Numerical Simulation of Density Currents over a Slope under the Condition of Cooling Period in Lake Biwa

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1 Nmerical Simlation of Densit Crrents oer a Slope nder the Condition of Cooling Period in Lake Bia Takashi Hosoda Professor, Department of Urban Management, Koto Uniersit, C1-3-65, Kotodai-Katsra, Nishiko-k, Koto , hosoda.takashi.4@koto-.ac.jp Frederick P. Malembeka, Lectrer, Department of Ciil Engineering, Arsha Technical College, fmalembeka@ahoo.com ABSTRACT: This paper describes the densit crrents ith thermal conection oer a slope nder the condition of cooling period in the northern part of Lake Bia, hich is considered as one of the main mechanisms of ertical miing in the ater colmn of the Lake. 3-D simlations of the flo indced b thermal conection are condcted nder the to-laered stratified ater condition to std the effect of a slope representing the shore part of the Lake. It is pointed ot throgh the eamination of simlated reslts that densit crrents are generated oer a slope attached to a rectanglar solid de to the rapid cooling near the ater srface, bt are not strong enogh to intrde nder the loer laer ith lo temperatre. KEY WORDS: Lake hdrodnamics, Densit crrents, Thermal conection, Etrophication 1 INTRODUCTION Lake Bia is the largest monomictic lake in Japan, haing a total srface area of abot 670 sqare km and 104m in maimm depth. Its role as sorce of ater is paramont in spporting the ciiliation in nearb prefectres. Alarming deterioration of ater qalit is a main concern, ith climate change effects sch as increased air temperatre being noted to be linked ith ccles of arm inters hich are related to lo leels of dissoled ogen (DO) arond the bottom laer of the Lake. Under these circmstances, a lot of researches hae been done b man researchers to clarif the fndamental mechanism of mass and heat transfer and to propose effectie contermeasres (Kmagai et al., 003). In this std, e deal ith the densit crrents ith thermal conection oer a slope representing the shore part of lake nder the condition of cooling period in the northern part of Lake Bia, becase thermal conection is considered as one of the main mechanisms of ertical miing in Lake Bia. As shon in Figre 1, the cooling period starts in September creating the pper laer ith constant ater temperatre de to thermal conection, and ends in the beginning of March. When the interface beteen the pper and loer laer, so-called thermo-cline, reaches the lake bottom, DO near the bottom increases discontinosl as shon in Figre. The large temporal ariation of temperatre ith small oscillations is also detected jst before the thermo-cline reaches the bottom. In this std, e focs on the generation mechanism of large temporal change of temperatre detected jst before the interface reaches the bottom. Althogh preios stdies indicated that the basic featres of temporal change and ertical profiles of ater qalit indices mentioned aboe can be reprodced b sing 1-D simlation model (Hosoda et al., 004) and 3-D simlation of thermal conection ith a rectanglar solid (Hosoda, 007), the large temporal ariation of temperatre shon in Figre is not reprodced et. So, nmerical simlation of thermal conection is condcted b sing a

2 rectanglar solid tank ith a slope to erif the effects of the shore area on temperatre ariations near the bottom. APPLICATION OF NUMERICAL MODEL Flo mechanism describing lake ndergoing thermal cooling is simlated considering conseration las of mass, momentm and heat (Eqation 1, and 3 respectiel). = 0 (1) ) ( 1 ν ρ = p t (a) ) ( 1 ν ρ = p t (b) ) ( 1 p g t = ν ρ (c) Figre 1 Monthl change of ertical temperatre distribtions in 1998 b Shiga Prefectre Figre Temporal ariations of DO and Water Temperatre obsered 1 m aboe the bottom of the northern part of Lake Bia beteen Dec. 001 and Mar. 00 b the former Research Institte of Lake Bia, Shiga Prefectre Temp. DO (deg.) (mg/l)

3 Figre 3 Plan-ie (top) and side-ie (bottom) of the comptational domain ith initial temperatre conditions T T T T T T T = λ (3) t Folloing the geometrical shape of the lake arond shores, an idealied comptation domain assmes shape of a Right Trapeoidal Prism. To replicate the thermal laers in the ater colmn of December, niform temperatre is assigned as shon in Figre 3. The figre shos the plan and side-ies of the comptational domain as ell as the initial temperatre conditions. L assmes ales of 1 km and 4 km, making to corresponding simlation cases ith slopes 8 % and %, respectiel. Temperatre in the pper laer ( 0 m 80 m ) as kept at 9 deg. hile in the loer thermal laer (0 m < 0 m ) 7 deg. as assmed. Total nmber of cells in to of the spatial dimensions is maintained at 40. These are aes Y-Y and -. Nmber of cells in the third spatial dimension decreases from 40 to 0 (X-X ais). A dark spot in the plan ie (Figre 3) shos the cold ater cells inclded to instigate the flo. 4 Water srface is assmed to ndergo cooling at a constant rate of cal/cm /sec dring the entire simlation time. The eqations ere soled nmericall sing finite olme method ith H-SMAC algorithm for pressre iteration method and QUICK scheme as a finite difference scheme for conectie terms. 3 FLOW MECHANISM In general, the reslts sho that the model as capable of reprodcing some basic featres of 3

