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

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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: December 31, 2016

Revised: July 31, 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)

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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.

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

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

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

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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

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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$^{-1}$

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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

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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

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