Introduction
The tidal system in the Indonesian seas is the most complex one in the world,
due to its rugged bottom topography, complicated coastline, and the
interference of tidal waves propagating from the Pacific Ocean, Indian Ocean,
and South China Sea (SCS). The earliest reports of tidal characteristics in
the Indonesian seas can be traced back to the colonial period in the early
twentieth century, which were recompiled by Wyrtki (1961) to construct
diurnal and semidiurnal cotidal charts based on all available coastal and
island observations. Although the results of Wyrtki (1961) are impressively
reasonable in the Indonesian seas, mapping of the Indonesian tides are still
incomplete owing to lack of observations. During the past decades,
remarkable progress of investigations about tidal phenomena is benefited by
use of satellite altimeter measurements and high-resolution numerical
simulation, and with no exception in the Indonesian seas. Based on tide
gauge observations and TOPEX/Poseidon (T/P) satellite altimeter data,
Mazzega and Berge (1994) have produced the cotidal charts of M2 and
K1 in the Indonesian seas using an inversion method. Using a barotropic
tide model, Hatayama et al. (1996) investigated the characteristics of
M2 and K1 tides and tidal currents in the Indonesian seas, which
shows that the tidal currents in the Java Sea (JS) and in the vicinities of
narrow straits, i.e., the Lombok and Malacca straits, are relatively strong.
Egbert and Erofeeva (2002) have assimilated satellite altimeter data into an
inverse barotropic ocean tide model, providing the cotidal charts and tidal
currents for M2 and K1 constituents in the Indonesian seas. Their
results are further reported by Ray et al. (2005), showing that there are
three types of tides in the Indonesian seas: semidiurnal tides dominated but
with significant diurnal inequality in the eastern Indonesian seas and the
adjoining region of the Pacific Ocean; mixed diurnal tides in the region
west of 118∘ E; and diurnal tides west of the Kalimantan Island.
Using the Regional Ocean Modeling System (ROMS), Robertson and Ffield (2005,
2008) have simulated the barotropic and baroclinic tides in the Indonesian
seas for four tidal constituents M2, S2, K1, and O1. The
results show that semidiurnal tides originate from both the Pacific and
Indian oceans; whereas the diurnal tides are mainly from the Pacific Ocean.
These results are confirmed by Teng et al. (2013), which suggests that the
M2 tide mainly propagates from the Indian Ocean into the Pacific Ocean
through the eastern Indonesian seas, whereas the K1 and O1 tides
propagate in an opposite direction. Although the characteristics of
Indonesian tides have been simulated with more and more accurate geometry,
and the results are indeed better than before, the tides in the southern SCS
and JS, particularly in the junction region between the SCS and JS, are
still not well determined as reflected by the fact that the simulated
results are model dependent.
The junction area between the SCS and the JS, comprising the southern Natuna
Sea, the Karimata Strait, and the Gaspar Strait, is a throat connecting the
SCS and the Indonesian seas (Fig. 1). Furthermore, this area is also the
convergent region of tidal waves that propagate from the SCS or the JS
(Hatayama et al., 1996). It is worth noting that the simulated tidal
currents in this area are discrepant among different models, even when the
satellite altimeter data have been assimilated into the models. This is most
possibly due to the coarse altimeter track separation (only one ascending
track and one descending track pass through this region; Ray et al., 2005).
Therefore, offshore observations are needed to provide a clearer recognition
about the Indonesian tides and to assess the existing model results.
The map of the Indonesian seas (upper) and observational stations
(lower). Isobaths are in meters.
In this study, long-term water level and current profile observations at
five stations (Fig. 1) are used to investigate the characteristics of tidal
elevation, current, and energy flux between the SCS and JS. The results are
not only important for understanding local dynamics but also useful for the
determination of open boundary condition in tidal simulation of the SCS or
Indonesian seas. The rest of the paper is organized as follows: Sect. 2
gives a description of the observed data; Sect. 3 presents the analyzed
results of tides, tidal currents, and tidal energy fluxes; finally, a summary
and discussion are given in Sect. 4.