4 Vel. (m/s) Vel. (m/s) Vel. (m/s) Vel. (m/s) (a) after 186 hrs (top: 8% slope, bottom: % slope) (b) after 399 hrs (top: 8% slope, bottom: % slope) Figre 4 Side ies of ater temperatre and flo field indced b thermal conection and densit crrents (a) 8% slope case (b) % slope case Figre 5 Plan ie shoing simlated flo from simlation ith 8 % and % slope. Data are taken from cells adjacent to lake s bottom or slope. 4

5 =1, X=7.3(km) =60, X=7.3(km) =1, X=6.7(km) =60, X=6.7(km) (a) time series obsered in the pper laer (b) time series obsered near the bottom Figre 6 Simlated temperatre time-series from simlation ith % slope flo field in the lake dring cooling period. Figre 4 shos the side ie from simlation cases ith % and 8% slopes after 186 hors and 399 hors. The figre shos that as the time progresses the entire ater colmn loses heat energ as the lake is cooled on its srface at constant rate, and thermo-cline moes donards toards the bottom, becase cooled ater particles near the icinit of srface become denser, sink donards and in trn the hotter and less dense particles replace them. The figre also shos that the densit crrents are generated on a slope de to the rapid cooling of ater particle near the srface of a slope. The densit crrents ith % slope seem to be stronger than the crrents ith % slope, and go donard nder ater colmn ith higher temperatre generating internal aes near the bottom. It can be pointed ot that the internal aes are cased b the flo indced b the densit crrents on the slope as ell as the impact of falling cooled ater bodies as the reslts of thermal conection. Figre 5 shos plan ies of the simlated flo from the simlation case ith 8 % slope. The data ere taken from a laer of cells adjacent to the lake s bed. Dring the earl stages of the simlation, temperatres at the slopes ere recorded to be higher than the remaining portion of the comptational domain. After a period of time the temperatre conditions ere reersed ith those at the slopes being loer than other parts of the domain, indicating that the ater colmn as losing heat from the srface don to the bottom ith the loering of thermo-cline. Similar reslts can be seen in the reslts ith % slope. Simlated temperatre time-series data ere sampled at different depths. The oerall ariation shos the decrease in temperatre as simlation time progresses. In general, the simlated temperatre as recorded to be sstematicall decreasing at the pper thermal laer indicating the oerall temperatre decrease as shon in Figre 6(a). The simlated temperatre as recorded to be flctating at different magnitdes hen sampled from points at the loer thermal laer. Figre 6(b) shos some simlated data sampled at seeral points at the location near the bottom. It can be pointed ot that these flctations are generated b the internal aes mentioned aboe. Althogh the time series in Figre 6(b) sho almost the same featres as the time series in Figre, the large temporal ariation of temperatre cannot be detected in Figre 6(b) before the interface reaches the bottom. This reslt indicates that another mechanism sch as rotating internal aes (Akitomo et al., 004) shold be considered to eplain the 5

6 featre of obsered time series. 4 CONCLUSIONS In order to inestigate the effect of the shore part of Lake Bia on the flo field dring the cooling period e shoed the simlation reslts of thermal conection sing a rectanglar solid ith a slope representing the shore of the lake. The simlation reslts indicate that the densit crrents are generated de to rapid cooling near the srface on a slope, and the internal aes near the bottom are cased b the flo indced b the densit crrents on the slope as ell as the impact of falling cooled ater bodies as the reslts of thermal conection. Bt, the large temporal ariation of temperatre obsered cannot be detected in the simlated time series before the interface reaches the bottom. This reslt indicates that another mechanism sch as rotating internal aes shold be considered. References Akitomo, K., Krogi, M., and Kmagai, M., 004, Nmerical std of a thermall indced gre sstem in Lake Bia, Limnolog, Vol.004(5), Kmagai, M. and Vincent, W.F., 003, Freshater Management, Springer, Toko, 1-. Hosoda, T., 007, Nmerical eperiments on thermal conection dring cooling period in the northern part of Lake Bia, CD-Proc. of Fifth Int. Smposim on Enironmental Hdralics, Tempe, Ariona. Hosoda, T. and Hosomi, T., 004, A simplified model for long term prediction on ertical distribtion of ater qalities in Lake Bia, Sstainable Deelopment of Energ, Water and Enironmental Sstems, Afgan, Bogdan & Dic(eds), Sets & eitlinger, Lisse,