C3, Δτ C2− C4 and Δτmean are the differential travel-times for station pairs C1–C3, C2C1–C4 and the mean differential travel time
for station pairs C1–C3 and C2–C4, respectively. Since Eq. (6) gives
the best results among the three equations, we prefer to use themean differential travel time to estimate the QCAT; when the differential travel-time data for only one station pair of C1–C3 or C2–
C4 are available, we use either Eqs. (4) or (5) to estimate QCAT.
As expected, QCAT agreed well with QADCP, with a 0.045 Sv RMSD

(Fig. 4g). Consequently, the empirical formulae Eqs. (4)–(6) accurately estimated the volume transport through the QS with Δτ forthe entire experiment period.
QCAT varied temporally with tide (Fig. 4g). Its value ranged from 0.710 to 0.859 Sv, with  0.044 Sv mean residual volumetransport. Time dependent volume transport due to residual current is deﬁned as the difference between the volume transportmeasurement (black line in Fig. 4g) and prediction (blue line in
Fig. 4g), estimated by a tidal harmonic analysis using T_TIDE Matlab software (Pawlowicz et al., 2002). This transport also variedwith diurnal tide (Fig. 4h). The mean volume transport of subtidal
current averaged in each diurnal tide period (gray bars in Fig. 4h)
was small when the diurnal tide was dominant (March 17–24) but
large when the semidiurnal tide increased (after March 25).
3.3. Tidal current and residual current structures
As introduced in Section 2, the inverse method applied to CAT
data (Park and Kaneko, 2001; Kaneko et al., 2005; Yamaguchi
et al., 2005) gave the depth-averaged current (eastward andnorthward components) at the given grid points (Fig. 1c). The raw
CAT data was acquired in 5-min intervals and directly used in apoint-by-point tidal harmonic analysis (Pawlowicz et al., 2002)
that involves 5 tidal constituents (O1, K1, M2, S2 and MSF) and a thetemporal mean of the residual current.
Fig. 6 presents tidal ellipse spatial distributions of tidal constituents O1, K1, M2, S2 and MSF (Fig. 6a–e), as well as the temporalmean of the residual current (Fig. 6g). The diurnal tidal constituent
O1 (Fig. 6a) is the greatest among the ﬁve tidal constituents, while
the MSF (Fig. 6e) is the smallest among them. The spatially averaged amplitudes and standard deviations (i.e., spatial variation) of

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X.-H. Zhu et al. / Continental Shelf Research 108 (2015) 65–75

15’

C3

O1

C3

C4

K1

C4

M2

C3

C4

12’

20oN
9.00’
C1

6’

C1

C2
1
0

C3

1

m/s

3’

15’

C1
C2

C2
2

S2

C3

C4

0.5

m/s
0

MSF

2

m/s

0

C4

1

(K1+O1)/2

C3

C4

12’

20oN
9.00’
C1

C1

6’
0.5

3’

C2

0.5

m/s

0

1

12’

Residual Current
C3
C4

110oE
14.00’

1

m/s

0

10’
15’

C1

C2

C2

1

16’

m/s
0

18’

10’

12’

110oE
14.00’

2

16’

18’

12’

20oN
9.00’
C1

6’
C2
3’

0.1 m/s

10’

12’

110oE
14.00’

16’

18’

Fig. 6. Depth-averaged tidal current ellipses for the tidal constituents. (a) O1, (b) K1, (c) M2, (d) S2, (e) MSF and (f) diurnal ((K1 þ O1)/2) , and (g) the temporal mean of residual
current. The red tidal current ellipses in (c) and (f) are from Shi et al. (2002). Note that the length scale for the diurnal tidal current ellipses is different from the others. (Forinterpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

O1, K1, M2, S2 and MSF tidal constituents are summarized in Table 1. The spatially averaged amplitudes of O1, K1, M2, S2 and MSF
follow the proportions 1.00:0.60:0.47:0.21:0.11. The major axis
directions (anticlockwise from due east) of the tidal ellipses for theﬁve constituents are generally along the strait direction of 8°(Table 1). The ellipticities, or ratios of minor axis to the major axis,of the ﬁve tidal constituents were generally small, ranging from0.06 to 0.18, indicating that all tidal currents ﬂowed dominantly
along the strait. The temporal mean of the residual current waslarger in the northern area than in the southern area; the spatiallyaveraged westward speed was 7.37 1.7 cm s  1 and the spatiallyaveraged direction (anticlockwise from the east) was 227.3734.7°(Fig. 6g). The maximum temporal mean of the westward residual
current speed was 12.4 cm s  1 in the northern QS.
In order to examine spatial structure variation of the residual
currents during the experiment period, the residual current distributions were averaged over each diurnal cycle and plotted inFig. 7. The residual currents changed in the magnitudes and directions. The residual currents were small when the diurnal tide
dominated but large when the semidiurnal tide increased (after