Data
The data used in this study were obtained under the trilateral collaborative
project “The South China Sea – Indonesian seas Transport/Exchange (SITE)
and Impacts on Seasonal Fish Migration” which was established in 2006 by the
First Institute of Oceanography (FIO), State Oceanic Administration, China;
the Agency for Marine and Fisheries Research and Development (AMFRD),
Ministry of Marine Affairs and Fisheries, Indonesia; and the Lamont-Doherty
Earth Observatory (LDEO), Columbia University, USA. The study area of the
project was extended to the Sunda Strait in 2008, and the title of the
collaborative program was changed to “The South China Sea – Indonesian seas
Transport/Exchange (SITE) and Dynamics of Sunda and Lombok straits, and
Their Impacts on Seasonal Fish Migration”.
Current and sea level measurements were made from December 2007 to September
2011 in the southern Natuna Sea, Gaspar Strait and Karimata Strait by using
trawl-resistant bottom mounts (TRBMs). The TRBMs were equipped with acoustic
Doppler current profilers (ADCPs) and pressure gauges for measuring current
profiles and sea levels. The volume, heat, and freshwater transports between
the SCS and the Indonesian seas have been previously reported by Fang et al. (2010)
and Susanto et al. (2013). In the present paper we focus on the tides
and tidal currents in the area as shown in the lower panel of Fig. 1. The
measurements were conducted along three sections. Section A is located in
the southern Natuna Sea between the Bangka Island and Kalimantan Island.
Section B1 is in the Gaspar Strait between the Bangka Island and Belitung
Island. Section B2 is located in the Karimata Strait between the Belitung
Island and Kalimantan Island. The mean water depths of the five TRBM
stations labeled A1, A2, B1, B2, and B3 are 36.6, 48.0, 44.2, 42.8, and 49.0 m, respectively (Table 1).
The vertical bin size of ADCP measurements is 1 m for Station A1 and 2 m for other stations. The observational lengths of
the sea level and current profile vary from 33 to 960 days as listed in
Table 2.
Locations and water depths of the observational stations.
Station
Longitude
Latitude
Depth
Bin size
(m)
(m)
A1
106∘50.1′ E
1∘40.0′ S
36.6
1
A2
107∘59.2′ E
1∘05.5′ S
48.0
2
B1
107∘09.6′ E
2∘46.8′ S
44.2
2
B2
108∘15.0′ E
2∘17.0′ S
42.8
2
B3
108∘33.0′ E
1∘54.9′ S
49.0
2
Record length of the obtained data.
Station
Measuring
Starting and ending dates
Length
parameter
(yr/mm/dd)
(d)
A1
Current profile
2008.01.13–2008.02.14
33
Sea level
2008.01.13–2008.05.05
114
A2
Current profile
2007.12.04–2008.01.12
301
2008.02.15–2008.11.01
Sea level
2007.12.02–2008.05.05
156
B1
Current profile
2008.05.12–2008.10.11
168
2008.11.07–2008.11.15
2009.10.19–2009.10.24
Sea level
2008.05.12–2008.11.03
176
B2
Current profile
2008.11.02–2010.11.11
740
Sea level
2009.10.18–2010.11.11
390
B3
Current profile
2008.11.07–2009.10.17
960
2009.10.19–2010.11.12
2011.02.17–2011.09.29
Sea level
2008.11.06–2009.09.09
308
Analyzed results from observations
Tides
Based on the observed sea level data, we extract the harmonic constants of
six principle tidal constituents K1, O1, Q1, M2,
S2, and N2 using the conventional harmonic analysis method
developed by Wang and Fang (1981), which is nearly of the same performance
as those developed by Foreman (1977) and Pawlowicz et al. (2002). Since the
shortest record length is 33 days (current observation at Station A1), the
Rayleigh criterion for separating these six constituents is satisfied.
According to Rayleigh criterion, to separate P1 from K1 and
K2 from S2 requires 182.6 days (e. g., Pugh, 1987, p. 113), thus the
influences of P1 on K1 and K2 on S2 are corrected
through introducing inference quantities (amplitude ratios and phase-lag
differences between P1 and K1, and between K2 and S2) in
this study. Moreover, a nearest tidal gauge station at Keppel Harbour
(103.82∘ E, 1.26∘ N) was used as an inference station,
where the amplitude ratio and phase-lag difference of P1 vs. K1
are equal to 0.296 and -10∘, respectively, and those of K2
vs. S2 are equal to 0.286 and -2∘, respectively.
