Yasunori Totsuka1, Hitoshi Tanaka2, Yutaka Fujita3,
Hiroto Yamaji4, Manabu Takuwa5 and Masaki Sawamoto2
1Graduate Student, Department of Civil Engineering, Tohoku University, Japan
2Professor, Department of Civil Engineering, Tohoku University, Japan
3Associate professor, Department of Civil Engineering, Nihon University, Japan
4Laboratory Assistant, Department of Civil Engineering, Tohoku University., Japan
5Undergraduate Student, Department of Civil Engineering, Tohoku University, Japan
Aoba-yama 06, Sendai 980-8579 JAPAN
Phone: +81-22-217-7515, Fax: +81-22-217-7516
E-mail: totsuka@kaigan.civil.tohoku.ac.jp
Abstract: Lake Inawashiro is a typical deep acid lake in Japan. The water from Nagase river is the cause of acid water in this lake. Deep lakes have a possibility of occurring internal seiche, which is very important in terms of not only hydraulics but also water quality, because it contributes to the water purification and mixture.
Field observation was carried out in autumn season when the thermocline breaks down gradually. From the result of water temperature observation, the existence of internal seiche was confirmed. The wave height and difference of water temperature attain to about 15m and 8 degree, respectively and internal seiche continued about 6 days. After this event the thermocline was disappeared. It thus suggests that the internal seiche mixed the lake water all over the depth.
To examine three dimensional interface response to wind stress, two layer model using a constant depth was applied. The nonlinear term and the interface stress are neglected to calculate this model. The input external force is only wind stress, which is obtained at AMeDAS (Automated Meteorological Data Acquisition System, Japan Meteorological Agency) station located about 3km north from Lake Inawashiro. From the result of calculation, computation and observation shows fairly good agreement, and the existence of internal Kelvin wave was confirmed in Lake Inawashiro. However there are some differences between observed and calculated. One of the reasons about it, the difference of actual wind and AMeDAS data is regarded.
Keywords: stratification, thermocline, internal seiche, two layer model, Colioris force, internal Kelvin wave
The thermocline is formed at the deep temperate lake during summer according to the rise of temperature, and the flow shows different pattern from the shallow lake because of stratification. Especially, the probability of occurring big internal wave becomes high in autumn because of decay of stratification and the effect of seasonal wind in Japan. Muraoka and Hirata [1][2] and Hirata and Muraoka[3] observed internal wave in Lake Chuzenji in Japan. When the upper layer attains to a certain thickness, the wind destroys only isotherm near the surface even if strong wind continues for a long time, and the most of energy by wind changed to the potential energy caused by slope of interface. If the erosion effect for thermocline caused by wind force is small, the response of internal wave plays an important role for water mixture. However, internal wave effects in only limited area. Compare with internal wave, internal seiche introduces large oscillation all over the lake and has a effect of change the water quality largely. Therefore, it is very important to make clear the characteristics of internal seiche and its effect to lake water movement and water quality. In Lake Chuzenji, not only internal wave but also internal seiche was observed. However, the examples of observation are very scarce yet.
In the present study, Lake Inawashiro which has one of the greatest area and depth in Japan was paid attention. The observation of water temperature was carried out at this lake for grasping the internal structure in autumn season. In addition to the observation, a two layer model was applied for examining the three dimensional interface movement and the effect of Colioris force.
Lake Inawashiro located at the center of Fukushima Prefecture in Japan (Figure 1) has the fourth largest area. The maximum depth attains to 94.6m, therefore the noticeable thermocline is formed in summer. This lake is quite closed because the most of inflow depends on Nagase river and residence time is about 5.4 years.
The oscillation of internal seiche was expected predominating in the direction of long axis. Therefore, the water temperature gauges were set at the three points shown in Figure 2. The vertical distance between the gauge is 5m from the surface to the depth of 40m and below 40m the distance is 10m. Time interval of the temperature measurement is 10 minutes because the period of internal seiche is expected more than 10 hours, and observation was carried out in October 1999.
