PROPAGATION OF LONG-PERIOD WAVES
INTO AN ESTUARY THROUGH A NARROW INLET
Takumi Okabe, Shin-ichi Aoki and Shigeru Kato
Department of Civil Engineering
Toyohashi University of Technology
Toyohashi, Aichi, JAPAN
[email protected]
ABSTRACT
In order to estimate sediment exchange between an estuary and the open sea through a narrow
inlet, it may be important to investigate the behavior of the long-period waves with periods of
several minutes generated by wave groups in the sea. In this study, characteristics of the longperiod waves and associated flows outside and inside an inlet is discussed based on the field
data. The results show that long-period waves are developed almost linearly to both significant
wave heights and periods outside the inlet, but they are reduced about half in the inlet. The longperiod fluctuation in the flow velocity seems to consist of two components; one is associated
with the long-period waves and the other is generated by tidal currents.
INTRODUCTION
In the vicinity of an inlet, tidal and wave actions and their interactions yield complex current
fields. Hence the sediment transport and associated topographic change around an inlet are very
complicated and little is known about them. For example, sediment exchange between an estuary
and the sea through an inlet and influence of tidal currents on longshore sediment transport are
difficult to estimate. External forces that induce sediment transport at around the inlet have wide
range of periods from those of wind waves (seconds) to tides (hours). However, predominant
factor or relative importance between the forces with different time scale has not been well
investigated. In this paper, we focus on long-period components having periods of several
minutes and investigate their characteristics based on the field data.
Although previous research (eg. Sato, 2002) have shown that long-period waves with periods of
several minutes can influence sediment transport, studies on long-period waves and associated
flows near an inlet are very limited. It remains unclear either qualitatively or quantitatively
whether long-period waves influence on sediment transport near the inlet. Detailed investigations
of behavior of the long-period waves generated by wave groups in the sea may be necessary in
order to estimate influences of the long-period waves on sediment transport near the inlet.
152
In this study, characteristics of the long-period waves and associated flows inside and outside the
inlet channel where tidal current is predominant are discussed based on the field data. We focus
on wave components with periods larger than 30s. We discuss the relations between long-period
waves and significant wave heights and periods, and propagation properties of the long-period
waves into the inlet. Generations of long-period fluctuations in the currents induced by tidal
action are also shown in the paper.
FIELD OBSERVATIONS
Field measurements were conducted at two stations outside and inside the inlet of Hamana Lake,
Japan, for about two months from August to November 2006.
Figure 1 and Photo 1 show the locations of the inlet and the observation stations. In the figure,
Stn-in denotes the station inside the inlet, where two-dimensional horizontal flow velocities and
water pressure were measured at the bottom. At the station Stn-out located outside the inlet, twodimensional bottom velocities, water pressure and water surface elevation were measured. All
the data were recorded continuously at sampling rate of 0.5s except for the data loss occurred in
the period of maintenance of the equipments. Mean water depths during the measurements were
5.6m at Stn-in and 4.8m at Stn-out, respectively. As Stn-out was sometimes in the surf zone due
to high waves (Photo 2), water surface elevation measured by the ultrasonic wave gauge could
not be used because of noisy signals under storm conditions. The details of the field
measurements are shown in Table 1.
Photo 1. Aerial view of Hamana Lake inlet
153
Hamana Lake
JAPAN
N
2m
Stn-in
4m
6m
2m
4m
Stn-out
Hamana Lake
6m
8m
10m
the Pacific Ocean
12m
(a) Location of Hamana Lake in Japan
(b) Inlet of Hamana Lake
Figure 1. Location of inlet of Hamana Lake and observation stations
Photo 2. Surf zone near Stn-out under storm conditions
Table 1. Summary of field measurement
Station
Mean Water depth
Equipment
Sampling Frequency
Stn-in
5.6(m)
WaveHunter-04
2(Hz)
Measurement Items
Water Pressure,
Two dimensional Flow
Velocity
Height of Sensors from sea bottom
0.7 to 0.9 (m)
154
Stn-out
4.8(m)
WaveHunter-04 Σ
2(Hz)
Water Surface
Elevation, Water
Pressure,
Two dimensional Flow
Velocity
0.7 to 0.9 (m)
CHARACTERISTICS OF LONG-PERIOD WAVES
In this study, long-period waves are classified into two categories that have different nature in
generation mechanism: wave-group-induced long-period waves and meteorologically-generated
long-period waves. Judging from the power spectra under high wave conditions, we defined the
frequency range of wave-group-induced waves as 30s – 300s. Wave components that have larger
periods than 300s are classified as meteorologically-generated long-period waves. These longperiod wave components were extracted from the raw wave data by the numerical filtering using
FFT.