The obtained amplitudes and Greenwich phase lags for the constituents
K1, O1, Q1, M2, S2, and N2 at five stations
are listed in Table 3. The harmonic constants of P1 and K2 can be
derived from those of K1 and S2, respectively, listed in the table
using the inference relations. It can be seen from the table that the
constituent K1 has the largest amplitude, exceeding 50 cm. The second
largest amplitude is that of constituent O1, exceeding 30 cm. For
semidiurnal tides, the amplitudes are all smaller than 5 cm for M2,
while they are greater than 5 cm for S2 at Stations B1, B2 and B3. For
all of the five stations, it is found that the amplitudes of diurnal tides
are much greater than those of semidiurnal tides, suggesting that diurnal
tides are the dominant constituents in this area. Meanwhile, the results
also show that the phase lags of the diurnal tides slightly increase from
Section A to Sections B1 and B2. On the contrary, the phase lags of the
semidiurnal tides dramatically increase from the eastern segment of Section
A (represented by Station A2) to Section B2, and from Section B1 to the
western segment of Section A (represented by Station A1). These results
suggest that the study area is located in the antinodal band of the diurnal
tidal waves but in the nodal band of the semidiurnal tidal waves. As a
result, the amplitudes of diurnal tides are greater than those of
semidiurnal tides, whereas the phase lags of diurnal tides change less than
those of semidiurnal tides. The semidiurnal tidal waves in this area appear
as a superposition of the incident waves propagating from the SCS and Indian
Ocean (Ray et al., 2005; Teng et al., 2013). These two incident waves happen
to have similar intensity and opposite phase, resulting in a nodal band.
In contrast to the semidiurnal tides, the diurnal tidal waves in this area
appear as a superposition of the incident waves propagating from the SCS and
the Pacific Ocean (Ray et al., 2005; Teng et al., 2013). These two incident
waves have basically the same phase, resulting in an antinodal band.
Tidal harmonic constants at the observation stations.
A1
A2
B1
B2
B3
Constituent
H
G
H
G
H
G
H
G
H
G
(cm)
(deg)
(cm)
(deg)
(cm)
(deg)
(cm)
(deg)
(cm)
(deg)
K1
59.1
30.0
50.8
27.0
59.6
33.3
54.4
45.4
57.2
36.2
O1
42.4
329.1
37.4
326.8
39.6
344.7
36.5
354.7
35.2
343.9
Q1
7.8
306.1
7.2
305.4
7.4
324.3
7.3
335.0
11.7
339.4
M2
3.8
341.3
4.4
322.9
4.3
236.4
1.9
117.5
2.2
68.5
S2
2.6
82.3
2.7
62.2
5.3
160.0
5.6
123.8
8.7
96.5
N2
0.4
306.6
0.8
284.9
2.0
206.6
0.6
192.6
0.5
8.7
Tidal currents
The conventional harmonic method is applied to the current data analysis for
extracting the harmonic constants of principle tidal constituents, as done
in the analysis of tidal elevation in Sect. 3.1. The vertical structures
of current ellipses of the constituents K1, O1, Q1, M2,
S2, and N2 at each station are shown in Fig. 2. It can be seen
that there are little vertical changes in the ellipses for all constituents
at all stations, except for the top layer where the currents suffer strong
disturbances due to winds, at some stations for some constituents. This
vertical uniformity indicates that the tidal currents are basically of
barotropic nature at all stations. Thus, we only use vertically averaged
currents to reveal the characteristics of tidal currents in this study.
Parameters of the vertically averaged current ellipse, including semi-major and
-minor axes (W and w), ellipticity (r), Greenwich phase-lag (ϕ), and
direction (λ) of the maximum current speed, are given in Table 4a–e.
In the table, signs of the ellipticity represent the sense of the
current vector rotation, positive for counterclockwise and negative for
clockwise (the term of ellipticity generally refers to the flatness of an
ellipse; here, it is defined as the ratio of the minor axis vs. major axis as
done in Fang and Ichiye (1983) and Beardsley et al., 2004). We can see that
at Station A1 the tidal currents rotate counterclockwise, except N2. At
Station A2 the tidal currents rotate counterclockwise, except S2. At
Station B1 the tidal currents rotate counterclockwise, except M2 and
N2. At Station B2 the diurnal tidal currents rotate counterclockwise,
the semidiurnal currents rotate clockwise. At Station B3 the diurnal tidal
currents rotate counterclockwise, while the semidiurnal currents rotate
clockwise except N2. Figure 3 shows the current ellipses of K1,
O1, M2, and S2. We can see that all stations show the
characteristics of rectilinear tidal currents. The semi-major axes lengths
of tidal current ellipses are 10 cm s-1 for diurnal constituents
K1 and O1, with O1 slightly smaller than K1 in the most
cases. At Station B1, however, which is located in the Gaspar Strait,
diurnal tidal currents are significantly increased by the narrowing effect
of the strait. In particular, the maximum speed of O1 can approach
20 cm s-1. The semi-major axes lengths of tidal current ellipses of the
semidiurnal constituents M2 and S2 are generally smaller than 5 cm s-1 at all stations.
