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W. R. Peltier and T. L. Clark

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K. S. Gage and W. L. Clark

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Sensitive Doppler radars have recently been developed that can routinely observe wind in the free atmosphere up to stratospheric heights. One of these, the 40 MHz Sunset radar, was used to observe the three-dimensional wind field associated with a polar front jet stream near Boulder on 15–16 April 1976. The south wind, which was the strongest component (of the wind), was sampled about once a minute for over 14 h over altitudes ranging from 5 to 13 km MSL at 1 km intervals. The temporal variability of the south wind at each height is presented along with the average variability for all heights over the 14 h period. The average variability closely follows a 1/3 power law at least out to 4 h lag time. Since the 1/3 power law is consistent with inertial range turbulence theory, and since turbulence cannot possibly be three-dimensionally isotropic on these scales, it is suggested that the observations might be interpreted as evidence for a two-dimensional, −5/3 inertial range.

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Wojciech W. Grabowski and Terry L. Clark

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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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Terry L. Clark and W. D. Hall

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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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Wojciech W. Grabowski and Terry L. Clark

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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 W. D. Hall

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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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W. R. Peltier and T. L. Clark

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The transient evolution of internal waves which are forced by the flow of stably stratified fluid over two-dimensional topography exhibits several pronounced nonlinear effects for geophysically relevant values of the governing parameters. For homogeneous flows in which the internal Froude number is constant, the importance of nonlinearity is determined by the aspect ratio of the topography and the flow in the steady-state regime is as predicted by Long's model. When the background flow is inhomogeneous, Long's model no longer applies and new nonlinear effects may occur. One example of such an effect is the marked increase in the efficiency with which resonant lee waves are excited beyond the linear efficiency. A second example concerns the possibility of the trapping and subsequent amplification of the internal wave beneath its own level of supercritical steepening. The latter process appears to be important in understanding the strong downslope windstorm which occurred at Boulder, Colorado, on 11 January 1972.

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Wojciech W. Grabowski and Terry L. Clark

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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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T. L. Clark and W. R. Peltier

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We describe a series of fixed Froude number numerical simulations of the generation of internal gravity waves by the flow of stably stratified fluid over an isolated obstacle. Upstream of the obstacle the parallel flow is shear free and the Brunt-Väisälä frequency is independent of height. Under these conditions the nonhydrostatic model which we employ does not support resonance modes. In this model the nonlinear lower boundary condition is treated via a general tensor transformation which maps the domain with an irregular lower boundary into a rectangle. We explore the characteristics of the wave field as a function of the aspect ratio of the topography and show that there exists a critical aspect ratio which, if exceeded, results in the generation of internal waves which are subject to a local convective instability. In the long time limit we compare the numerically determined wave drag, the vertical profile of Reynolds stress and the downslope wind amplification to the corresponding predictions of linear steady-state theory. In the limit of small aspect ratio the analytic and numerical results coincide; in particular the Eliassen-Palm theorem is recovered. In the unstable regime the drag on the obstacle increases drastically, the strength of the downslope flow is enhanced and the vertical profile of Reynolds stress is strongly divergent. We discuss the implications of these results to the understanding of certain characteristics of mountain waves in the atmosphere.

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T. L. Clark and W. R. Peltier

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We examine the evolution of a field of internal waves launched by stratified flow over symmetric topography in mean flows which reverse direction at some height above the surface. With the gradient Richardson number at this “critical level” in the undisturbed flow restricted to values greater than 0.25, the nonlinear interaction in the region is such that the surface strongly reflects large amplitude internal waves incident upon it. When the critical level is located near certain discrete heights above the ground the incident and reflected waves interfere constructively and the wave amplitude in the low levels is resonantly enhanced by a large factor. These results are related to our previous analyses of the process by which breaking internal waves are able to induce intense downslope windstorms.

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