Parameter identification and numerical simulation for the exchange coefficient of dissolved oxygen concentration under ice in a boreal lake

Qinxi Bai; Zhijun Li; Lei Wang; Bing Tan; Enmin Feng

doi:10.3934/jimo.2018016

Article Contents

2018, Volume 14, Issue 4: 1463-1478. Doi: 10.3934/jimo.2018016

This issue Previous Article Optimal liability ratio and dividend payment strategies under catastrophic risk Next Article Scheduling family jobs on an unbounded parallel-batch machine to minimize makespan and maximum flow time

Parameter identification and numerical simulation for the exchange coefficient of dissolved oxygen concentration under ice in a boreal lake

†.
School of Mathematical Sciences, Dalian University of Technology, Dalian 116024, Liaoning Province, China
‡.
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, Liaoning Province, China

* Corresponding author: Zhijun Li
* Corresponding author: Zhijun Li

Received: January 2017

Revised: August 2017

Early access: January 2018

Published: October 2018

The second author is supported by the National Natural Science Foundation of China (NNSFC) (Nos.51579028,41376186), the third author is supported by the NNSFC (No.11401073), and the fourth author is supported by the NNSFC (No.41306207)

Abstract Full Text(HTML) Figure(4) / Table(2) Related Papers Cited by

Abstract

Dissolved oxygen (DO) is one of the main parameters to assess the quality of lake water. This study is intended to construct a parabolic distributed parameter system to describe the variation of DO under the ice, and identify the vertical exchange coefficient K of DO with the field data. Based on the existence and uniqueness of the weak solution of this system, the fixed solution problem of the parabolic equation is transformed into a parameter identification model, which takes K as the identification parameter, and the deviation of the simulated and measured DO as the performance index. We prove the existence of the optimal parameter of the identification model, and derive the first order optimality conditions. Finally, we construct the optimization algorithm, and have carried out numerical simulation. According to the measured DO data in Lake Valkea-Kotinen (Finland), it can be found that the orders of magnitude of the coefficient K varying from 10^-6 to 10^-1 m² s^-1, the calculated and measured DO values are in good agreement. Within this range of K values, the overall trends are very similar. In order to get better fitting, the formula needs to be adjusted based on microbial and chemical consumption rates of DO.

Keywords:

Mathematics Subject Classification: Primary: 90C90; Secondary: 62P12.

Citation:

Full Text(HTML)

Figure 1. Schematic diagram of model identification area and mesh generation

Download: Full-size image PowerPoint slide

Figure 2. DO concentration curves at different depths at No. 2 station

Download: Full-size image PowerPoint slide

Figure 3. Comparison curves of the measured and the calculated DO concentration at depth 0.70 m when $K$ varying from 10$^{ - 8}$ to 10 m$^{2}$ s$^{-1}$ with a total of ten orders of magnitude

Download: Full-size image PowerPoint slide

Figure 4. Comparison curves of the measured and the calculated DO concentrations at different depths when the order of magnitude $ K $ = 10$^{-4}$ m$^{2}$ s$^{-1}$

Download: Full-size image PowerPoint slide

Table 1. Relative errors (%) of the measured and the calculated DO concentrations at different depths when $K$ varying from 10$^{ - 8}$ to 10 m$^{2}$ s$^{-1}$ with ten orders of magnitude

Relerr	10$^{-8}$	10$^{-7}$	10$^{-6}$	10$^{-5}$	10$^{-4}$	10$^{-3}$	10$^{-2}$	10$^{-1}$	10$^{0}$	10$^{1}$
0.45 m	0.6112	0.2582	0.3161	0.3270	0.3287	0.3294	0.3324	0.3419	0.3609	0.4018
0.70 m	0.6537	0.1574	0.1586	0.1675	0.1689	0.1694	0.1720	0.1627	0.1691	0.2215
0.95 m	1.6933	0.2649	0.1927	0.2279	0.234	0 0.2355	0.2370	0.2473	0.3164	0.9236
1.95 m	8.5907	4.0214	19.431	28.932	30.002	30.126	30.058	27.763	23.361	22.972
2.95 m	21.037	13.605	4.5830	3.7184	3.7672	3.7480	3.5150	3.4109	3.2001	6.6497

| Show Table

DownLoad: CSV

Table 2. Correlation coefficients of the measured and the calculated DO concentrations at different depths when $K$ varying from 10$^{ - 8}$ to 10 m$^{2}$ s$^{-1}$ with ten orders of magnitude

Corcoef	10$^{-8}$	10$^{-7}$	10$^{-6}$	10$^{-5}$	10$^{-4}$	10$^{-3}$	10$^{-2}$	10$^{-1}$	10$^{0}$	10$^{1}$
0.45 m	0.9012	0.9247	0.9530	0.9552	0.9552	0.9550	0.9540	0.9506	0.9438	0.9310
0.70 m	0.9248	0.9475	0.9693	0.9702	0.9700	0.9698	0.9676	0.9609	0.9451	0.9257
0.95 m	0.9361	0.9429	0.9518	0.9458	0.9448	0.9444	0.9439	0.9417	0.9282	0.8409
1.95 m	0.9431	0.9276	0.9799	0.9728	0.9695	0.9687	0.9666	0.9637	0.9575	0.9383
2.95 m	0.8850	0.8883	0.9535	0.9706	0.9719	0.9723	0.9749	0.9684	0.9671	0.9279