Table 1
Spatially-averaged amplitude, direction (rotating anticlockwise from the east), errors of the major and minor axis lengths and their standard deviations for the O1, K1, M2, S2
and MSF tidal current constituents.
Tidal constituents

Amplitudes 7 STDs (cm s  1)

Direction7 STDs (deg)Errors of major axis length 7STDs (cm s  1)

Errors of minor axis length 7 STDs (cm s  1)

O1
K1
M2
S2
MSF

57.5 7 9.2
34.3 7 4.1
27.0 7 2.6
12.0 7 1.7
6.6 7 1.2

16.7 710.0
16.1 78.6
15.1 76.0
12.8 78.2
15.8 713.5

7.4 7 1.9
7.0 7 1.6
3.6 7 0.9
3.8 7 0.9
6.0 7 1.5

3.4 71.3
3.3 71.2
2.0 70.7
2.0 70.8
2.8 71.0

X.-H. Zhu et al. / Continental Shelf Research 108 (2015) 65–75

15’

C3

C3

C4

C3

C4

71

C3

C4

C4

12’

20oN
9.00’
C1

6’

C1

C2
3’

C3

C2

0.1 m/s

C3

C4

C1

C2

0.1 m/s

15’

C1

C2

0.1 m/s

C3

C4

0.1 m/s

C3

C4

C4

12’

20oN
9.00’
C1

6’

C1

C2
3’

C3

C2

0.1 m/s

C3

C4

C1

C2

0.1 m/s

15’

C1

C2

0.1 m/s

C3

C4

0.1 m/s

C3

C4

C4

12’
o

20 N
9.00’

C1

6’

C1

C2
3’

C3

C2

0.1 m/s

C3

C4

C1

C2

0.1 m/s

15’

C1

C2

0.1 m/s

0.1 m/s

C4

12’
o

20 N
9.00’

C1

6’

C1

C2
3’

C2
0.1 m/s

10’ 12’o 16’ 18’

Tidal level (m)110 E
14.00’

1.5

0.1 m/s

10’ 12’o 16’ 18’110 E
14.00’

(a)

(b)

(c)

(d)

(e)

(f)

03/18

03/19

03/20

03/21

03/22

03/23

(g)

(h)

(i)

(j)

(k)

(l)

(m)

(n)

0.5
−0.5
−1.5
03/17

03/24 03/25 03/26
Date (month/day)

03/27

03/28

03/29

03/30

03/31

04/01

04/02

Fig. 7. Vector plots of the residual current distributions averaged during each diurnal cycle. (a)–(n) Distributions corresponding to the cycles separated by blue broken lines
in the bottom panel, where the tidal gauge data are also plotted with the black line.

March 25), consistent with mean residual transport trends (shaded horizontal bars in Fig. 4h). The time series of the spatial mean
residual current averaged in the tomography domain (red line in
Fig. 4h) also agreed well with residual transport (black line in
Fig. 4h), verifying the inverse calculation.
3.4. CAT compared with shipboard ADCPThe tidal current velocity data obtained from inverse analysis of
CAT data were interpolated to the shipboard ADCP track and thenvalidated by the shipboard ADCP data. The current velocities obtained by the CAT measurements agreed well with the depthaveraged shipboard ADCP data, with 0.094 m s  1 and 0.066 m s  1
RSMDs, and correlation coefﬁcients r2 ¼ 0.980, 0.847 for the alongand cross-strait velocity components, respectively (Fig. 8).

