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## Abstract

Lagragian parcel calculations of condensation and coalescence theory are presented where both a distribution function approach as well as a conventional finite-difference approach are compared. The comparisons suggest that the use of series of log-normal distributions to represent the water droplet spectra may be a practical approach to treating cloud physical process in multi-dimensioned cloud models.

## Abstract

Lagragian parcel calculations of condensation and coalescence theory are presented where both a distribution function approach as well as a conventional finite-difference approach are compared. The comparisons suggest that the use of series of log-normal distributions to represent the water droplet spectra may be a practical approach to treating cloud physical process in multi-dimensioned cloud models.

## Abstract

Two cumulus cloud models in two space dimensions are presented; one is a bulk physical model without microphysics and the other includes the microphysical processes such as nucleation, diffusional growth of cloud droplets, stochastic coalescence, and fallout of raindrops. Some numerical aspects of the models are discussed; in particular, a method is described which allows long time steps to be used for the calculation of condensation in regions of relatively old droplet populations. The bulk-physical and microphysical models are compared for a non-precipitation case. The numerical results revealed that the inclusion of microphysics had little effect on the whole cloud dynamics, a result which considerably differs from that of Árnason and Greenfield.

The microphysical model produced hi-modal spectra of droplets through the interaction of a mid-level nucleation region, which corresponds to the base of an upper intense thermal, and the vertical advection and diffusion of existing droplets from a lower thermal. Supersaturations calculated ranged from approximately 0.4 to 1% in non-nucleation regions to slightly over 2% in nucleation regions for a non-coalescence run in which Warner's nuclei distribution was assumed. Some calculations with coalescence included resulted in unrealistically high values of supersaturation, which were caused by the model's inability to replenish scavenged droplets fast enough. The results indicate not only that breakup will have to be included (as is physically clear), but that the nuclei resolution may have to be extended to relatively high values of critical supersaturation to account for droplet replenishment in the presence of scavenging raindrops.

## Abstract

Two cumulus cloud models in two space dimensions are presented; one is a bulk physical model without microphysics and the other includes the microphysical processes such as nucleation, diffusional growth of cloud droplets, stochastic coalescence, and fallout of raindrops. Some numerical aspects of the models are discussed; in particular, a method is described which allows long time steps to be used for the calculation of condensation in regions of relatively old droplet populations. The bulk-physical and microphysical models are compared for a non-precipitation case. The numerical results revealed that the inclusion of microphysics had little effect on the whole cloud dynamics, a result which considerably differs from that of Árnason and Greenfield.

The microphysical model produced hi-modal spectra of droplets through the interaction of a mid-level nucleation region, which corresponds to the base of an upper intense thermal, and the vertical advection and diffusion of existing droplets from a lower thermal. Supersaturations calculated ranged from approximately 0.4 to 1% in non-nucleation regions to slightly over 2% in nucleation regions for a non-coalescence run in which Warner's nuclei distribution was assumed. Some calculations with coalescence included resulted in unrealistically high values of supersaturation, which were caused by the model's inability to replenish scavenged droplets fast enough. The results indicate not only that breakup will have to be included (as is physically clear), but that the nuclei resolution may have to be extended to relatively high values of critical supersaturation to account for droplet replenishment in the presence of scavenging raindrops.