Fig.1 Location of Lake Inawashiro
Fig.2 Observation stations of water temperature
Figure 3 shows wind vectors and
water temperature variation from October 21st to 30th in 1999. The AMeDAS data
is used for wind data. AMeDAS observing station locates about 3km north from
Lake Inawashiro, and time interval of observation is 1 hour. From the result of
water temperature observation, both St.1 and St.3 isotherms vary with antiphase.
On the other hand, the temperature at St.2 hardly changes compare to St.1 and
St.3. This measured data implies that the internal seiche occured in Lake
Inawashiro during this period, and it suggests that St.2 represents the node of
internal seiche. During this season, the temperature of upper layer reduces
gradually with decrease of air temperature, therefore the difference of water
temperature between upper layer and lower layer becomes small, which caused
such a big internal seiche under the normal wind condition. On October 27th the
wind speed attains to 14m/s because of a typhoon,
resulting in a large amount of water discharge into the lake. At St.1 the
bottom water rose up to the surface. This internal seiche was so big that the
change of water temperature attained about 8 degree.
The isotherm of 13 degree, where the gradient of water temperature is the steepest, is assumed to be thermocline, and the oscillation period is calculated by the analysis of Fourier power spectra. Figure 4 shows the power spectrum, in which remarkable peaks appears at the period of 20 hours. Meanwhile, theoretical period of internal seiche in the square lake with constant depth can be expressed by Eq.(1).
(1)
where T: period, ci: celerity of wave expressed by Eq.(2),
(2)
![文本框:
Fig. 4 Power spectrum of interface elevation
Table 1?Parameters in Lake Inawashiro
and calculated period
h1 [m] h2 [m] T [hr]
St.1 0.000831 22 43 19.4
St.3 0.000964 18 52 18.7](images/A%20STUD11.gif)
where m,n: oscillation mode, a,b:
the length and width of a lake respectively,
: the difference of relative density, g: the acceleration of gravity, h1,h2: the thickness of upper and
lower layer, respectively. Table 1 shows each parameters in Lake Inawashiro and
the result of calculation. The period of observation and calculation show good
agreement. Therefore, it can be conclude that this oscillation is internal
seiche of the basic mode in this lake.
After this internal
seiche, the interval of isotherms extended and thermocline was disappeared. It
means that internal seiche contributed to the mixture of the lake water.
From the result of observation, the existence of internal seiche in the direction of long axis was confirmed. However, this result is only two dimensional one, and it is therefore necessary to grasp three dimensional behavior for discussing about flow structure in the lake. Consequently two layer model using constant depth was applied for Lake Inawashiro. Kanari [4] calculated the response of two layer model Lake Biwa to a suddenly imposed wind. However its result was not compared with actual data of interface elevation. Therefore, this model was applied to Lake Inawashiro and compared with observation results.
Governing equations are equation of motion and continuity. Definition of the variables are illustrated in Figure 5. For calculating this model, the nonlinear terms of elevation and interface stress are neglected. Lake Inawashiro is divided into 400m interval square meshes with constant depth of 60m and thermocline is set at the depth of 25m according to the observation results. For initial condition, it is assumed that there is no motion in the computation domain.