3.5. Error evaluation
We estimated the ﬁve major tidal current constituents from
CAT data using inverse analysis. The CAT measurement error, inverse error, or non-tidal energy can inﬂuence the results of thetidal current constituents and non-tidal current. Here, we consider
the ‘error of tidal currents’ to be the velocity signal not attributable
to the tidal constituents and calculated as in Pawlowicz et al.
(2002):σξ =⎛ ∂F ⎞2 2⎛ ∂F ⎞2 2⎛ ∂F ⎞2 2⎛ ∂F ⎞2 2⎟σ ,⎟ σBr + ⎜⎟ σ Ai + ⎜⎟ σ Ar + ⎜⎜⎝ ∂Bi ⎠ Bi
⎝ ∂Br ⎠⎝ ∂Ai ⎠
⎝ ∂Ar ⎠

(7)

where F is a non-linear function of Ak , Bk , i.e., ξ = F (Ak , Bk ) , whose
partial derivatives with respect to Ar , Ai , Br , Bi can be determinedexactly (Pawlowicz et al., 2002). Ak = Ar + iAi , Bk = Br + iBi are

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X.-H. Zhu et al. / Continental Shelf Research 108 (2015) 65–75

91%, respectively.
The errors of the residual current velocities were generally
small, with somewhat larger values in the northern area than in
the southern area (Fig. 9f). The maximum error of the residual
current velocity was 2.1 cm s  1, appearing at the northern part of
the QS. The area mean of error for the residual currents is
1.2 cm s  , which is approximately 17 % of the temporal mean of
the residual currents.

1
r2=0.980
rms=0.094(m/s)

(m/s)

0.5

U

ADCP

0

−0.5

4. Discussion
4.1. Comparisons with previous studies

−1
−1

−0.5
U

0
0.5
(m/s)

1

0
0.5
(m/s)

1

CAT

1
2

r =0.847
rms=0.066(m/s)

(m/s)

0.5

V

ADCP

0

−0.5

−1
−1

−0.5
V

CAT

Fig. 8. Scatter plots of the velocity. (a) Along- and (b) across-strait components
obtained between the shipboard ADCP data (VADCP) and the inversion results using
CAT data (VCAT). The solid line indicates the regression line derived from the leastsquares method. The two gray broken lines indicate the standard deviation of thedifference.

harmonic constants of tidal constituent k. The subscripts r and idenote the real and imaginary parts. σ A2r , σ A2i , σB2r , σB2i are the total
residual powers for Ar , Ai , Br , Bi , respectively.
The ‘error of residual currents’, not included in the tidal harmonic analysis, for a mean residual current isN(∑m = 1 v0 ( tm ) −N−1

v0 ( tm )

2

),

(8)

where v0 is the time dependent residual current, m denotes thetime series sequence number and N denotes the time series
length. v0(m) denotes the temporal mean of the residual current.
The ‘error of tidal currents’ in Eq. (7) and ‘error of residual
currents’ in Eq. (8) were calculated using T_Tide Matlab software.
The error of the depth-averaged tidal currents along the major
and minor axes of the ﬁve tidal constituent tidal ellipses and theresidual current velocity are shown in Fig. 9. The error of thedepth-averaged tidal currents was larger in the northern area thanin the southern area. The error of the major axis length was also
larger than the error of the minor axis length. The area mean (i.e.,
average over the tomography domain) errors and standard deviations of the O1, K1, M2, S2 and MSF tidal current constituentsmajor and minor axes lengths are summarized in Table 1. The
errors of the major axis lengths of the O1, K1, M2, S2 and MSF tidal
current constituents was approximately 13%, 20%, 13%, 31% andTo evaluate the accuracy of our estimated tidal currents, we
compared our results with previous independent measurements.
Estimated tidal current data from this study agreed well with results reported by Shi et al. (2002). They collected all available
current data from observations in the QS from 1963 to 1999, which
included the four data sites covered by our CAT observations. The
diurnal ((O1 þK1)/2) and semidiurnal tidal currents estimated from
our CAT data (black ellipses) are consistent with those reported by
Shi at al. (2002) (red ellipses) (Fig. 6c and f). The RMSD of the
major axes, minor axes and diurnal tidal current ellipses orientations at the four sites are 14.3 cm s  1, 4.0 cm s  1 and 12.3°, respectively; the RMSD of the major axes, minor axes and semidiurnal tidal current ellipses orientations at the four sites are
1.8 cm s  1, 2.8 cm s  1 and 9.6°, respectively.
Shi et al. (2002) showed that mean residual currents in the QS
during spring were stronger in the northern strait than in thesouthern strait (Fig. 8 in their paper). Using a three-dimensional
numerical model, Chen et al. (2009) showed that the residual
currents in the QS ranged between 3 and 30 cm s  1, were stronger
in the northern strait than in the southern strait, and were directed west in the northern area but south in the southern area
(Fig. 8b1–b3 in their paper). Our results (Fig. 6g) show the same
spatial variation between the northern area and the southern area
as found previously, and residual current magnitudes that fellwithin the range of the model results. These comparisons afﬁrm
the validity of our method of estimating tidal currents by inversion
analysis of CAT data.
Based on different observations, i.e., non-synchronous data, thevolume transport due to residual currents in the QS have been
estimated as 0.2–0.4 Sv in winter and spring (Shi et al., 2002),
0.055 Sv (Chen et al., 2007), 0.020–0.048 Sv (Yan et al., 2008), and0.065 Sv (Zhu et al., 2014). Although these estimates vary inmagnitude, they are consistently directed westward. Based onsynchronous data and realistic bathymetry, we estimate a 0.044 Sv
westward volume transport due to residual current in the spring,
within the range of previously reported values.
4.2. Dynamics of residual currents
In a simulation of the QS including only tidal current forcing, awestward residual current was produced by tidal rectiﬁcation (Shiet al., 2002). Inclusion of more forcing, such as winds, buoyant
ﬂux, and boundary currents in a tide-resolving model indicatedthat the sea level difference between the two entrances of the QS
also signiﬁcantly drives a throughﬂow in the QS (Gao et al., 2013).
The spatial distributions of tidal currents given by our CAT data
allow us to examine the possible factors controlling the residual
currents in the QS.
By integrating the 3-dimensional momentum equations fromthe sea bottom up to the sea surface, we obtained the depthaveraged momentum equations in the across-strait direction