## Abstract

A technique for the treatment of the pressure in anelastic, nonhydrostatic terrain-following coordinates is described. It involves the use of two levels of pressure in such a manner so as to ensure that the anelastic mass-continuity equation is satisfied to round-off level. This procedure significantly improves model stability and accuracy. In the presence of modestly steep topography, the computational burden of the diagnostic elliptic pressure solver is equivalent to that of a direct solver. The two-level pressure approach is viewed as inappropriate for iterative schemes. A pressure truncation error analysis is described for calculating the second-order truncation error fields Γ associated with kinetic energy conservation for arbitrary formulations of the pressure gradient terms. The full transformed equation set is used, such that the combined effect of all of the equations contributing to the error is considered. Truncation error equations are derived for two specific formulations containing terms of *O*(Δ*x*
^{2}, Δ*y*
^{2}, Δ*z*
^{2}). These equations are used to validate a more general field analysis technique applicable for any numerical formulation. The kinetic energy errors that result specifically from the application of the two-level pressure technique are compared with the second-order Γ errors and are shown to be 5–10 times as small. Simulations show the stabilizing effect of the two-level pressure technique where comparisons between the two-level approach using a single block iteration and the same approach using a fully converged solution show negligible differences. The particular cases chosen were numerically unstable using a single block iteration without the two-level approach. The error analysis showed modest errors in the kinetic energy budget resulting from the numerical formulation of the pressure gradient terms with little difference between the formulations tested. The cases presented all had well-resolved topography.

## Abstract

A technique for the treatment of the pressure in anelastic, nonhydrostatic terrain-following coordinates is described. It involves the use of two levels of pressure in such a manner so as to ensure that the anelastic mass-continuity equation is satisfied to round-off level. This procedure significantly improves model stability and accuracy. In the presence of modestly steep topography, the computational burden of the diagnostic elliptic pressure solver is equivalent to that of a direct solver. The two-level pressure approach is viewed as inappropriate for iterative schemes. A pressure truncation error analysis is described for calculating the second-order truncation error fields Γ associated with kinetic energy conservation for arbitrary formulations of the pressure gradient terms. The full transformed equation set is used, such that the combined effect of all of the equations contributing to the error is considered. Truncation error equations are derived for two specific formulations containing terms of *O*(Δ*x*
^{2}, Δ*y*
^{2}, Δ*z*
^{2}). These equations are used to validate a more general field analysis technique applicable for any numerical formulation. The kinetic energy errors that result specifically from the application of the two-level pressure technique are compared with the second-order Γ errors and are shown to be 5–10 times as small. Simulations show the stabilizing effect of the two-level pressure technique where comparisons between the two-level approach using a single block iteration and the same approach using a fully converged solution show negligible differences. The particular cases chosen were numerically unstable using a single block iteration without the two-level approach. The error analysis showed modest errors in the kinetic energy budget resulting from the numerical formulation of the pressure gradient terms with little difference between the formulations tested. The cases presented all had well-resolved topography.

## Abstract

One-dimensional, kinematical microphysical cloud models are used to study two numerical aspects associated with modelling the initial microphysical stages of cloud growth in Eulerian spatial domain. First, the high-frequency oscillations in the spatially integrated nucleation rate and maximum supersaturation which became apparent in Clark’s model calculations are reproduced in the present paper and their cause and effect studied. Second, the spatial and radius resolution requirements for calculations of the initial phases of cloud development are studied. It is found that rather high spatial as well as radius resolution are required to obtain a reasonable degree of convergence for the solution of the droplet spectrum coefficient of dispersion for a case where an eddy mixing coefficient *K*=2 m^{2} sec^{−1} was used.

The effect of eddy mixing on droplet spectral broadening is investigated where adequate spatial and radius resolution are used. The results indicate that mixing has a rather strong effect on the coefficient of dispersion for the droplet spectrum. Arbitrary values of *K*=0, 1, 2 and 4 m^{2} sec^{−1} were Used where it was found that results similar to those of Warner were obtained for the *K*=0 case only.

The gamma distribution parameterization of Clark has been generalized to include condensation coefficient effects. The original parameterization scheme has been given a far more thorough comparison with a finite-difference model in one spatial dimension.

## Abstract

One-dimensional, kinematical microphysical cloud models are used to study two numerical aspects associated with modelling the initial microphysical stages of cloud growth in Eulerian spatial domain. First, the high-frequency oscillations in the spatially integrated nucleation rate and maximum supersaturation which became apparent in Clark’s model calculations are reproduced in the present paper and their cause and effect studied. Second, the spatial and radius resolution requirements for calculations of the initial phases of cloud development are studied. It is found that rather high spatial as well as radius resolution are required to obtain a reasonable degree of convergence for the solution of the droplet spectrum coefficient of dispersion for a case where an eddy mixing coefficient *K*=2 m^{2} sec^{−1} was used.