In the case of application of a two layer model, the selection of water temperature at each layer causes large effects to calculation results. In the present study, observation results are used for verification of calculation. According to Figure 3, the water temperature in the upper layer at St.1 is almost constant about 15 degree, on the other hand, in lower layer the water temperature decreases gradually with increase of depth. The averaged water temperature in lower layer is calculated and its value is about 6 degree. The calculation result shows in Figure 6, and for comparison, the other cases with different temperature in lower layer were calculated and shown in Figure 6. Generally, it is thought that the amplitude of interface elevation becomes higher corresponding to the decrease of the difference of relative density. In fact the amplitude of interface elevation becomes higher as shown in Figure 7 which describes the change of interface by each water temperature in lower layer with constant wind strength. From Figure 6, however, the amplitude of interface is minimum at 10 degree in lower layer and is maximum at 8 or 6 degree. Comparing with observed results, the case of 6 degree in lower layer shows best agreement. From this result, in the deep lake there is a thick low temperature layer under the thermocline, so this low temperature layer may be quite dominant to the change of interface elevation. And it seems that the change of amplitude is not necessarily determined corresponding to the relative density if wind magnitude and its directions are not constant. Again comparing observation and calculated result at 6 degree in lower layer, the amplitude and period shows good agreement. The period of oscillation is calculated about 23.3 hours by power spectrum. This value is a little larger than observed. It is suggested that some assumption shown following causes these errors: (1) difference between actual wind at lake and AMeDAS one, (2) assumption of constant depth and (3) effect of nonlinear terms which was neglected in the present computation.
After determining the water temperature at each layer, the fundamental response of interface to the wind on this model will be investigated. For meteorological conditions, constant NW5m/s wind, which is predominant in autumn, is imposed and stops after 12 hours. Figure 8 shows the interface configuration at the time interval of 6 hours after wind stops. In this figure, “+” and “-” shows the area of water level higher and lower compared with hydrostatic condition, respectively. Just after the wind stops, the interface tilts to the up wind direction caused by the balance of hydrostatic pressure. Then the spatial pattern rotates in the anticlockwise direction, with the rotating period of about 25 hours. The Kelvin wave propagation velocity is calculated by Eq.(2), and its value is 0.35m/s. The equivalent radius of this lake is estimated to be 5.8km. The rotating period of Kelvin wave is expressed by Eq.(3).
(3)
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Fig. 6 Results of calculation at each water
temperature in lower layer
Time[hr]
Fig. 7 Response of interface for NW5m/s
wind at St.1 with each temperature](images/A%20STUD15.gif)
where Tk: the rotating period of Kelvin wave, r: the equivalent radius of lake. Consequently, the rotating period of Kelvin wave will be 28.9 hours. Though there are some differences between theoretical value and calculation, this rotation seen in Figure 8 may be Kelvin wave, because if Colioris parameter is neglected in this model, such rotation does not occur. Therefore, in the results shown in Figure 8, Colioris force effect is predominant and it should be included in the computation for reproducing rotational behavior in the lake and we could not discuss about flow structure without it in this lake.
In this study, the
observation of water temperature in Lake Inawashiro was carried out. It is found
that the big internal seiche occurs in autumn when the stratification decays,
and the amplitude attains about 15m. After this event, the thermocline was
disappeared. It means that internal seiche contributed to the water mixture. In
addition, two layer model was applied to grasp the interface configuration. From
the result of calculation, the existence of
Kelvin wave in Lake Inawashiro was confirmed, and it shows that Colioris
force is predominant in this lake.
Acknowledgements
The authors would
like to express their grateful thanks to the members of Hydraulic Engineering
Laboratory, Tohoku University, for their sincere cooperation during the
observations.
[1] Muraoka, S. and Hirata, K.: Internal waves in Lake Chuzenji, Annual Journal of Hydraulic Engineering, JSCE, vol.27, (1983), pp.179-184. (in Japanese).
[2] Muraoka, S. and Hirata, K.: Internal waves in Lake Chuzenji (2), Annual Journal of Hydraulic Engineering, JSCE, vol.28, (1984), pp.327-332. (in Japanese).
[3] Hirata, K. and Muraoka, S.: Temperature variation in nearshore of Lake Chuzenji, Annual Journal of Hydraulic Engineering, JSCE, vol.29, (1985), pp.377-382. (in Japanese).
[4] Kanari, S.: Internal waves in Lake Biwa (2), Bulletin of Disaster Prevention Research Institute, Kyoto University, vol.22, part2, No.202, (1973), pp.69-96.