The numerical scheme development of a simplified frozen soil model

In almost all frozen soil models used currently, three variables of temperature, ice content and moisture content are used as prognostic variables and the rate term, accounting for the contribution of the phase change between water and ice, is shown explicitly in both the energy and mass balance equations. The models must be solved by a numerical method with an iterative process, and the rate term of the phase change needs to be pre-estimated at the beginning in each iteration step. Since the rate term of the phase change in the energy equation is closely related to the release or absorption of the great amount of fusion heat, a small error in the rate term estimation will introduce greater error in the energy balance, which will amplify the error in the temperature calculation and in turn, cause problems for the numerical solution convergence. In this work, in order to first reduce the trouble, the methodology of the variable transformation is applied to a simplified frozen soil model used currently, which leads to new frozen soil scheme used in this work. In the new scheme, the enthalpy and the total water equivalent are used as predictive variables in the governing equations to replace temperature, volumetric soil moisture and ice content used in many current models. By doing so, the rate terms of the phase change are not shown explicitly in both the mass and energy equations and its pre-estimation is avoided. Secondly, in order to solve this new scheme more functionally, the development of the numerical scheme to the new scheme is described and a numerical algorithm appropriate to the numerical scheme is developed. In order to evaluate the new scheme of the frozen soil model and its relevant algorithm, a series of model evaluations are conducted by comparing numerical results from the new model scheme with three observational data sets. The comparisons show that the results from the model are in good agreement with these data sets in both the change trend of variables and their magnitude values, and the new scheme, together with the algorithm, is more efficient and saves more computer time.     

A Simplified Scheme of the Generalized Layered Radiative Transfer Model

In this paper, firstly, a simplified version （SGRTM） of the generalized layered radiative transfer model （GRTM） within the canopy, developed by us, is presented. It reduces the information requirement of inputted sky diffuse radiation, as well as of canopy morphology, and in turn saves computer resources. Results from the SGRTM agree perfectly with those of the GRTM. Secondly, by applying the linear superposition principle of the optics and by using the basic solutions of the GRTM for radiative transfer within the canopy under the condition of assumed zero soil reflectance, two sets of explicit analytical solutions of radiative transfer within the canopy with any soil reflectance magnitude are derived： one for incident diffuse, and the other for direct beam radiation. The explicit analytical solutions need two sets of basic solutions of canopy reflectance and transmittance under zero soil reflectance, run by the model for both diffuse and direct beam radiation. One set of basic solutions is the canopy reflectance αf （written as α1 for direct beam radiation） and transmittance βf （written as β1 for direction beam radiation） with zero soil reflectance for the downward radiation from above the canopy （i.e. sky）, and the other set is the canopy reflectance （αb） and transmittance βb for the upward radiation from below the canopy （i.e., ground）. Under the condition of the same plant architecture in the vertical layers, and the same leaf adaxial and abaxial optical properties in the canopies for the uniform diffuse radiation, the explicit solutions need only one set of basic solutions, because under this condition the two basic solutions are equal, i.e., αf = αb and βf = βb. Using the explicit analytical solutions, the fractions of any kind of incident solar radiation reflected from （defined as surface albedo, or canopy reflectance）, transmitted through （defined as canopy transmittance）, and absorbed by （defined as canopy absorptance） the canopy and other properties per     

A Generalized Layered Radiative Transfer Model in the Vegetation Canopy

In this paper, a generalized layered model for radiation transfer in canopy with high vertical resolution is developed. Differing from the two-stream approximate radiation transfer model commonly used in the land surface models, the generalized model takes into account the effect of complicated canopy morphology and inhomogeneous optical properties of leaves on radiation transfer within the canopy. In the model, the total leaf area index （LAI） of the canopy is divided into many layers. At a given layer, the influences of diffuse radiation angle distributions and leaf angle distributions on radiation transfer within the canopy are considered. The derivation of equations serving the model are described in detail, and these can deal with various diffuse radiation transfers in quite broad categories of canopy with quite inhomogeneons vertical structures and uneven leaves with substantially different optical properties of adaxial and abaxial faces of the leaves. The model is used to simulate the radiation transfer for canopies with horizontal leaves to validate the generalized model. Results from the model are compared with those from the two-stream scheme, and differences between these two models are discussed.     

安徽省淮南森林冠层辐射传输过程的特征

本文利用淮南森林观测站2018年7月1日至2019年6月30日冠层辐射观测,分析了淮南栎树森林下垫面冠层内外辐射变化特征。结果表明:（1）从春季到夏季,栎树冠层之上向下的太阳短波辐射增加,到冬季逐渐减少。从早春开始,由于叶片生长增多,冠层中间和冠层之下向下的太阳短波辐射下降,从秋季到冬季树叶凋落,其向下的太阳辐射增加,与冠层之上的变化趋势相反;对于向上的短波辐射,无论冠层之上、冠层中间还是冠层之下,随季节的变化都与向下的短波辐射相似,只是数值小很多。（2）冠层之上、冠层中间和冠层之下向下的长波辐射,随时间的变化从春季逐渐开始增大至夏季达到最大,随后逐渐减小并在冬季达到最小;就空间变化而言,冠层中间和冠层之下向下的长波辐射值比冠层之上的辐射值高,使得冠层对长波辐射的振幅增大,晴空条件最高可达1.3倍。（3）淮南森林区冠层之上（距地面25 m）年平均反照率为0.14,比中国北方地区（35°N）温带季风气候区（混交林为主）反照率的整体水平低0.01,表明淮南的森林茂密、灌丛更多些。（4）冠层上部分和整层的短波辐射透射率主要受叶片的影响。夏季,冠层的短波透射率平均为0.1。到了冬天,叶子凋落,透射率增加并趋于一个平稳的波动。冠层的短波辐射吸收率在夏季最高,秋季逐渐降低,随着叶子凋落在冬季迅速减小,趋于一常值。     

A Comparison of Two Canopy Radiative Models in Land Surface Processes

This paper compares the predictions by two radiative transfer models-the two-stream approximation model and the generalized layered model （developed by the authors） in land surface processes -for different canopies under direct or diffuse radiation conditions. The comparison indicates that there are significant differences between the two models, especially in the near infrared （NIR） band. Results of canopy reflectance from the two-stream model are larger than those from the generalized model. However, results of canopy absorptance from the two-stream model are larger in some cases and smaller in others compared to those from the generalized model, depending on the cases involved. In the visible （VIS） band, canopy reflectance is smaller and canopy absorptance larger from the two-stream model compared to the generalized model when the Leaf Area Index （LAI） is low and soil reflectance is high. In cases of canopies with vertical leaf angles, the differences of reflectance and absorptance in the VIS and NIR bands between the two models are especially large. Two commonly occurring cases, with which the two-stream model cannot deal accurately, are also investigated. One is for a canopy with different adaxial and abaxial leaf optical properties; and the other is for incident sky diffuse radiation with a non-uniform distribution. Comparison of the generalized model within the same canopy for both uniform and non-uniform incident diffuse radiation inputs shows smaller differences in general. However, there is a measurable difference between these radiation inputs for a canopy with high leaf angle. This indicates that the application of the two-stream model to a canopy with different adaxial and abaxial leaf optical properties will introduce non-negligible errors.