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Ronald L. Drake

Abstract

This paper is concerned with complementing the known analytical studies of the pure coalescence equation. There is still a need for better analytical tools for the analysis of this problem even though high-speed computers have contributed much to the knowledge of this system. Specifically, when the detailed microphysics is incorporated into a large-scale, three-dimensional, moist, deep-convection model, it is currently impossible to solve the coalescence equation numerically for several size categories. Hence, there is a need for better analytical tools. In particular, we are concerned with the relationships between integral power moments of the size spectrum and the collection kernel, relationships between Friedlander's similarity solutions and the kernel, bounds for the size spectrum, and various power-moment inequalities. The results we obtained will allow us to make reasonable approximations for spectra which can, in turn, be used in large-scale convection models.

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Ronald L. Drake and Thomas J. Wright

Abstract

This paper is concerned with extending the list of the known analytical solutions of the pure coalescence equation. Through the use of Laplace transforms three families of exact solutions are obtained for arbitrary initial conditions and for kernels which are certain linear combinations of the constant, sum and product kernels. The solutions which are given in this paper contain, as special cases, the solutions obtained by Scott. Other Special cases of these families of solutions are given for certain initial spectra, namely, the well-known gamma distributions and their limiting function, the Dirac delta function.

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Ronald L. Drake, Patrick D. Coyle, and Daniel P. Anderson

Abstract

This paper is concerned with the effects of nonlinear eddy coefficients on rising, dry fine thermals. The base atmosphere in which these thermals are embedded is hydrostatic, horizontally homogeneous, nearly neutral, and without an ambient wind. By identifying the turbulent transfer terms with the subgrid–scale motions, a nonlinear formulation, based upon the worn. of Lilly and Smagorinsky, is obtained for the eddy coefficients. The mixing length in these terms is based upon a vorticity formulation rather than the size of the numerical grid used in the computations. Using numerical techniques based upon the work of Arakawa, and Adams and Bashforth, we tested these formulations by following the evolution of rising line thermals. We concluded that the eddy formulation used in this paper is more realistic than using constant coefficients throughout the field of computation. In addition, we found that the density stratification is important if the convective layer is deeper than 3 km.

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Ronald L. Drake, Patrick D. Coyle, and Daniel P. Anderson

Abstract

This paper is concerned with the time evolution of interactive dry line thermals in a convective layer. The temperature perturbations which produce these line thermals are randomly chosen. Since the domain of computation is the x-z plane, the evolving flow field is described by the streamfunction, vorticity, and the potential temperature. The nonlinear acceleration terms were differenced by an Arakawa scheme and the time differencing was the second-order, explicit, two-step Adams-Bashforth scheme. The turbulent transfer terms were given by a nonlinear formulation based on the work of Lilly and Smagorinsky. The convective layers in our numerical experiments were simulated by releasing a single set of thermals and by successive releases of thermals. Even though our work is a two-dimensional simulation, our results were consistent with the gross properties of real convective fields reported by several investigators. Hence, our system is a relatively inexpensive model that can be used to study convective layers over irregular surfaces and terrain.

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