Development of long-period waves
(cm)
L
RMS
η
H1/3 (m)
(cm)
L
RMS
Stn-out
H1/3
T1/3
15
10
5
30-300s
300s-
0.6
H1/3
0.4
T1/3
15
Stn-in
10
0.2
T1/3 (s)
η
3
2
1
0
16
12
8
4
0
T1/3 (s)
H1/3 (m)
Figure 2 shows time series of significant wave height H1/3, significant wave period T1/3, and root
mean square (RMS) value of long-period water surface elevation ηLRMS. In the period of
measurement, some low-pressures passed near the coast and high waves were observed. From
the comparison between the figures, the wave-group-induced long-period waves (30s – 300s)
clearly developed under high wave conditions. Although similar trend can be seen for the
meteorologically-generated long-period waves with periods over 300s, the increase in the
amplitude was smaller than the wave-group-induced long-period waves.
5
8
6
4
2
30-300s
300s-
06/09/11
06/10/02
06/10/23
Date
Figure 2. Time series of significant wave height, significant wave period and RMS value of longperiod waves at Stn-out and Stn-in
Figure 3 plots the RMS values of the long-period waves as functions of the product of the
significant wave heights and periods observed at Stn-out. The RMS values of the wave-group155
induced long-period waves (30s – 300s) increase almost linearly to H1/3T1/3 except for large
values, while those of the meteorologically-generated long-period waves seem (300s – ) to have
little relation with wave heights and wave periods of short waves. Aoki (2002) and Aoki et
al.(2002) show that the linear relationship is seen in the wave data obtained on the mild-slope
sandy coast with wide surf zone.
Propagation of long-period waves into an inlet
η RMS=0.0034H1/3T1/3
L
-4
η RMS=1.54*10 (H1/3T1/3)
L
Stn-out 30-300s
Stn-out 300s-
0.10
η
L
RMS
(m)
0.15
2
0.05
0.00
0
10
20
30
H1/3T1/3 (m*s)
40
50
Figure 3. Relationship between RMS values of long-period waves and products of significant
wave heights and periods at Stn-out
Figure 4 shows relationship between the RMS values of long-period waves observed inside and
outside the inlet, Stn-in and Stn-out, in which the broken lines indicate that the same values were
observed both inside and outside of the inlet. For the wave-group-induced long-period waves
(Fig. 4(a)), the RMS values in the inlet show almost half of those measured outside. On the other
hand, the meteorologically-generated long-period waves (Fig. 4(b)) were not reduced in the inlet
and even amplified for some data.
156
20
(cm) : Stn-in
3
15
10
RMS
2
L
η
η
L
RMS
(cm) : Stn-in
30-300s
5
1
over 300s
0
0
0
5
10
15
L
η RMS (cm) : Stn-out
20
0
(a) Wave-group-induced long waves
1
L
RMS
η
2
3
(cm) : Stn-out
(b) Meteorologically-generated long waves
Figure 4. Relationship between long-period waves inside and outside the inlet
The transmission of waves into an infinitely-long narrow channel with no reflection is
theoretically given by Mei (1989) as
B=
−2 A
[1 + ka + (2ika / π ) ln(2γ ka / π e)]
(1)
where A is complex amplitude of the incident waves from the ocean, B is complex amplitude of
waves in the narrow channel, k is wave number, γ  is the Euler’s constant (approximate value is
0.5772156) and 2a is width of the channel. The inlet of Hamana Lake can be assumed to be
narrow but there should be some reflection from the lake. The transmission coefficient is given
as |B/A|.
Figure 5 shows |B/A| as functions of ka. In the case of 5m water depth (h=5m) and 200m channel
width (a=100m), Eq.(1) gives the transmission coefficient 0.498 for the wave with period of 30s,
which corresponds to the point P-1 (Fig. 5). For the wave with period of 300s, the transmission
coefficient yields 1.472 as indicated by P-2. The bar charts show results estimated from field
data, which is calculated from the ratio of the energy spectra, for calm and high wave conditions,
Figures 5(a) and (b) respectively. Under calm wave conditions, observed transmission
coefficients show closer values to the theoretical ones while factors show smaller values under
high wave conditions. One of the reasons why the transmission becomes small may be that longperiod free waves generated near the surf zone tend to be reflected offshore in storm conditions.
157
Eq.(1) : Mei(1989)
Eq.(1) : Mei(1989)
1/2
10/1 05:00-06:10
H1/3=0.67m
1.2
P-1
0.8
1.6
0.4
1/2
( Iin / Iout )
9/18 12:30-13:40
H1/3=2.49m
P-2
1/2
1/2
| B/A | , ( Iin / Iout )
( Iin / Iout )
9/10 12:00-13:10
H1/3=0.77m
P-2
| B/A | , ( Iin / Iout )
1.6
1.2
9/24 00:20-01:30
H1/3=2.76m
0.8
P-1
0.4
0
1
2
3
4
0
1
ka
2
3
4
ka
(a) Calm wave condition
(b) High wave condition
Figure 5. Transmission coefficient in the inlet – comparison with the theory
LONG-PERIOD VELOCITY FLUCTUATION
10
5
10
4
10
3
10
2
10
1
10
0
2
S(f) (cm /s)
Figure 6 shows power spectra of the east-west-ward horizontal velocities measured at Stn-out
under calm and high wave conditions. Under high wave conditions, not only the energy in the
frequency range of wind waves (less than 30s in wave period), but also low-frequency range,
especially between 30s and 300s in the period, were increased.