The effect of eddy mixing on droplet spectral broadening is investigated where adequate spatial and radius resolution are used. The results indicate that mixing has a rather strong effect on the coefficient of dispersion for the droplet spectrum. Arbitrary values of *K*=0, 1, 2 and 4 m^{2} sec^{−1} were Used where it was found that results similar to those of Warner were obtained for the *K*=0 case only.

The gamma distribution parameterization of Clark has been generalized to include condensation coefficient effects. The original parameterization scheme has been given a far more thorough comparison with a finite-difference model in one spatial dimension.

## Abstract

Simulations with a three-dimensional numerical cloud model are presented for airflow over a bell-shaped mountain and for a multicellular severe storm.

A comparison of results using the Orlanski (1976) and Klemp and Wilhelmson (1978) treatments for the normal velocities shows that physical modes can be computationally excited using the latter's treatment with the result of very large horizontally averaged vertical velocities.

Cell splitting occurs for the model calculations and the analysis indicates the splitting is caused by an entrainment effect which may be an artifact of the experimental design.

An analysis of subgrid/resolved scale kinetic energy shows that this ratio is much smaller for the current severe storm simulations than that found by Lipps (1977) for his trade wind cumuli simulations.

A comparison of some general features of the multicellular severe storm with observational data is presented.

## Abstract

Simulations with a three-dimensional numerical cloud model are presented for airflow over a bell-shaped mountain and for a multicellular severe storm.

A comparison of results using the Orlanski (1976) and Klemp and Wilhelmson (1978) treatments for the normal velocities shows that physical modes can be computationally excited using the latter's treatment with the result of very large horizontally averaged vertical velocities.

Cell splitting occurs for the model calculations and the analysis indicates the splitting is caused by an entrainment effect which may be an artifact of the experimental design.

An analysis of subgrid/resolved scale kinetic energy shows that this ratio is much smaller for the current severe storm simulations than that found by Lipps (1977) for his trade wind cumuli simulations.

A comparison of some general features of the multicellular severe storm with observational data is presented.

## Abstract

A relatively sophisticated cloud phase parameterization scheme based on the gamma distribution is presented which, it is hoped, will eventually make it possible for cloud modellers to include the effects of microphysics more realistically than has been so far possible.

Cloud phase calculations are presented using Lagrangian parcel theory, one-dimensional Eulerian formulation in the vertical, and two-dimensional Eulerian formulation in the horizontal and vertical directions. The solutions obtained using the parameterized scheme were compared with the more conventional finite-difference microphysical calculations of Clark and there was found to be very good agreement for all cases treated.

The efficiency of the scheme allowed a one-dimensional study on the effect of vertical spatial resolution on the prediction of microphysical parameters such as droplet number concentration, mean droplet radius and supersaturation. It was found that poor spatial resolution results in a rather slight under-estimation of the droplet number concentration.

## Abstract

A relatively sophisticated cloud phase parameterization scheme based on the gamma distribution is presented which, it is hoped, will eventually make it possible for cloud modellers to include the effects of microphysics more realistically than has been so far possible.

Cloud phase calculations are presented using Lagrangian parcel theory, one-dimensional Eulerian formulation in the vertical, and two-dimensional Eulerian formulation in the horizontal and vertical directions. The solutions obtained using the parameterized scheme were compared with the more conventional finite-difference microphysical calculations of Clark and there was found to be very good agreement for all cases treated.

The efficiency of the scheme allowed a one-dimensional study on the effect of vertical spatial resolution on the prediction of microphysical parameters such as droplet number concentration, mean droplet radius and supersaturation. It was found that poor spatial resolution results in a rather slight under-estimation of the droplet number concentration.

