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Tetsuji Yamada

Abstract

A turbulent closure model is analyzed under the condition that the turbulent flow is steady in its ensemble average and both the advection and diffusion terms, i.e., third moments of turbulence, are neglected in the turbulent Reynolds stress and heat flux equations. The critical flux Richardson number is defined as a limiting value beyond which physically correct solutions are no longer possible. All the turbulence moments are suppressed completely when the Richardson number exceeds the critical value. The validity of making such an assumption is tested against the numerical results which were obtained by utilizing a more complete set of equations.

The critical flux Richardson numbers of 0.18σ0.27 are obtained from the different proposed empirical constants. The ratio of the eddy transport coefficient of heat to that of momentum have values of 0.5σ1.2 at the critical condition of stability. A review is made to clarity the differences between the present model and the earlier works of Ellison, Townsend, and Arya.

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Tetsuji Yamada

Abstract

The similarity functions A, B and C are computed based on the various scales previously proposed for wind, temperature and height in the planetary boundary layer. The vertically averaged geostrophic wind recently proposed by Arya and Wyngaard is found to be a better choice as a wind scale for the Wangara experiment than a local wind at a specified height. A similar conclusion is drawn for the temperature scale. As for the height scale the similarity functions A, B and C scaled by the height of the surface inversion layer during the nighttime resulted in relatively less scatter than those scaled by u*/|f|.

Nomograms for the geostrophic drag and the beat transfer coefficients are presented by utilizing the approximate expressions for A, B and C deduced from the Wangara data. Agreement between the values predicted and those deduced from the data appears to support the appropriateness of the present choice of the scales originally suggested by Arya and Wyngaard.

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Tetsuji Yamada and George Mellor

Abstract

Previously, the authors have studied a hierarchy of turbulent boundary layer models, all based on the same closure assumptions for the triple turbulence moments. The models differ in complexity by virtue of a systematic process of neglecting certain of the tendency and diffusion terms in the dynamic equations for the turbulent moments. Based on this work a Level 3 model was selected as one which apparently sacrificed little predictive accuracy, but which afforded considerable numerical simplification relative to the more complex Level 4 model.

An earlier paper had demonstrated that the model produced similarity solutions in near agreement with surface, constant flux data. In this paper, simulators from the Level 3 model are compared with two days of Wangara atmospheric boundary layer data (Clarke et al., 1971). In this comparison, there is an easily identified error introduced by our inability to include advection of momentum in the calculation since these terms were not measured. Otherwise, the calculated results and the observational data appear to he in close agreement.

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George L. Mellor and Tetsuji Yamada

Abstract

Turbulence models centered on hypotheses by Rotta and Kolmogoroff are complex. In the present paper we consider systematic simplifications based on the observation that parameters governing the degree of anisotropy are small. Hopefully, we shall discern a level of complexity which is intuitively attractive and which optimizes computational speed and convenience without unduly sacrificing accuracy.

Discussion is focused on density stratified flow due to temperature. However, other dependent variables—such as water vapor and droplet density—can be treated in analogous fashion. It is, in fact, the anticipation of additional physical complexity in modeling turbulent flow fields that partially motivates the interest in an organized process of analytical simplification.

For the problem of a planetary boundary layer subject to a diurnally varying surface heat flux or surface temperature, three models of varying complexity have been integrated for 10 days. All of the models incorporate identical empirical constants obtained from neutral flow data alone. The most complex of the three models requires simultaneous solution of 10 partial differential equations for turbulence moments in addition to the equations for the mean velocity components and temperature; the least complex eliminates all of the 10 differential equation whereas a “compromise” model retains two differential equations for total turbulent energy and temperature variance.

We conclude that all of the models give nearly the same results. We find the two-differential-equation model particularly attractive.

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Chih-Yue Jim Kao and Tetsuji Yamada

Abstract

Detailed observations of both mean and turbulence fields of an anticyclonic, quasi-steady state, stratocumulus-capped boundary layer obtained with ground-based and balloonborne equipment during the night of 19/20 November 1976 at Cardington, Bedford, UK, are simulated in relation to large-scale subsidence, longwave radiative model cooling, and large-scale moisture supply from sea to land, using a simplified second-order turbulence-closure radiative model.

Using a one-dimensional version of the model, most of the observed features are well simulated, including the large temperature “jump” in a thin layer at cloud top, thermodynamic profiles within the boundary layer, cloud depth and cloud liquid water content, turbulence in the cloud layer, and radiative fluxes and their associated cooling (heating) rates. The results also show that in order to reproduce the observed features, the large-scale subsidence rate and horizontal moisture input should be properly incorporated.

In addition to the one-dimensional simulations for the observed balloon profiles, we used a three-dimensional version of the model to investigate the mechanisms which resulted in a cloudless band embedded in this large sheet of stratocumulus, observed during the same night around the north shore of the English Channel. The physics derived from the one-dimensional simulations applies well in the three-dimensional model. The sensitivity tests show that the terrain effects, which induce larger downward vertical motion, are primarily responsible for this clear band.

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