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Terry L. Clark
and
W. D. Hall

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.

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Gary P. Klaassen
and
Terry L. Clark

Abstract

We employ a two-dimensional numerical model with interacting nested domains to simulate the evolution of a small nonprecipitating cumulus cloud in the absence of shear. Grid nesting permits the use of a realistic boundary layer forcing to initiate cloud growth and, at the same time, the specification of very high spatial resolution in the vicinity of the cloud. The finest mesh employed in this study (5 m) gives about 160 points across the base of the cloud. Initially, the model produces a cloud which has a smooth upper surface. About eight to nine minutes after the onset of condensation, nodes appear on the upper cloud boundary. These nodes have a characteristic tangential length scale which is small compared to the width of the cloud base. In one of our simulations, a down-draft forms above the center of the cloud top and penetrates into the interior of the cloud. The entrainment of this unsaturated air reduces the liquid water content of the cloud below the adiabatic value and curtails growth of the cloud. In the present series of simulations, a penetrative downdraft is observed to form only in a cloud which develops a particular configuration of boundary nodes, a characteristic which is probably due to the assumed environmental conditions. Experiments were performed to assess the role which eddy mixing plays in the formation of the nodes and the entrainment process. It was found that while eddy mixing does not significantly affect the early nodal development, it does tend to inhibit the penetration of the downdraft. Our simulations indicate that entrainment in a growing cumulus is a well-ordered laminar phenomenon driven by inviscid dynamical processes rather than a turbulent phenomenon driven by mixing.

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Piotr K. Smolarkiewicz
and
Terry L. Clark

Abstract

A surface boundary layer model was developed which utilizes the single-level surface mesonet data and the results of a surface energy and moisture budget calculation. The heat and moisture fluxes calculated using this model were employed in the three-dimensional simulation of a cumulus cloud field. The analytical treatment of the surface layer represents a significant computational advantage.

The cloud field simulation represents seven hours of the 20 June 1981 case of the Cooperative Convective Precipitation Experiment (CCOPE) conducted in Montana. The model results are compared .with the available radar and aircraft data, with the record of hourly surface observations and with the available cloud photographs.

The results of the numerical experiments suggest that dynamical inhomogeneities imposed on the flow by the terrain plays a leading role in the rate of development of the convective cloud field. The thermodynamical inhomogeneities generated by such things as the type of soil and vegetation coverage are equally important in the early stage of the cloud field formation, while later they primarily affect the local properties of the cloud field, i.e., temporal and spatial location of single clouds For this case the global characteristics of the cloud field, i.e., geometrical structure of the field, cloud size distribution, and cloud sky coverage are mostly determined by the surface orography rather than by the thermodynamics of the surface layer.

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Jean-Luc Redelsperger
and
Terry L. Clark

Abstract

Two- and three-dimensional numerical simulations were performed to investigate the scale selection and initiation of both moist and dry convection over gentle western and gentle eastern slopes where the latter represents an idealization of the eastern Colorado region of the Great Plains of North America. This work extends earlier studies of thermally forced convection by considering a model framework that is large enough to resolve both the convective scale dynamics as well as the larger scale dynamics within which the convection is embedded. As a result, the scale interaction problem leading to the selection of the dominant deep modes of the troposphere and consequent convection initiation is more realistically treated. The main physical mechanisms involved in the initiation of convection in these studies are the usual boundary-layer instabilities leading to the development of eddies and/or shear-aligned rolls, the excitation of gravity waves by the boundary-layer motions interacting with the free atmosphere, and the eventual development of coherent vertical structures that link the boundary layer motions and the overlying gravity waves into larger horizontally spaced modes than typically obtained from an isolated boundary layer.

It has previously been shown that the mean wind shear spanning the region between the top of the boundary layer and the overlying stable layer plays an important role in producing energetic deep modes in the presence of thermal forcing. In the present simulation this shear results from a combination of initial baroclinicity associated with the westerlies and production by the differential thermal gradients formed by heating gently sloping terrain. Westerly geostrophic shears of either 3 or 5 m s−1 km−1 over the first 5.5 km above sea level were used as initial conditions. A balance is maintained between shear production through large-scale forcing and shear destruction through boundary-layer mixing that results in significant shear. The experiments showed a broad range of responses as a consequence of the horizontal variability of the shear structures. The preferred region of both dry and moist convection was found to be the eastern slope where the terrain effects result in an enhancement of the low-level shear. In response to the directional structure of the shear spanning the boundary layer and free atmosphere both a banded and a less coherent scattered organization were obtained for the waves and clouds.

Dominant deep modes were found to organize and initiate moist convection. West–east horizontal scales of the deep modes in the dry experiments were found to range from about 11 to 28 km with either a banded or a cellular structure with scales between 4 to 6 km in the south–north direction. The timing of the onset of the moist convection appeared to affect the final horizontal-scale selection in the moist experiments. The moist convection appears to lock onto the scales of the dry modes that initiate the convection for these particular experiments. The largest horizontal scales of dominant modes in the dry experiments were about 28 km and developed rather slowly as compared with the 11 km scale dominant modes. These largest horizontal scales did not develop in the moist experiments where clouds appeared early but did develop in those moist experiments where moist convection took longer to develop.

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Wojciech W. Grabowski
and
Terry L. Clark

Abstract

High resolution two-dimensional numerical experiments of rising thermals in a stably stratified environment were performed to study the cloud boundary instability. Unstable modes develop on the leading edge of the rising thermal, which are driven by the buoyant production of vorticity and lead to the type of entraining eddies that are thought to be responsible for observed dilution of convective clouds. These instabilities develop on the complex and evolving base state characterized by a nonparallel flow near the interface with a contractional component across the interface and a stretching component along it.