10
-1
10
-2
10
-3
300s
30s
Stn-out: VE
09/24 00:20-01:30 H1/3=3.4m
10/25 20:10-21:20 H1/3=0.4m
0.001
0.01
F (Hz)
0.1
1
Figure 6. Power spectra of bottom velocities for calm and high wave conditions
158
In Figure 7, the RMS values of long-period north-south-ward velocities observed at Stn-out are
plotted as functions of mean current velocities obtained by averaging the velocity data over 70
minutes. Figures 7(a) and (b) correspond to frequency ranges 30s – 300s and 300s – 2100s,
respectively. The size of the circle markers represents the magnitude of RMS value of longperiod waves discussed above. Two data groups are indicated in Figure 7(a): G-1 and G-2. The
group G-1 is comprised of small RMS values of long-period waves and the long period velocity
components seem to be generated by tidal currents because the RMS values of long-period
velocity increase as the mean velocities increase. On the other hand, the group G-2 consists of
large RMS values of long-period waves and shows no correlation with mean velocities. This
shows that there seem two different sources for the generation of long-period velocity
fluctuations around an inlet with strong tidal current.
Stn-out (N-S)
Mean:70min-data
RMS: 30s-300s
L
18
η RMS = 10 (cm)
16
G-2
12
10
VN
L
RMS
(cm/s)
14
8
6
4
2
G-1
-20
-10
0
VN - MEAN (cm/s)
10
(a) Velocity components in the range between 30s to 300s
Stn-out (N-S)
Mean:70min-data
RMS: over 300s-data
10
L
η RMS = 1.5 (cm)
6
VN
L
RMS
(cm/s)
8
4
2
-20
-10
0
VN - MEAN (cm/s)
10
(b) Velocity components in the range between 300s to 2100s
Figure 7. Relationship between mean velocities and RMS values of long-period velocities at Stn-out
159
Figure 8 shows an example of time series of the RMS value of the long-period velocity
comparing with tidal fluctuation of water surface at Stn-out. Increase in the long-period velocity
appears at the phase of ebb tide indicated by ellipses in the figure, which implies that the strong
tidal current has some influence on the generation of long-period velocities under calm wave
conditions.
Tide Level
L
VN RMS : 30-300s
Stn-out
50
2.0
1.5
VN
L
RMS
0
1.0
(cm/s)
Tide Level (cm)
100
-50
0.5
-100
06/11/04
06/11/05
Date
06/11/06
06/11/07
Figure 8. Time series of tide level and RMS value of the long-period velocity at Stn-out
CONCLUSION
The conclusions obtained in the study are summarized as follows:
1)
Outside the inlet, the RMS values of the wave-group-induced long-period waves
(30s – 300s) are increased almost proportionally to the products of the significant
wave heights and periods.
2)
The RMS values of wave-group-induced long-period waves are reduced half in the
inlet. The transmission coefficients of the long-period waves into the inlet show good
agreement with the theory under calm wave condition, while show smaller values
under high wave conditions.
3)
The long-period velocity components are increased corresponding to the development
of long-period waves. Under calm wave conditions long-period velocity fluctuations
seem to be caused by strong tidal currents.
160
ACKNOWLEDGMENTS
This study was carried out as a part of research project “Dynamic sediment management and
coastal disaster prevention by advanced technologies” supported by the Special Coordination
Funds for promoting Science and Technology of Ministry of Education, Culture, Sports, Science
and Technology. The authors are grateful to the financial support and collaboration with the
project team.
REFERENCES
Aoki, S. 2002. Generation and Propagation of Coastal Long Waves. Proceedings of Civil
Engineering in the Ocean VOL.18, pp.155-160. (in Japanese)
Aoki, S., T. Okabe and I. Deguchi. 2002. Characteristics of Long Wave Generation and
Secondary Oscillation in a Harbor:Comparison between Two Coasts with Different Wave
Conditions. Proceedings of Coastal Engineering, JSCE, Vol.49, pp. 231-235. (in Japanese)
Mei, C.C. 1989. The Applied Dynamics of Ocean Surface Waves. Singapore:World Scientific
Publishing. pp. 199-202.
Sato, S. 2002. Interaction between Infragravity Waves and Coastal Sedimentary Processes.
Proceedings of Civil Engineering in the Ocean VOL.18, pp.161-166. (in Japanese)
161