X.-H. Zhu et al. / Continental Shelf Research 108 (2015) 65–75

15’

C3

O1

C3

C4

K1

73

C3

C4

M2

C4

12’

20oN
9.00’
C1

6’

C1

C2
3’

0.2 m/s

15’

C3

C1

C2

S2

C3

C4

MSF

C2
0.2 m/s

0.1 m/s

C4

Residual Current
C3
C4

12’

20oN
9.00’
C1

6’

3’

0.1 m/s

10’

12’

C1

C1
C2

C2

C2

110oE
14.00’

16’

0.1 m/s

18’

10’

12’

110oE
14.00’

16’

0.1 m/s

18’

10’

12’

110oE
14.00’

16’

18’

Fig. 9. Error of the ﬁve tidal constituents. The thin and thick lines indicate the error in the major and minor axes, respectively. Note that the length scale for the diurnal tidal
current ellipses is different from the others.

u (∂v/∂x ) + v (∂v/∂y ) + fu

(

)

(

= − g (∂η/∂y) + Ah ∂ 2v/∂x 2 + Ah ∂ 2v/∂y 2)− τ by/ρh + τ wy/ρh ,(9)and in the along-strait direction:(= − g (∂η/∂x) + Ah ∂ u/∂x

) + A ( ∂ u/∂y ) − τ /ρh + τ /ρh (. 10)2hτbx = ρCd u2 + v 2 u and τby = ρCd u2 + v 2 v , where Cd is 0.0025. The
wind stresses were calculated as τwx = Cw ρa uwind2 + vwind2 uwindu (∂u/∂x ) + v (∂u/∂y ) − fv
2data. In these calculations, the Coriolis parameter f was
5.0  10  5 s  1, corresponding to the latitude of the QS, and thehorizontal eddy viscosity coefﬁcient (Ah) was 90 m2 s  1, as used byCa´ceres et al. (2003). The bottom stresses were calculated as

