# The Distribution and Uncertainty Quantification of Wind Profile in the Stochastic General Ekman Momentum Approximation Model

• The general Ekman momentum approximation boundary-layer model (GEM) can be effectively used to describe the physical processes of the boundary layer. However, eddy viscosity, which is an approximated value, can lead to uncertainty in the solutions. In this paper, stochastic eddy viscosity is taken into consideration in the GEM, and generalized polynomial chaos is used to quantify the uncertainty. The goal of uncertainty quantification is to investigate the effects of uncertainty in the eddy viscosity on the model and to subsequently provide reliable distribution of simulation results. The performances of the stochastic eddy viscosity and generalized polynomial chaos method are validated based on three different types of eddy viscosities, and the results are compared based on the Monte Carlo me-thod. The results indicate that the generalized polynomial chaos method can be accurately and efficiently used in uncertainty quantification for the GEM with stochastic eddy viscosity.
• Fig. 1.  The flow chart for solving the GEM with stochastic eddy viscosity based on gPC. The left chart is the stochastic Galerkin method; the right chart is the stochastic collocation method. The number of samples should be at least twice the number of expansion terms [$N > 2(P + 1)$].

Fig. 2.  Probability density functions of (a) δ and (b) ${z_{\rm{m}}}$. The approximate range of δ is (0, 0.4), and that of ${z_{\rm{m}}}$ is (300, 700).

Fig. 3.  Average Ekman spiral comparison for (a) cyclonic and (b) anticyclonic shear flows in experiment I. The solid and dotted lines represent the results obtained by using the gPC and Monte Carlo methods, respectively.

Fig. 4.  As in Fig. 3, but for the standard deviation of u.

Fig. 5.  As in Fig. 3, but for the standard deviation of v.

Fig. 6.  The 99% confidence interval of u for (a) cyclonic and (b) anticyclonic shear flow in experiment I. The grey shaded area is the confidence interval of wind speed, and the dotted line is the mean of wind distribution.

Fig. 7.  As in Fig. 6, but for v.

Fig. 8.  The histogram of K. The abscissa is the eddy viscosity, and the ordinate is the frequency. The approximate range of K is (3, 13), and the mean is 8.

Fig. 9.  As in Fig. 3, but for experiment Ⅱ.

Fig. 10.  As in Fig. 5, but for experiment Ⅱ.

Fig. 11.  As in Fig. 5, but for experiment Ⅱ.

Fig. 12.  As in Fig. 6, but for experiment Ⅱ.

Fig. 13.  As in Fig. 7, but for experiment Ⅱ.

Fig. 15.  As in Fig. 3, but for experiment Ⅲ.

Fig. 14.  As in Fig. 8, but for α. The abscissa is the value of α, and the ordinate is the frequency. The approximate range of α is (–10, 10), and the mean is 0.

Fig. 16.  As in Fig. 4, but for experiment Ⅲ.

Fig. 17.  As in Fig. 5, but for experiment Ⅲ.

Fig. 18.  As in Fig. 6, but for experiment Ⅲ.

Fig. 19.  As in Fig. 7, but for experiment Ⅲ.

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###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

## The Distribution and Uncertainty Quantification of Wind Profile in the Stochastic General Ekman Momentum Approximation Model

###### Corresponding author: Sixun HUANG, huangsxp@163.com;
• 1. College of Meteorology and Oceanography, National University of Defense Technology, Nanjing 211101
• 2. Center for Computational Science and Finance, Shanghai University of Finance and Economics, Shanghai 200433
Funds: Supported by the National Natural Science Foundation of China ( 91730304, 41575026, and 61371119)

Abstract: The general Ekman momentum approximation boundary-layer model (GEM) can be effectively used to describe the physical processes of the boundary layer. However, eddy viscosity, which is an approximated value, can lead to uncertainty in the solutions. In this paper, stochastic eddy viscosity is taken into consideration in the GEM, and generalized polynomial chaos is used to quantify the uncertainty. The goal of uncertainty quantification is to investigate the effects of uncertainty in the eddy viscosity on the model and to subsequently provide reliable distribution of simulation results. The performances of the stochastic eddy viscosity and generalized polynomial chaos method are validated based on three different types of eddy viscosities, and the results are compared based on the Monte Carlo me-thod. The results indicate that the generalized polynomial chaos method can be accurately and efficiently used in uncertainty quantification for the GEM with stochastic eddy viscosity.

Reference (39)

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