## Abstract

Numerical simulations of airflow over two different choices of mountainous terrain and the comparisons of results with aircraft observations are presented. Two wintertime casts for flow over Elk Mountain, Wyoming where surface heating is assumed to be zero and one case for airflow over Mt. Withington, New Mexico where surface heating is strong are considered.

In the Elk Mountain simulations the flow becomes approximately steady state since the upstream conditions are assumed to be constant and the surface heating is assumed to be zero. The response is significantly different in the two cases. In one case (dynamic Elk) strong lee waves formed with a horizontal separation of ∼10 km whereas in the second case (microphysical Elk) mainly weak untrapped waves formed with a vertical wavelength of ∼2.5 km. Because of the presence of the lee waves in the first case it is shown that the ridges south of Elk Mountain affect the flow near Elk Mountain. In the second case where there were no strong lee waves, the ridges to the south had very little effect on the flow near Elk Mountain so Elk acted as an isolated peak. The comparison between the simulation and the observations of the Elk Mountain experiments was good. In particular, the model's prediction of the location and intensity of trapped lee waves in the dynamic Elk case was good.

In the Mt. Withington simulations, the presunrise response was very weak though there were some weak lee waves. After sunrise, strong longitudinal rolls developed in the lower 1 km. These rolls were parallel to the mean wind direction in the lowest first kilometer and had an initial cross roll separation of 4–5 km for a mixed layer depth of 1.5 km. Later in the morning, after additional surface heating, the longitudinal rolls tended to increase their cross roll separation distance and to break up into a more cellular pattern although still retaining a well-defined roll structure. The ratio of cross roll separation to mixed layer depth was within the typically observed ratio of ∼2–3.

The overall comparison between the observations and the simulated flow fields in the Mt. Withington case was reasonable although detailed comparisons between individual features met with mixed success. The low-level observations appeared to represent cellular patterns as opposed to the simulated roll patterns although the horizontal scales perpendicular to the simulated rolls compared favorably. This difference in convective regime between the model and observations may be due in part to the very crude surface layer treatment of the model used to treat the unstable boundary layer as well as due to difficulties in choosing representative low-level winds. In the upper levels the comparison was successful in that the observations corroborate the presence of the trapped lee waves simulated by the model.

## Abstract

Numerical simulations of airflow over two different choices of mountainous terrain and the comparisons of results with aircraft observations are presented. Two wintertime casts for flow over Elk Mountain, Wyoming where surface heating is assumed to be zero and one case for airflow over Mt. Withington, New Mexico where surface heating is strong are considered.

In the Elk Mountain simulations the flow becomes approximately steady state since the upstream conditions are assumed to be constant and the surface heating is assumed to be zero. The response is significantly different in the two cases. In one case (dynamic Elk) strong lee waves formed with a horizontal separation of ∼10 km whereas in the second case (microphysical Elk) mainly weak untrapped waves formed with a vertical wavelength of ∼2.5 km. Because of the presence of the lee waves in the first case it is shown that the ridges south of Elk Mountain affect the flow near Elk Mountain. In the second case where there were no strong lee waves, the ridges to the south had very little effect on the flow near Elk Mountain so Elk acted as an isolated peak. The comparison between the simulation and the observations of the Elk Mountain experiments was good. In particular, the model's prediction of the location and intensity of trapped lee waves in the dynamic Elk case was good.

In the Mt. Withington simulations, the presunrise response was very weak though there were some weak lee waves. After sunrise, strong longitudinal rolls developed in the lower 1 km. These rolls were parallel to the mean wind direction in the lowest first kilometer and had an initial cross roll separation of 4–5 km for a mixed layer depth of 1.5 km. Later in the morning, after additional surface heating, the longitudinal rolls tended to increase their cross roll separation distance and to break up into a more cellular pattern although still retaining a well-defined roll structure. The ratio of cross roll separation to mixed layer depth was within the typically observed ratio of ∼2–3.

