American Institute of Mathematical Sciences

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October  2018, 14(4): 1463-1478. doi: 10.3934/jimo.2018016

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 ‡, and  Enmin Feng †,

†. 

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

Received  January 2017 Revised  August 2017 Published  January 2018

Fund Project: 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)

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: Dissolved oxygen, parameter identification, numerical simulation, partial differential equation, frozen lakes.
Mathematics Subject Classification: Primary: 90C90; Secondary: 62P12.
Citation: Qinxi Bai, Zhijun Li, Lei Wang, Bing Tan, Enmin Feng. Parameter identification and numerical simulation for the exchange coefficient of dissolved oxygen concentration under ice in a boreal lake. Journal of Industrial & Management Optimization, 2018, 14 (4) : 1463-1478. doi: 10.3934/jimo.2018016
References:
[1]

K. Addy and L. Green, Dissolved oxygen and temperature, PLA Notes, 22 (2004), 48-65. Google Scholar

[2]

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. Google Scholar

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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. Google Scholar

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[5]

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. Google Scholar

[6]

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. Google Scholar

[7]

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. Google Scholar

[8]

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. Google Scholar

[9]

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. Google Scholar

[10]

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. Google Scholar

[11]

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. Google Scholar

[12]

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. Google Scholar

[13]

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. Google Scholar

[14]

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. Google Scholar

[15]

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. Google Scholar

[16]

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. Google Scholar

[17]

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. Google Scholar

[18]

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. Google Scholar

[19]

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. Google Scholar

[20]

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. Google Scholar

[21]

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. Google Scholar

[22]

Y. Wang, L2 Theory of Partial Differential Equations, 1st edition, Peking University Press, Beijing, 1989.Google Scholar

[23]

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. Google Scholar

show all references

References:
[1]

K. Addy and L. Green, Dissolved oxygen and temperature, PLA Notes, 22 (2004), 48-65. Google Scholar

[2]

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. Google Scholar

[3]

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. Google Scholar

[4]

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. Google Scholar

[5]

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. Google Scholar

[6]

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. Google Scholar

[7]

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. Google Scholar

[8]

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. Google Scholar

[9]

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. Google Scholar

[10]

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. Google Scholar

[11]

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. Google Scholar

[12]

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. Google Scholar

[13]

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. Google Scholar

[14]

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. Google Scholar

[15]

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. Google Scholar

[16]

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. Google Scholar

[17]

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. Google Scholar

[18]

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. Google Scholar

[19]

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. Google Scholar

[20]

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. Google Scholar

[21]

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. Google Scholar

[22]

Y. Wang, L2 Theory of Partial Differential Equations, 1st edition, Peking University Press, Beijing, 1989.Google Scholar

[23]

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. Google Scholar

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 m0.61120.25820.31610.32700.32870.32940.33240.34190.36090.4018
0.70 m0.65370.15740.15860.16750.16890.16940.17200.16270.16910.2215
0.95 m1.69330.26490.19270.22790.2340 0.23550.23700.24730.31640.9236
1.95 m8.59074.021419.43128.93230.00230.12630.05827.76323.36122.972
2.95 m21.03713.6054.58303.71843.76723.74803.51503.41093.20016.6497
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Relerr 10$^{-8}$ 10$^{-7}$ 10$^{-6}$ 10$^{-5}$ 10$^{-4}$ 10$^{-3}$ 10$^{-2}$ 10$^{-1}$ 10$^{0}$ 10$^{1}$
0.45 m0.61120.25820.31610.32700.32870.32940.33240.34190.36090.4018
0.70 m0.65370.15740.15860.16750.16890.16940.17200.16270.16910.2215
0.95 m1.69330.26490.19270.22790.2340 0.23550.23700.24730.31640.9236
1.95 m8.59074.021419.43128.93230.00230.12630.05827.76323.36122.972
2.95 m21.03713.6054.58303.71843.76723.74803.51503.41093.20016.6497
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
Corcoef10$^{-8}$10$^{-7}$10$^{-6}$10$^{-5}$10$^{-4}$10$^{-3}$10$^{-2}$10$^{-1}$10$^{0}$10$^{1}$
0.45 m0.90120.92470.95300.95520.95520.95500.95400.95060.94380.9310
0.70 m0.92480.94750.96930.97020.97000.96980.96760.96090.94510.9257
0.95 m0.93610.94290.95180.94580.94480.94440.94390.94170.92820.8409
1.95 m0.94310.92760.97990.97280.96950.96870.96660.96370.95750.9383
2.95 m0.88500.88830.95350.97060.97190.97230.97490.96840.96710.9279
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Corcoef10$^{-8}$10$^{-7}$10$^{-6}$10$^{-5}$10$^{-4}$10$^{-3}$10$^{-2}$10$^{-1}$10$^{0}$10$^{1}$
0.45 m0.90120.92470.95300.95520.95520.95500.95400.95060.94380.9310
0.70 m0.92480.94750.96930.97020.97000.96980.96760.96090.94510.9257
0.95 m0.93610.94290.95180.94580.94480.94440.94390.94170.92820.8409
1.95 m0.94310.92760.97990.97280.96950.96870.96660.96370.95750.9383
2.95 m0.88500.88830.95350.97060.97190.97230.97490.96840.96710.9279
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