The vertical distributions of current ellipses of tides
constituents K1, O1, Q1, M2, S2, and N2 at
Stations A1 (a), A2 (b), B1 (c), B2 (d), and B3 (e).
The vertically averaged tidal current ellipses of principle tidal
constituents K1, O1, M2, and S2 at the observational
stations. Dots on the ellipses represent the tips of the tidal current
vectors at 00:00 GMT.
Vertically averaged tidal current ellipse.
Constituent
W
w
r
ϕ
λ
(cm s-1)
(cm s-1)
(deg)
(deg)
(a) Station A1
K1
9.63
0.70
0.07
136.6/316.6
12.6/192.6
O1
8.02
3.34
0.42
114.2/294.2
355.2/175.2
Q1
2.25
0.58
0.26
108.0/288.0
338.0/158.0
M2
2.34
0.38
0.16
128.9/308.9
153.4/333.4
S2
1.83
0.16
0.09
92.1/272.1
158.1/338.0
N2
0.97
0.19
-0.19
113.3/293.3
158.1/338.1
(b) Station A2
K1
11.51
1.89
0.16
144.9/324.9
348.8/168.8
O1
10.31
1.97
0.19
120.3/300.3
339.9/159.9
Q1
2.41
0.16
0.06
103.4/283.4
335.8/155.8
M2
3.00
0.58
0.19
8.7/188.7
176.7/356.7
S2
2.28
0.79
-0.35
18.5/198.5
163.3/343.3
N2
0.80
0.30
0.37
164.3/344.3
0.2/180.2
(c) Station B1
K1
13.32
0.05
0.00
82.4/262.5
167.4/347.4
O1
19.08
0.77
0.04
13.5/193.5
172.2/352.2
Q1
4.32
0.39
0.09
162.9/342.9
354.0/174.0
M2
5.41
1.07
-0.20
89.0/269.0
177.4/357.4
S2
4.34
0.67
0.16
112.6/292.5
188.0/8.0
N2
1.40
0.30
-0.21
89.1/269.1
180.7/0.7
(d) Station B2
K1
12.25
1.27
0.10
109.7/289.7
119.3/299.3
O1
11.56
1.55
0.13
25.9/205.9
128.5/308.5
Q1
2.32
0.36
0.16
169.4/349.4
309.5/129.5
M2
4.05
0.31
-0.08
37.8/217.8
127.9/307.9
S2
1.10
0.04
-0.04
157.4/337.4
265.3/85.3
N2
0.86
0.05
-0.05
5.0/185.0
125.5/305.5
(e) Station B3
K1
7.77
0.15
0.02
84.0/264.0
145.5/325.5
O1
10.26
0.24
0.02
4.8/184.8
146.9/326.9
Q1
2.25
0.04
0.02
146.4/326.4
327.7/147.7
M2
4.30
0.06
-0.01
25.1/205.1
144.5/324.5
S2
1.10
0.40
-0.36
7.8/187.8
116.1/296.1
N2
0.86
0.08
0.10
178.6/358.6
324.6/144.6
W – length of the semi-major axis (i.e., maximum speed); w – length of the semi-minor
axis (i.e., minimum speed); r – ellipticity, equal to the ratio w/W with
signs representing the direction of the current vector rotation
(positive/negative for counterclockwise/clockwise); ϕ – Greenwich
phase lag of the maximum current speed; λ – direction of the semi-major
axis measured clockwise from north. Both ϕ and λ have
two values with a difference of 180∘, respectively.
Tidal energy flux density
The energy flux across a section of unit width is called flux density. For a
specific constituent it can be calculated from harmonic constants of tidal
elevation and tidal current by the following formula:
Fx,Fy=ρghT∫0Tζu,vdt=12ρghHUcosξ-G,Vcosη-G,
where Fx,Fy are the east and north
components of the tidal energy flux density, respectively, T is the period of
the tidal constituent, ρ is the water density (taken to be 1021 kg m-3
for a temperature of 28 ∘C and a salinity of 33 which are
roughly equal to the mean temperature and salinity in the study area), g is
the gravity acceleration,
h is the undisturbed water depth, ζ is the tidal elevation, u,v are the east and north components of vertically
averaged tidal currents, t is the time, H and G are the amplitude and
phase lag of the tide, U,V are the amplitudes of the east
and north components of vertically averaged tidal current, and ξ,η are the phase lags of the corresponding components of tidal
current.
