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

†. 

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.

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

[3]

L. ArvolaK. SalonenJ. KeskitaloT. TulonenM. 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. BaiR. LiZ. LiM. LeppärantaL. 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. FoleyI. D. JonesS. 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. GolosovO. A. MaherE. SchipunovaA. TerzhevikG. 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. JylhäM. LaapasK. RuosteenojaL. ArvolaA. DrebsJ. KersaloS. SakuH. GregowH. 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. KirillinM. LeppärantaA. TerzhevikN. GraninJ. BernhardtC. EngelhardtT. EfremovaS. GolosovN. PalshinP. SherstyankinG. 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. PattersonB. 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. RankovićJ. RadulovićI. RadojevićA. 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. RuuhijärviM. RaskS. VesalaA. WestermarkM. OlinJ. 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. StefanM. HondzoX. FangJ. 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. VuorenmaaK. SalonenL. ArvolaJ. MannioM. 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. ZhangZ. WuM. LiuJ. HeK. ShiY. ZhouM. 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. ArvolaK. SalonenJ. KeskitaloT. TulonenM. 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. BaiR. LiZ. LiM. LeppärantaL. 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. FoleyI. D. JonesS. 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. GolosovO. A. MaherE. SchipunovaA. TerzhevikG. 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. JylhäM. LaapasK. RuosteenojaL. ArvolaA. DrebsJ. KersaloS. SakuH. GregowH. 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. KirillinM. LeppärantaA. TerzhevikN. GraninJ. BernhardtC. EngelhardtT. EfremovaS. GolosovN. PalshinP. SherstyankinG. 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. PattersonB. 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. RankovićJ. RadulovićI. RadojevićA. 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. RuuhijärviM. RaskS. VesalaA. WestermarkM. OlinJ. 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. StefanM. HondzoX. FangJ. 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. VuorenmaaK. SalonenL. ArvolaJ. MannioM. 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. ZhangZ. WuM. LiuJ. HeK. ShiY. ZhouM. 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
Figure 2.  DO concentration curves at different depths at No. 2 station
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
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$^{-1}$
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
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
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
[1]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[2]

Yila Bai, Haiqing Zhao, Xu Zhang, Enmin Feng, Zhijun Li. The model of heat transfer of the arctic snow-ice layer in summer and numerical simulation. Journal of Industrial & Management Optimization, 2005, 1 (3) : 405-414. doi: 10.3934/jimo.2005.1.405

[3]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[4]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[5]

Xu Zhang, Xiang Li. Modeling and identification of dynamical system with Genetic Regulation in batch fermentation of glycerol. Numerical Algebra, Control & Optimization, 2015, 5 (4) : 393-403. doi: 10.3934/naco.2015.5.393

[6]

Christina Surulescu, Nicolae Surulescu. Modeling and simulation of some cell dispersion problems by a nonparametric method. Mathematical Biosciences & Engineering, 2011, 8 (2) : 263-277. doi: 10.3934/mbe.2011.8.263

[7]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[8]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056

[9]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068

[10]

Daoyin He, Ingo Witt, Huicheng Yin. On the strauss index of semilinear tricomi equation. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4817-4838. doi: 10.3934/cpaa.2020213

[11]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409

[12]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437

[13]

Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109

[14]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[15]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[16]

Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309

[17]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[18]

Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511

[19]

Ka Luen Cheung, Man Chun Leung. Asymptotic behavior of positive solutions of the equation $ \Delta u + K u^{\frac{n+2}{n-2}} = 0$ in $IR^n$ and positive scalar curvature. Conference Publications, 2001, 2001 (Special) : 109-120. doi: 10.3934/proc.2001.2001.109

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (106)
  • HTML views (1185)
  • Cited by (0)

Other articles
by authors

[Back to Top]