| Show Table

DownLoad: CSV

Related Papers

Cited by

References

	K. Addy and L. Green , Dissolved oxygen and temperature, PLA Notes, 22 (2004) , 48-65.
	V. Z. Antonopoulos and S. K. Gianniou , Simulation of water temperature and dissolved oxygen distribution in Lake Vegoritis, Greece, Ecol. Model., 160 (2003) , 39-53. doi: 10.1016/S0304-3800(02)00286-7.
	L. Arvola , K. Salonen , J. Keskitalo , T. Tulonen , M. Järvinen and J. Huotari , Plankton metabolism and sedimentation in a small boreal lake -a long-term perspective, Boreal Environ. Res., 19 (2014) , 83-97.
	J. Babin and E. E. Prepas , Modelling winter oxygen depletion rates in ice-covered temperate zone lakes in Canada, Can. J. Fish. Aquat. Sci., 42 (1985) , 239-249. doi: 10.1139/f85-031.
	Q. Bai , R. Li , Z. Li , M. Leppäranta , L. Arvola and M. Li , Time-series analyses of water temperature and dissolved oxygen concentration in Lake Valkea-Kotinen (Finland) during ice season, Ecol. Inform., 36 (2016) , 181-189. doi: 10.1016/j.ecoinf.2015.06.009.
	J. Barica and J. A. Mathias , Oxygen depletion and winterkill risk in small prairie lakes under extended ice cover, J. Fish. Res. Board Can., 36 (1979) , 980-986. doi: 10.1139/f79-136.
	F. Evrendilek and N. Karakaya , Monitoring diel dissolved oxygen dynamics through integrating wavelet denoising and temporal neural networks, Environ. Monit. Assess., 186 (2014) , 1583-1591. doi: 10.1007/s10661-013-3476-9.
	X. Fang and H. G. Stefan , Simulated climate change effects on dissolved oxygen characteristics in ice-covered lakes, Ecol. Model., 103 (1997) , 209-229. doi: 10.1016/S0304-3800(97)00086-0.
	B. Foley , I. D. Jones , S. C. Maberly and B. Rippey , Long-term changes in oxygen depletion in a small temperate lake: Effects of climate change and eutrophication, Freshwater Biol., 57 (2012) , 278-289. doi: 10.1111/j.1365-2427.2011.02662.x.
	S. Golosov , O. A. Maher , E. Schipunova , A. Terzhevik , G. Zdorovennova and G. Kirillin , Physical background of the development of oxygen depletion in ice-covered lakes, Oecologia, 151 (2007) , 331-340. doi: 10.1007/s00442-006-0543-8.
	K. Jylhä , M. Laapas , K. Ruosteenoja , L. Arvola , A. Drebs , J. Kersalo , S. Saku , H. Gregow , H. R. Hannula and P. Pirinen , Climate variability and trends in the Valkea-Kotinen region, southern Finland: Comparisons between the past, current and projected climates, Boreal Environ. Res., 19 (2014) , 4-30.
	U. T. Khan and C. Valeo , A new fuzzy linear regression approach for dissolved oxygen prediction, Hydrolog. Sci. J., 60 (2015) , 1096-1119. doi: 10.1080/02626667.2014.900558.
	G. Kirillin , M. Leppäranta , A. Terzhevik , N. Granin , J. Bernhardt , C. Engelhardt , T. Efremova , S. Golosov , N. Palshin , P. Sherstyankin , G. Zdorovennova and R. Zdorovennov , Physics of seasonally ice-covered lakes: A review, Aquat. Sci., 74 (2012) , 659-682. doi: 10.1007/s00027-012-0279-y.
	J. A. Mathias and J. Barica , Factors controlling oxygen depletion in ice-covered lakes, Can. J. Fish. Aquat. Sci., 37 (1980) , 185-194. doi: 10.1139/f80-024.
	M. E. Meding and L. J. Jackson , Biological implications of empirical models of winter oxygen depletion, Can. J. Fish. Aqua. Sci., 58 (2001) , 1727-1736. doi: 10.1139/f01-109.
	J. C. Patterson , B. R. Allanson and G. N. Ivey , A dissolved oxygen budget model for Lake Erie in summer, Freshwater Biol., 15 (1985) , 683-694. doi: 10.1111/j.1365-2427.1985.tb00242.x.
	V. Ranković , J. Radulović , I. Radojević , A. Ostojić and L. Čomić , Neural network modeling of dissolved oxygen in the Gruža reservoir, Serbia, Ecol. Model., 221 (2010) , 1239-1244.
	J. Ruuhijärvi , M. Rask , S. Vesala , A. Westermark , M. Olin , J. Keskitalo and A. Lehtovaara , Recovery of the fish community and changes in the lower trophic levels in a eutrophic lake after a winter kill of fish, Hydrobiologia, 646 (2010) , 145-158.
	H. G. Stefan , M. Hondzo , X. Fang , J. G. Eaton and J. H. Mccormick , Simulated long-term temperature and dissolved oxygen characteristics of lakes in the north-central United States and associated fish habitat limits, Limnol. Oceanogr., 41 (1996) , 1124-1135. doi: 10.4319/lo.1996.41.5.1124.
	H. G. Stefan and X. Fang , Dissolved oxygen model for regional lake analysis, Ecol. Model., 71 (1994) , 37-68. doi: 10.1016/0304-3800(94)90075-2.
	J. Vuorenmaa , K. Salonen , L. Arvola , J. Mannio , M. Rask and P. Horppila , Water quality of a small headwater lake reflects long-term variations in deposition, climate and in-lake processes, Boreal Environ. Res., 19 (2014) , 47-65.
	Y. Wang, L² Theory of Partial Differential Equations, 1^st edition, Peking University Press, Beijing, 1989.
	Y. Zhang , Z. Wu , M. Liu , J. He , K. Shi , Y. Zhou , M. Wang and X. Liu , Dissolved oxygen stratification and response to thermal structure and long-term climate change in a large and deep subtropical reservoir (Lake Qiandaohu, China), Water Res., 75 (2015) , 249-258. doi: 10.1016/j.watres.2015.02.052.