Table 5 lists the east component of energy flux density Fx, north component of
energy flux density Fy,
magnitude F, and direction θ (in degrees
measured clockwise from the true north) at Stations A1, A2, B1, B2, and B3
from observed harmonic constants. Moreover, the direction differences
between the current major axis and the energy flux vector Δθ
are also given in Table 5 (since the current ellipse has two semi-major axes
with opposite directions, in the calculation of Δθ we choose
the one that is aligned with the energy flux). Figure 4 shows the tidal
energy flux densities of the principal diurnal tidal constituents K1
and O1 and the principal semidiurnal tidal constituents M2 and
S2.
Horizontal tidal energy flux density.
Tidal energy flux density.
Constituent
Fx
Fy
F
θ
Δθ
(kW m-1)
(kW m-1)
(kW m-1)
(deg)
(deg)
(a) Station A1
K1
0.0628
-3.0800
3.0806
178.8
-13.8
O1
1.9216
-5.0011
5.3576
159.0
-16.2
Q1
0.1394
-0.2759
0.3091
153.2
-4.8
M2
-0.0746
0.1175
0.1392
327.6
-5.8
S2
0.0312
-0.0807
0.0865
158.9
0.8
N2
-0.0023
0.0066
0.0069
340.7
2.6
(b) Station A2
K1
3.2910
-6.0846
6.9176
151.6
-17.2
O1
3.6167
-7.5581
8.3789
154.4
-5.5
Q1
0.1690
-0.3507
0.3893
154.3
-1.5
M2
-0.0310
-0.2249
0.2270
187.9
11.2
S2
-0.0032
-0.1135
0.1135
181.6
18.3
N2
-0.0050
-0.0079
0.0093
212.6
32.4
(c) Station B1
K1
2.4623
-11.2900
11.5554
167.7
0.3
O1
1.6738
-14.6383
14.7337
173.5
1.3
Q1
0.0506
-0.6735
0.6754
175.7
1.7
M2
-0.0752
0.4329
0.4394
350.1
-7.3
S2
0.0098
-0.3511
0.3512
178.4
-9.6
N2
-0.0115
0.0289
0.0311
338.3
-22.4
(d) Station B2
K1
4.7790
-4.2226
6.3772
131.5
12.2
O1
5.6926
-5.3376
7.8035
133.2
4.7
Q1
0.2630
-0.2354
0.3530
131.8
2.3
M2
0.0157
-0.0282
0.0323
150.8
22.9
S2
-0.1103
-0.0120
0.1109
263.8
-1.5
N2
-0.0089
0.0065
0.0110
305.9
0.4
(e) Station B3
K1
4.0403
-6.1473
7.3562
146.7
1.2
O1
4.4794
-7.0172
8.3251
147.4
0.5
Q1
0.3395
-0.5330
0.6319
147.5
-0.2
M2
0.0966
-0.1394
0.1696
145.3
0.8
S2
-0.0328
-0.0787
0.0853
202.6
86.5
N2
0.0062
-0.0084
0.0104
143.7
-0.9
Fx – east component of energy flux density; Fy – north component of energy
flux density; F – magnitude of energy flux density; θ – direction of
energy flux density, measured clockwise from north; Δθ –
direction of energy flux density, measured clockwise from the major axis of
the current ellipse (=θ-λ).