2

2

bx

wx

Here, u and v are the depth-averaged total currents given by CAT
in the along-strait (x axis) and across-strait (y axis) directions, respectively; τbx and τby are the bottom friction in the along-strait andacross-strait directions, respectively; τwx and τwy are the windstresses in the along-strait and across-strait directions, respectively;
f is Coriolis parameter; ρ is the water density (1022 kg m  3); g is theacceleration of gravity; η is the elevation from the mean sea surface;
Ah is the horizontal eddy viscosity coefﬁcient; h is the water depth.
Bars over each term denote temporal averages over 15 diurnal tidal
cycles. The temporal variations of ū and v̄ are assumed to be negligible to the other terms in Eqs. (9) and (10).
The ﬁrst two terms on the left hand sides of Eqs. (9) and (10)denote the nonlinear effects of tidal and residual currents that cause
tidal rectiﬁcation (Tadv); the third term on the left hand side is the
Coriolis term (Tcor); the ﬁrst term on the right hand side of Eq. (9) is
the water elevation gradient in the across-strait direction; the ﬁrst
term on the right hand side of Eq. (10) is the water elevation gradient in the along-strait direction, partly corresponding to the sea
levels at the two entrances of the QS; the second and third terms on
the right hand side of the two equations are horizontal mixing (Thor);
the fourth and ﬁfth terms on the right hand side of the two equations are the bottom friction (Tbf) and wind stresses (Twf). We calculated the bold font terms shown in Eqs. (9) and (10) using ourand τwy = Cw ρa uwind2 + vwind2 vwind ; Cw denotes the drag coefﬁcient of the wind speed given by
Cw = 10−3 ×(0.63 + 0.066 × uwind2 + vwind2 ) ; ρa = 1.29 kgm −3 is air density(Smith and Banke, 1975). uwind and vwind denote the eastward andnorthward wind velocity components from reanalysis of data with
six hour intervals (http://apps.ecmwf.int/datasets/data/interim-fulldaily/).
In the across-strait direction (Fig. 10), u (∂v/∂x ) (Fig. 10a) was thelargest andτby/ρh the smallest component (Fig. 10f).
v (∂v/∂y ) (Fig. 10b) was of the same order as Tcor (Fig. 10c) and Thor
(Fig. 10d and e). The area mean of the absolute values of TCor, Tadv
( u (∂v/∂x ) + v (∂v/∂y )), Thor ( Ah (∂ 2v/∂x 2) + Ah (∂ 2v/∂y2 )), Tbf and Twf
were (2.42, 8.67(7.02 þ1.65), 6.03(2.89 þ3.14), 0.85 and 0.30)(  10  6 m s  2), respectively. Then, the approximate ratios ofthese terms were TCor:Tadv:Thor:Tbf :Twf ¼ 1.00:3.58:2.50:0.35:0.12.
Tbf and Twf were negligible in the across-strait direction momentum balance. The water elevation gradient was approximately
4.3  10  6 m s  2 by Eq. (9). Using the width of the QS (19.5 km),this value corresponded to a sea level difference of 0.009 m acrossthe strait, close to 0.01 m reported by Shi et al. (2002). Apparently,
the geostrophic balance does not hold exactly for the residual
current in the along-strait direction because of large nonlinear
effect contributions (Tadv) and frictional forcing due to horizontal
shear (Thor). In other words, our synchronous data supported tideinduced residual currents, as suggested earlier by non-synchronous data (Shi et al., 2002).

74

X.-H. Zhu et al. / Continental Shelf Research 108 (2015) 65–75

4

10

15

−2

o

20 N
9.00’

−4

−2

−1

−3

−3

−5

−10

5

6’

1

−1

5

12’

2

15’

−1

3’

15’

2

−0
.5

4
6

−1

2

−2

6’

.5
−0

4

20oN
9.00’

2
4

−4
−2

−2

12’

−2

3’
10’

12’

110oE
14.00’

16’

18’

10’

12’

16’

110oE
14.00’

18’

10’

12’

16’

110oE
14.00’

18’

Fig. 10. Contour plots for the terms in Eq. (5). (a) u (∂v/∂x ) , (b) v (∂v/∂y ) , (c) fu , (d) Ah (∂ 2v/∂x 2) , (e) Ah (∂ 2v/∂y2) and (f) τby/ρh . The units in all panels is 10  6 m s  2.

(  10  6 m s  2), respectively. The approximate ratios of theseterms were TCor:Tadv:Thor:Tbf:Twf ¼ 1.00:4.63:3.55:1.34:0.19. The Twf
was negligible in the along-strait direction momentum balance.
Eq. (10) estimated the water elevation gradient to be approximately 2.2  10  6 m s  2, which corresponds to a sea level

In the along-strait direction (Fig. 11), the Tadv ofu (∂u/∂x ) (Fig. 11a) was the largest and the TCor is the smallest(Fig. 11c). The area mean for the absolute values of the TCor, Tadv
( u (∂u/∂x ) + v (∂u/∂y )), Thor ( Ah (∂ 2u/∂x 2) + Ah (∂ 2u/∂y2 )), Tbf and Twf
were [1.70, 7.88 (5.50 þ2.38), 6.03 (2.07þ 3.96), 2.27 and 0.33]

15’

10

20oN
9.00’

−4
−2

−6

5

4

8
6
2

2.

