A Study On The Inference Of Eddy Viscosity For Atmospheric Boundary Layer Models

The significant turbulent motion in the boundary layer has always been a key issue in atmospheric science.The turbulence of the boundary layer is mainly reflected by the eddy viscosity in the model.So accurate eddy viscosity coefficients are critical for characterizing dynamic and thermodynamic structure in the boundary layer.In this paper,we consider obtaining the eddy viscosity as an inverse problem,and retrieve its value by the observation of the mode output value.Using weak nonlinear Prandtl model and General Ekman momentum approximation model hereafter GEM),respectively calculate the eddy viscosity of the model by variational inversion and statistical inversion.Inversion of eddy viscosity coefficient in weak nonlinear Prandtl model is conducted according to variational inversion theory.The solution to the positive problem is calculated by the Wentzel–Kramers–Brillouin hereafter WKB)method,and then the objective function is constructed by the square of the difference between the observed and output values.The numerical experiments are conducted using simulated measurement data and as much as possible to make data eliminate model dependency.The results show that the variational method can reverse the eddy viscosity coefficient well.Moreover,the variational method can overcome the influence of certain observation errors,which proves the effectiveness and robustness of the method.The generalized polynomial chaos combined with the EnKF method hereafter PCEnKF)retrieve the low-dimensional eddy viscosity coefficient of GEM.The EnKF method is to use the Monte Carlo method to obtain the likelihood function.However,this method has a slow convergence rate and requires a large number of samples to make the sampling error small.In this paper,the likelihood function is solved by the polynomial chaos expansion.The convergence speed is accelerated by selecting the appropriate polynomial basis functions,and only need a small amount of calculation to get the likelihood value.The results in numerical experiments indicate that polynomial chaos expansion combined with regression method for solving coefficients can quickly and effectively solve wind speed distribution in GEM,and the calculation time is only onefifth of the Monte Carlo method.Based on this,the calculation process of parameters in GEM is constructed by PC-EnKF,and then carry out numerical experiments.The results show that the PC-EnKF is a very effective method for solving the posterior probability distribution of eddy viscosity of low-dimensional random variables.It is also found that the posterior distribution of eddy viscosity solved with wind observations in the big uncertainty area is more accurate,which provides an important guidance for selecting the location of observation points.The uncertainty quantifation and inverse problem of high dimensional eddy viscosity coefficient is solved.It can appear "curse of dimensionality " as polynomial chaos expansion for high dimensional parameters is performed.The linear dimension reduction space is obtained by using the projective resampling method and the slice inverse regression method,and then combined with the Monte Carlo method can avoid this problem.Thus,a new method for solving wind speed distribution is proposed: Projective Resampling inverse regression uncertainty quantification-based control variate method PRIRUQ-CV).This method can greatly reduce the computational complexity of polynomial chaos expansion with less loss of accuracy,while its results are better than those of the Monte Carlo method with the same number of samples.The PRIRCV-EnKF which is one new inversion method,is proposed by applying PRIRUQ-CV method to PCEnKF.Numerical experiments for verifying the feasibility of the method are conducted.The results show that it is difficult to determine wind speed statistics due to insufficient accuracy,although the reduced dimensional space can capture the main information of the original model.However,it can accurately obtain statistical information after combining with the Monte Carlo method,and the results are significantly better than those of the Monte Carlo method with the same number of samples.Therefore,it is proved that the method can avoid the problem that the polynomial chaos expansion method is too large in high-dimensionality and keeps better than Monte Carlo method.The validity of the PRIRCV-EnKF method for retrieving the eddy viscosity coefficient is further verified.It indicates that this method can effectively retrieve the parameters of the generalized Ekman momentum approximation model,and significantly reduce parameter uncertainty based on prior distribution.