An analytical model is presented which describes the temporal evolution of the shear layer prior to the onset of the instability. It is shown that the flow pattern associated with the thermal rise leads to an exponential increase of the shear normal to the interface and exponential decrease of the shear-layer depth, which at a certain stage can lead to the onset of shearing instabilities. The theoretical predictions are in good agreement with the numerical simulation results. A shearing velocity is found from this theory which is the product of the shear-layer vorticity and the shear-layer depth. This shearing velocity is independent of the diffusional mixing and represents at least one attractive parameter for field testing of the theoretical model.

Once the shear layer collapses to a depth of about 40 m, instabilities are typically excited with characteristic scales between 100 and 200 m and exponential growth rates of about 40 s. The Richardson number at the upper-thermal interface is negative and both buoyant, and shear terms contribute to the kinetic energy of the instability. The scale selection and growth rates are in rough agreement with those for classical shearing instability. While growing, the instabilities migrate sideways along the interface, increasing their tangential scale. The size of the eddies into which instabilities finally develop depends not only on the scale of initial excitation, but also on the growth rate, thermal size, further evolution of the shear layer (which may allow finer-scale instabilities to be excited), and interaction of instabilities excited at different times. The spectrum of eddy sizes observed in the simulations ranged from about 50 to about 250 m. These findings provide further evidence of cumulus entrainment being driven by an inviscid baroclinic process.

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Terry L. Clark
and
Frank B. Lipps

Abstract

No abstract available.

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Terry L. Clark
and
W. D. Hall

Abstract

A three-dimensional numerical model is used to study the effect of small-scale supersaturation fluctuations on the evolving droplet distribution in the first 150 m above cloud base. The primary purpose of this research is to determine whether the irreversible coupling between the thermodynamics and dynamics due to finite phase relaxation time scales τs is sufficient to produce significant small-scale horizontal variations in supersaturation. Thus, the paper is concerned only with this internal source for thermodynamic variability. All other source terms, such as the downgradient flux of the variance of thermodynamic fields, have purposely been neglected.

Lagrangian particle experiments were run in parallel with the basic Eulerian model. The purpose of these experiments is to relax some of the microphysical parameterization assumptions with respect to assumed distribution shape and as a result add credibility to the results of distribution broadening.

Model results of five cases are presented, representing the cloud condensation nuclei characteristics of typical continental and maritime cumulus with mean dissipation rate of −100 cm2 s−3. The results show that for a maritime case of N≈100 cm−3 and =0.5 m s−1 the standard deviation of the supersaturation is as large as its horizontal mean. The horizontal variability of all thermodynamic fields is shown to increase significantly with τs. The droplet broadening response to this irreversible coupling effect is found to be significant for the larger values of τs in the Eulerian experiments. The Lagrangian particle experiments showed a somewhat reduced but still significant effect.

Although the experiments do show a broadening effect caused by finite values of τs, in no case were we able to show a continual increase in distribution broadening with height as reported from cumulus observations.

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Terry L. Clark
and
William D. Hall

Abstract

This note describes how to generate vertically stretched grids within the context of vertical nesting that are consistent with the conservative interpolation formula used by Clark and Farley. It is shown that all nested grids derive their structure directly from the parent grid, where the only flexibility allowed for nested grids is their grid ratio relative to the parent grid. Formulas are presented that can he used to analyze resulting nested grid structures, and an example showing how these formulas were used to generate relatively smooth inner meshes is described. Suggestions for further improvements in grid design are also provided.

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Wojciech W. Grabowski
and
Terry L. Clark

Abstract

Three-dimensional numerical experiments were performed with thermals rising in a stably stratified environment to study the cloud-environment boundary instability. This work extends that reported in Part I. It is shown that the analytical theory developed in Part I, which describes the evolution of the laminar interface between the thermal and its environment, applies to the three-dimensional case with only minor modifications. As in the two-dimensional case, the scale selection and growth rate of the unstable modes appear to depend upon the depth and velocity change across the shear layer near the interface, which is in rough agreement with classical linear theory developed for the case of planar geometry.

Analysis is presented that indicates further evolution of the three-dimensional eddies results in a transition to turbulence. A decrease of the Taylor-microscale Reynolds number and leveling off of the average enstrophy and velocity-derivative skewness is observed in the numerical experiments, which is typical for the development of numerical isotropic homogeneous turbulence. This transition is also associated with an increase (from about 0.5 to about 2) in the ratio between the vortex stretching and baroclinic production term of the enstrophy equation, with the magnitude of the stretching term approaching a value close to that for isotropic homogeneous turbulence.

Implications for the problem of cumulus entrainment are discussed. A heuristic argument based on the results of this study is given to explain why entrainment in cumuli and in high Reynolds number laboratory thermals is associated with the presence of large structures, not much smaller than the size of a cloud or thermal.

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Wojciech W. Grabowski
and
Terry L. Clark

Abstract

The direct effect of vertical shear of the horizontal wind for the unperturbed environment on the cloud-environment interface instability is investigated. Results indicate that the direct influence of environmental shear typical of atmospheric magnitudes is negligible. This is explained as a result of the large difference between typical magnitudes of environmental shear (usually smaller than 10−2 s−1) and the magnitude of baroclinically generated interfacial shear (typically around 10−1 s−1 for small cumuli).

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