From Table 5 and Fig. 4, it is found that for diurnal tides, the tidal
energy flows from the SCS to the JS at all stations. Maximum energy flux
densities of 11.6 (for K1) and 14.7 (for O1) kW m-1 appear
at Station B1 in the Gaspar Strait. On the other hand, the tidal energy flux
for M2 tide is quite small and flows to the JS only in the eastern
passage of the study area, including the Karimata Strait. In the western
passage, including the Gaspar Strait, the M2 tidal energy flows
oppositely from the JS to the SCS. However, for S2 tide, the tidal energy
flux flows from SCS to JS at all stations except B2. In the Indonesian seas,
the magnitudes of tidal energy densities may exceed 100 kW m-1 (Ray et
al., 2005; Teng et al., 2013), thus the energy fluxes in the study area are
relatively small. Table 5 shows that direction differences between energy
flux and current major axis are generally small. From directions of energy
fluxes shown in Fig. 4 we can judge that (1) the southward incident diurnal
and S2 waves from the SCS are slightly stronger than the northward
incident diurnal and S2 waves from the JS and (2) the southward
incident M2 wave from the SCS is slightly stronger than the northward
incident M2 wave from the JS in the eastern passage, and is slightly
weaker than the latter in the western passage. The feature (2) further
indicates that the M2 amphidromic point should be located between the
A1 and B1 line and the A2 and B2 line, and the amphidromic system should rotate
clockwise.
Tidal elevation gradients
Based on the tidal currents, the gradients of sea surface height can be
derived from the shallow water equations, as done by Proudman and Doodson (1924).
The equations in the x (positive for eastward) and y (positive
for northward) directions are respectively:
∂u∂t=fv-ga-a‾-p∂v∂t=-fu-gb-b‾-q,
where f is the Coriolis parameter, and a=∂ζ∂x, a‾=∂ζ‾∂x,
b=∂ζ∂y, b‾=∂ζ‾∂y are elevation gradients of tides and
equilibrium tides, respectively. The vector of (a,b) is
called the tidal elevation gradient vector. The equilibrium tide
ζ‾ has been adjusted for the earth's elastic response,
and is equal to (see e.g., Fang et al., 1999)
ζ‾=Csin2ϕcos(ωt+λ)for diurnal tides(CK1=0.104m,CO1=0.070m)ζ‾=Ccos2ϕcos(ωt+2λ)for semidiurnal tides(CM2=0.168m,CS2=0.078m),
where λ and ϕ are longitude and latitude, respectively. In
the Eqs. (2) and (3), p and q represent the east and north components of
bottom friction:
p=1hCDu2+v21/2uq=1hCDu2+v21/2v,
where CD is the drag coefficient and is taken to be 0.0025 in this
study. The values of p and q can be obtained by inserting the
observed values of u and v into Eqs. (5) and (6), respectively, and can
be decomposed into various constituents with frequencies equal to
corresponding tidal constituents through harmonic analysis (similar to the
analysis of u and v). The amplitudes and phase lags of the obtained
constituents of p (q) are denoted as P and μ (Q and ν),
respectively.
The tidal elevation gradient ellipses of K1, O1, M2,
and S2 at the observational stations.
For a given constituent with angular speed equal to ω, we have
u=Ucos(ωt-ξ)=U′cosωt+U′′sinωtv=Vcos(ωt-η)=V′cosωt+V′′sinωtζ=Hcos(ωt-G)=H′cosωt+H′′sinωtζ‾=H‾cos(ωt-G‾)=H′‾cosωt+H′′‾sinωtp=Pcos(ωt-μ)=P′cosωt+P′′sinωtq=Qcos(ωt-ν)=Q′cosωt+Q′′sinωt,
where U′=Ucosξ, U′′=Usinξ (the rest are similar).
Inserting Eq. (7) into Eqs. (2) and (3) yields
A′=A‾′+(-ωU′′+fV′-P′)/gA′′=A‾′′+(ωU′+fV′′-P′′)/g
and
B′=B‾′+(-ωV′′-fU′-Q′)/gB′′=B‾′′+(ωV′-fU′′-Q′′)/g,
where A′,A′′,A‾′,A‾′′=∂∂xH′,H′′,H‾′,H‾′′, and B′,B′′,B‾′,B‾′′=∂∂yH′,H′′,H‾′,H‾′′. The elevation
gradients of equilibrium tides
(A‾′,A‾′′,B‾′,B‾′′) can be obtained
from Eq. (4). By inserting A‾′,A‾′′,B‾′,B‾′′ into Eqs. (8)
and (9), we can get the values of the tidal elevation gradients A′,A′′;B′,B′′. The tidal elevation gradient ellipse
parameters can be obtained from the values of A′,A′′;B′,B′′ in the same way as the calculation of tidal
current ellipse parameters from the values of U′,U′′,V′,V′′. The tidal elevation gradient ellipse has a close relationship to
the tidal regime, that is, the distribution of co-amplitude and co-phase-lag
contours (see Appendix B for detailed derivation). In particular, if the
tidal elevation gradient ellipse rotates counterclockwise (clockwise) the
angle from the vector grad H to the vector grad G on the cotidal chart lies
between 0 and 180∘ (0 and -180∘).