The overall comparison between the observations and the simulated flow fields in the Mt. Withington case was reasonable although detailed comparisons between individual features met with mixed success. The low-level observations appeared to represent cellular patterns as opposed to the simulated roll patterns although the horizontal scales perpendicular to the simulated rolls compared favorably. This difference in convective regime between the model and observations may be due in part to the very crude surface layer treatment of the model used to treat the unstable boundary layer as well as due to difficulties in choosing representative low-level winds. In the upper levels the comparison was successful in that the observations corroborate the presence of the trapped lee waves simulated by the model.

## Abstract

A numerical experiment on falling particles arranged in zones, with slab symmetry, constant air density, and initially still air is performed whereby single-shed particles are treated by a Lagrangian method and the air motion by an Eulerian method.

The results of this study of the zone's dynamics indicate that the zones fall considerably faster than their respective terminal velocity. This additional or convective velocity depends on loading and terminal speed. The spreading velocities, away from the axis of symmetry, appear to be dependent on terminal velocity such that a maximum occurs near the terminal velocity of 4 m sec^{−1}.

In all cases considered, the magnitude of the perturbation nonhydrostatic pressure gradient in the vertical is found to be an order of magnitude smaller than the perturbation hydrostatic pressure gradient.

## Abstract

A numerical experiment on falling particles arranged in zones, with slab symmetry, constant air density, and initially still air is performed whereby single-shed particles are treated by a Lagrangian method and the air motion by an Eulerian method.

The results of this study of the zone's dynamics indicate that the zones fall considerably faster than their respective terminal velocity. This additional or convective velocity depends on loading and terminal speed. The spreading velocities, away from the axis of symmetry, appear to be dependent on terminal velocity such that a maximum occurs near the terminal velocity of 4 m sec^{−1}.

In all cases considered, the magnitude of the perturbation nonhydrostatic pressure gradient in the vertical is found to be an order of magnitude smaller than the perturbation hydrostatic pressure gradient.

## Abstract

Deep cumulus dynamics has often been treated as an initial value problem where the long time effect of surface energy fluxes are neglected. Initiation is often assumed to follow from a strong localized deformation of the flow field, which is elsewhere quiescent. In nature, however, the atmosphere is rarely found in an undisturbed condition just prior to the inception of deep growth. One likely cause of widespread motions is the natural modal response of the environment to surface energy fluxes which results in a field of disturbances. Evidence is presented in this paper for the possible existence of a class of solutions when deep convection is allowed to evolve in the context of a thermally forced field of shallow convection. This class of solutions is neglected when one visualizes the growth of severe local storms in term of buoyant bubbles in an otherwise tranquil atmosphere. Considering deep cumulus initiation as a field problem severely limits the concept of an isolated cloud. Individual clouds may owe much of their structure to the existence of, and interaction with, the field of thermally forced deep normal modes. The importance of the local forcing terms is demonstrated here through a numerical simulation of the evolution of deep and severe convection out of a locally forced shallow cloud field in the absence of large scale forcing.

When convection is initiated over the entire domain locally through thermal forcing at the ground, the modal solution first observed corresponds to the Rayleigh solution which consist of modes confined to the boundary-layer. However, solutions in the deep linear equations show that a second modal solution also exists. The dominance of this solution, which consists of deep modes of longer horizontal wavelength, is shown here to lead to deep convection.

While the contribution of local forcing terms to the overall energy budget may be negligible at the severe convective stage, the mechanism of initiation appears to influence the pattern of evolution even into the mature stage. At the stage of shallow cumulus, the well-known phenomenon of upshear cumulus development is observed. As clouds grow deeper, an interesting phenomenon of phase-decoupled modal solutions is observed:. the growth of clouds appears to decouple the boundary-layer horizontal motions in phase from the stable layer motions, an effect that cyclically enhances and suppresses cloud growth. A characteristic time may be computed, and average cloud longevity inferred. Finally; the interaction of a moving storm system in its severe stage with the boundary-layer modes appears to provide one explanation for the spatial and temporal distribution of new convective cells in a multicellular storm system.

