The Distribution Of Dissolved Oxygen Under The Ice And Optimization Identification Of Its PDE System Parameters

Dissolved oxygen(DO)is an essential factor in the study of lakes.It is a vital environmental factor to maintain the dynamic balance of lake water environment and is an indispensable condition for the survival of aquatic organisms.Organisms in lakes,from fish to insects,to tiny planktonic animal,need oxygen to breathe.The concentration of DO level in lakes reflects the dynamic balance between the oxygen enrichment in atmosphere and oxygen evolution process of plant photosynthesis with the oxygen consumption process of biological respiration and biochemical reaction.And DO is more important to frozen lakes.When the lake freezes,lake water is isolated from the atmosphere,and oxygen supply from outside is blocked,oxygen depletion may occur in the lake water below the ice.Under anoxic conditions,there are negative ecological consequences,such as winter fish kills,benthic losses,and activation of the anaerobic processes that produce dangerous compounds.The change of the ecological environment under the ice will directly affect the survival and reproduction of organisms in the water.There is no doubt that DO is a most important and essential ecological indicator in the study on the ecological environment of water body under the ice.Based on the measured data of DO concentration and water temperature of a frozen lake,The variations of DO concentration under the ice are analyzed using data statistics,function fitting,wavelet analysis,spectrum analysis and other mathematical tools;And based on the parameter identification theory of distributed parameter systems and numerical method of partial differential equations,a deep study on the DO parameters is made,including the DO vertical exchange coefficients under the ice and the function expression of DO consumption rate.The main contents of this thesis and the results obtained are summarized as follows:1.Distribution characteristics analysis of DO under the ice.(1)Spatial variation characteristics.The typical spatial variation patterns of DO are analyzed and the maximum of DO concentration in the vertical direction and its causes are discussed.And the variation law of the bottom anaerobic zone is analyzed by statistical analysis and function fitting.(2)Temporal variation characteristics.The variation trend and fluctuation characteristics of DO concentration under the ice are analyzed by three methods respectively,including 24-h moving average method,Daubechies wavelet analysis method and Gauss fitting method,which the results of the front two methods are helpful to analyze detail change characteristics of DO and Gauss fitting method helps to predict the overall variation of DO concentration utilizing the trend function changing with the frozen time.And the periodic variation of DO at different depths under the ice is analyzed by means of fast Fourier transform and periodic spectrum analysis.These analyses help to deepen the understanding of the distribution variation characteristics of DO concentration in frozen lakes,and to further modeling on differential equation.2.Identification analysis and numerical simulation of the vertical exchange coefficient of DO system under the ice.According to the variation of DO in frozen lakes in winter,a parabolic distributed parameter system is constructed to describe the DO variation in three layers.First of all,the PDE system is equivalently transformed into a homogeneous boundary value system.Then the existence and uniqueness of the weak solution are proved based on the equivalent equations.Secondly,the parameter identification model with PDE constraints is structured taking the DO vertical exchange coefficients as the identification parameter and the square sum of error between simulated DO with measured DO as the performance index.And the existence of the parameter identification model is proved,and the first-order optimal condition of the existence of optimal parameters is given.Finally,the optimal algorithm is constructed that combined with the semi-implicit difference scheme according to the parameter identification model,and the numerical simulation is carried out with the measured data of DO concentration and water temperature under the ice.And the magnitude change range of the DO vertical exchange coefficient and the best vertical exchange coefficients of each layer are got by identification.The results can be compared with the previous empirical results,and can provide support in the quantitative calculation of the DO vertical exchange coefficient in frozen lakes.The analysis shows that the DO vertical exchange coefficient does not play a decisive role in the variation of DO concentration in the small frozen lake.3.Identification analysis and numerical simulation of the function parameter in the expression of DO consumption rate.Firstly,a layered distributed parameter system with nonhomogeneous boundary values is established,and it is transformed into a homogeneous boundary value system.Then the existence and uniqueness of the weak solution are proved based on the existence and uniqueness of the homogeneous boundary value system.Secondly,the parameter identification model with PDE constraints is structured taking the parameter ai3x3 + ai2x2?ai1x+ai0 in the function expression of the DO consumption as the identification parameter and the square sum of error between simulated DO with measured DO as the performance index.And the existence of optimal parameter identification model is proved.Then an adaptive genetic algorithm is constructed that combined with the semi-implicit difference scheme according to the parameter identification model,and the numerical simulation is carried out with the measured data of DO concentration and water temperature under the ice.The optimal function parameter of each layer is gained through identification,and the optimal expression of the DO consumption rate per layer is obtained.At last.Further analysis of the identification results shows that the optimal parameter function on the interval[0,1]is basically between the linear function x and the quadratic function x2,so there is no need to use polynomial function higher than 3 order to analyze in the frozen lake.These results provide an important reference for discussing DO consumption rate iin frozen lakes.