Figure 5 shows the tidal elevation gradient ellipses of K1, O1,
M2 and S2 at the observation stations. For K1 and O1
tides, the tidal elevation gradient vectors rotate counterclockwise at
Stations A1, A2, and B2, and rotate clockwise at Stations B1 and B3. For
M2 tide, the tidal elevation gradient vectors rotate counterclockwise
at Stations A1 and A2, and rotate clockwise at Stations B1, B2, and B3.
However, for S2 tide, the tidal elevation gradient vectors rotate
counterclockwise at Stations A1 and B1, and rotate clockwise at Stations A2,
B2, and B3.
From the known tidal elevation gradient we have calculated the directions of
the co-tidal and co-amplitude lines as done by Proudman and Doodson (1924)
in constructing co-tidal charts of the North Sea. Since the purpose of the
present work is not to construct co-tidal charts in the study area, the
obtained results are not shown here.
Summary and discussion
The sea level and current data obtained at five stations along three
sections between the SCS and JS are analyzed to reveal the characteristics
of tides and tidal currents in this region. The results show that the ratios
of diurnal vs. semidiurnal tides amplitudes
(HO1+HK1) / (HM2+HS2) are greater than 8 at all
stations, suggesting predominance of the diurnal tides in the study area.
The amplitudes of K1 are larger than 50 cm at all stations with the
phase lags being around 30∘. In comparison, the amplitudes of
M2 are smaller than 5 cm. It is worth mentioning that the amplitudes of
S2 may exceed M2 in the Karimata and Gaspar straits. The greater
amplitudes and smaller spatial phase-lag changes of diurnal tides compared
with those of semidiurnal tides indicate that the study area is located in
the antinodal band of the diurnal tidal waves but in the nodal band of the
semidiurnal tidal waves.
The tidal currents are analyzed based on the ADCP observations on board of
five TRBMs, showing that the tidal currents are of rectilinear type at all
stations. The semi-major axes lengths of tidal current ellipses are about
10 cm s-1 for diurnal tides, with O1 slightly smaller than K1 in
the most cases. However, in the Gaspar Strait, O1 may exceed K1 and
approaches to 20 cm s-1 at Station B1. The maximum speeds of semidiurnal
constituents M2 and S2 are generally smaller than 5 cm s-1
at all stations.
By examining the tidal energy fluxes at each station, we found that the
diurnal tidal energy flows from the SCS to the JS with the maximum energy
flux density of 14.7 kW m-1 appearing at Station B1. The tidal energy
flux distributions of semidiurnal tides are quite complicated; M2
energy flux flows southward in the Karimata Strait but northward in the
Gaspar Strait and S2 energy flux generally flows from the SCS to JS except
at Station B2.
With these long-term observational results, we can make an accuracy
assessment on the existing tidal models for the study area. Four
representative tidal models, TPXO7.2 (Egbert and Erofeeva, 2002;
0.25∘ × 0.25∘ resolution), GOT00.2 (Ray, 1999;
0.5∘ × 0.5∘), NAO.99b (Matsumoto et al., 2000;
0.5∘ × 0.5∘), and DTU10 (Cheng and Andersen,
2011; 0.125∘ × 0.125∘), are compared with our
observations for tides (see Appendix A). The comparison shows that the
amplitudes and phase lags of the model results are generally consistent with
the observations. However, discrepancies of the model results from the
observations are not ignorable. DTU10 is the best one in the area between
the South China Sea and Java Sea, due to use of more satellites and longer
altimeter measurements. Moreover, DTU10 has the highest resolution among
these four tide models. It indicates that if the open boundary of a tidal
model is located in the area between the South China Sea and Java Sea, DTU10
is the best choice for deriving open boundary condition. The tidal currents
of the model TPXO7.2 are also compared with observations in Appendix A (the
models GOT00.2, NAO.99b, and DTU10 do not contain tidal currents). The
comparison shows that the relative discrepancies are generally greater than
those for tidal elevations. Therefore, further effort of assimilating the in
situ observations into numerical models in the future is worthwhile in
providing more accurate knowledge of the tidal systems in the study area.
Since the study area is often chosen as an open boundary in simulating tides
in the SCS or Indonesian seas (e. g., Fang et al., 1999; Gao et al., 2015)
the observational results of this study are expected to be useful in
improving model results.