## Abstract

Deep cumulus dynamics has often been treated as an initial value problem where the long time effect of surface energy fluxes are neglected. Initiation is often assumed to follow from a strong localized deformation of the flow field, which is elsewhere quiescent. In nature, however, the atmosphere is rarely found in an undisturbed condition just prior to the inception of deep growth. One likely cause of widespread motions is the natural modal response of the environment to surface energy fluxes which results in a field of disturbances. Evidence is presented in this paper for the possible existence of a class of solutions when deep convection is allowed to evolve in the context of a thermally forced field of shallow convection. This class of solutions is neglected when one visualizes the growth of severe local storms in term of buoyant bubbles in an otherwise tranquil atmosphere. Considering deep cumulus initiation as a field problem severely limits the concept of an isolated cloud. Individual clouds may owe much of their structure to the existence of, and interaction with, the field of thermally forced deep normal modes. The importance of the local forcing terms is demonstrated here through a numerical simulation of the evolution of deep and severe convection out of a locally forced shallow cloud field in the absence of large scale forcing.

When convection is initiated over the entire domain locally through thermal forcing at the ground, the modal solution first observed corresponds to the Rayleigh solution which consist of modes confined to the boundary-layer. However, solutions in the deep linear equations show that a second modal solution also exists. The dominance of this solution, which consists of deep modes of longer horizontal wavelength, is shown here to lead to deep convection.

While the contribution of local forcing terms to the overall energy budget may be negligible at the severe convective stage, the mechanism of initiation appears to influence the pattern of evolution even into the mature stage. At the stage of shallow cumulus, the well-known phenomenon of upshear cumulus development is observed. As clouds grow deeper, an interesting phenomenon of phase-decoupled modal solutions is observed:. the growth of clouds appears to decouple the boundary-layer horizontal motions in phase from the stable layer motions, an effect that cyclically enhances and suppresses cloud growth. A characteristic time may be computed, and average cloud longevity inferred. Finally; the interaction of a moving storm system in its severe stage with the boundary-layer modes appears to provide one explanation for the spatial and temporal distribution of new convective cells in a multicellular storm system.

## Abstract

Numerical simulations of stochastic coalescence in a parcel framework are presented using a series of distribution functions. The equations governing the distribution parameter tendencies are derived using a variational approach with constraints. Solutions with two and three log-normal distribution functions are compared with a conventional benchmark model and the distribution model is shown to produce accurate solutions. Although only coalescence is considered within this paper, the procedures for including further physical processes is discussed. All of the simulations presented use the log-normal distribution although the method is general enough that it could be adapted to use other distributions such as the gamma distribution.

A decrease in the number of dependent variables by as much as by a factor of 10 as well as an equivalent reduction in computation time required for the treatment of the coalescence equation makes the distribution model attractive for multi-dimensional cloud model simulations. Further research in the direction of extending the distribution model for such purposes is currently in progress.

## Abstract

Numerical simulations of stochastic coalescence in a parcel framework are presented using a series of distribution functions. The equations governing the distribution parameter tendencies are derived using a variational approach with constraints. Solutions with two and three log-normal distribution functions are compared with a conventional benchmark model and the distribution model is shown to produce accurate solutions. Although only coalescence is considered within this paper, the procedures for including further physical processes is discussed. All of the simulations presented use the log-normal distribution although the method is general enough that it could be adapted to use other distributions such as the gamma distribution.

A decrease in the number of dependent variables by as much as by a factor of 10 as well as an equivalent reduction in computation time required for the treatment of the coalescence equation makes the distribution model attractive for multi-dimensional cloud model simulations. Further research in the direction of extending the distribution model for such purposes is currently in progress.