1.5

3

−2

5

1.

6’

0.5

2
3.5

12’

−8

−10

3’

15’

−2

2

−6

−1

6’

−4
−4

3

5

−3

2

20oN
9.00’

−3
−1 −2
1

6

4

3

12’

3’
10’

12’

110oE
14.00’

16’

18’

10’

12’

110oE
14.00’

16’

18’

10’

12’

110oE
14.00’

16’

18’

Fig. 11. Contour plots for the terms in Eq. (6). (a) u (∂u/∂x ) , (b) v (∂u/∂y ) , (c) fv , (d) Ah (∂ 2u/∂x 2) , (e) Ah (∂ 2u/∂y2) and (f) τbx/ρh . The unit in all panels is 10  6 m s  2.

X.-H. Zhu et al. / Continental Shelf Research 108 (2015) 65–75

difference of 0.015 m, using the length of the QS (70 km). A negative (westward) along-strait water elevation gradient balancing
momentum in the along-strait direction implies that tidal rectiﬁcation (Shi et al., 2002) is not the sole mechanism of westward
residual current formation in the QS. The sea level difference between the two entrances of the QS also importantly affects westward residual current formation in the QS (Gao et al., 2013).

5. Conclusions
We present the ﬁrst synchronous observations of tidal current,
residual current, and volume transport in the QS. A 15-day coastal
acoustic tomography experiment measured the tidal currents in
the QS using four acoustic stations during March, 2013. Throughout the experiment, reciprocal sound transmissions were performed along six transmission lines. Shipboard ADCP surveys
performed along the sound transmission lines provided independent transport estimates and comparisons with the CAT
data.
The range-averaged along-transect current velocities estimatedusing differential travel time data ( Δτ ) agreed well with ADCP
measurements. The differential travel time data correlate wellwith volume transport (QADCP) through the QS estimated from the
ADCP data. The QCAT for the entire CAT experiment period, predicted using the empirical regression formula between QADCP andΔτ , ranges from  0.710 to 0.859 Sv, with a 0.044 Sv mean westward volume transport of residual (time independent) current.
Inverse analysis was adopted to reconstruct tidal current distributions from Δτ obtained between paired CAT stations. Using
the current data from inverse analysis, we obtained harmonic
constants of ﬁve major tidal currents (K1, O1, M2, S2, and MSF), andthe residual current in the QS. The major axes of the tidal ellipsesof the ﬁve constituents were generally directed along the strait.
The ratios of O1, K1, M2, S2 and MSF tidal currents amplitudes were
1.00:0.60:0.47:0.21:0.11. The residual current ﬂowed westward,
stronger in northern area than in southern area and turned from awestward direction in the northern area to southward in southernarea of the strait.
The dynamic analysis based on the currents estimated from
CAT data was used to examine the controlling factors of the residual currents in the QS. The residual current varies signiﬁcantly
during the 15 days, such that it is stronger in the period of coexistence of semi-diurnal and diurnal tides than in the period of
diurnal tide. The westward residual current in the QS was causedby both tidal rectiﬁcation and the sea level difference between its
two entrances.
The volume transport of the residual current averaged overeach diurnal tide period was small during the diurnal tide andbecame larger when the semidiurnal tide was dominant. This increase implies that the temporal variation of residual currents
cannot be explained by tidal rectiﬁcation alone, which usually
gives a proportional relation between tidal current and residual
current. Therefore, a long-term (4 1 year) current monitoring the
QS is necessary for a full understanding of the mechanisms controlling transport through the QS. The easy handling of the CAT
system and its performance (as shown in this study) suggest that
the CAT system is potentially a highly viable way to achieve this
difﬁcult goal.

75

Acknowledgments
This study is supported by the National Basic Research Programof China (2011CB403503); the National Natural Science Foundation of China (41276095, 41476020, 41176021, 41576001 and41321004); the projects of the State Key Laboratory of Satellite
Ocean Environment Dynamics, Second Institute of Oceanography(No. SOEDZZ1403 and SOEDZZ1501) and the Global Change and
Air-Sea interaction (GASI-03-01-01-02), and the Scientiﬁc Research Fund of SIO under grants JT1402.

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