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R. A. Brown

Abstract

The neutral Ekman boundary layer is known to be dynamically unstable to infinitesimal perturbations under typical geophysical conditions. This paper discusses this instability to two-dimensional, simple-harmonic perturbations, for the stratified Ekman layer.

While viscosity and Coriolis forces are generally important in setting up the basic mean profile, the inflection point instability can be investigated in the inviscid, non-rotating system limit. However, the singular nature of the resulting second-order characteristic equation makes it necessary to solve the non-singular sixth-order, viscous stratified equation. Since typically occurring Reynolds numbers are much larger than critical, emphasis has been placed on investigating the behavior of maximum growth rates versus stratification for large Re. The appropriate dimensionless parameters are found to be: ξ=(2/Ro Re)½, and Ra= gSδ4/KmmKh [where δ=(2K/ f)½, Re= V gδ/K m, Ro=Vg/fδ and S=( z+g/ cp)/] for the general case, or ξ and RI=gS/V z for the inviscid case.

Unstable stratification shifts maximum growth rates toward a longitudinal orientation and shorter wave-lengths from the neutral stratification values of leftward orientation angle, ε=17°, and wavenumber, α=0.5. The local Richardson number at the inflection point is found to he the pertinent parameter for the effects of stratification. This instability is damped completely for values of Rii>0.25. Unstable stratification tends to support the dynamic instability such that the growth rate for this mode is dominant significantly into the convective instability regime.

The instability takes the form of counter-rotating circular motions which remain qualitatively similar for a wide range of the basic variables.

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R. A. Brown

Abstract

The equations of motion for a neutrally buoyant fluid are solved to produce an equilibrium flow consisting of a modified Ekman spiral mean flow plus a helical secondary flow. By considering the secondary flow to be a finite perturbation on a mean large-scale flow, approximate equations are obtained for the secondary flow and the modified mean flow an functions of the large-scale parameters. When the energetics of this system are considered, an equilibrium magnitude for the secondary flow can be obtained.

The finite perturbations are assumed to preserve the structure of the infinitesimal perturbation solutions for the dynamic instability of the Ekman boundary layer. In particular, helical rolls occur as finite perturbation solutions. These finite disturbances are found to alter the mean Ekman velocity profile such that it becomes stable. The rolls, with characteristic depths of 5–7 times the Ekman characteristic length and corresponding wavelengths of 4π times this parameter, may be frequent occurrences in both atmospheric and oceanic boundary layers.

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A. R. Brown and N. Wood

Abstract

Numerical simulations are used to investigate the impact on the stable boundary layer of moderate topography (with hill heights in some cases comparable to the undisturbed boundary layer depth). Area-averaged properties of the resulting boundary layers, which are often highly inhomogeneous, are diagnosed. The presence of the hills leads to enhanced turbulence and drag, and a deepening of the area-averaged boundary layers (over and above that due to a simple displacement effect). The ability of well-established formulas for the depth of the boundary layer over homogeneous terrain to predict this deepening is investigated. Finally, the implications of the results for the use in large-scale weather and climate prediction models of effective roughness length parameterizations of the effects of hills are discussed. While not capturing some of the more detailed effects, the simplest approach of using a roughness length independent of stability is found to perform reasonably well in predicting the total surface drag.

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K. A. Browning and R. J. Donaldson

Abstract

The thunderstorm that produced tornadoes near Geary, Okla., on 4 May 1961 is analyzed using data mainly from a vertically-scanning FPS-6 radar. The storm configuration was remarkably similar in many respects to the severe Wokingham hailstorm in England, analyzed by Browning and Ludlam. Each attained a fairly steady state during which certain characteristic features were displayed. Most important of these was the vault, a region of low reflectivity beneath the highest parts of the storm which is believed to be symptomatic of an intense and persistent updraft. This and other features of its structure are analyzed in conjunction with nearby soundings to give a model of the airflow associated with the Geary storm. Like the Wokingham storm, it comprised a persistent updraft which entered and left on the downshear side. This kind of airflow is believed to be representative of an important class of persistently intense cumulonimbus which travel within a strongly sheared environment.

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P. J. Mason and A. R. Brown

Abstract

Large eddy simulations use a subgrid model, which is characterized by a length scale that is often related to the scale of the computational mesh by a numerical constant, C s. Mason and Callen argued that this subgrid model and its length scale define and impose the filter operation of the simulation. They saw C s as a measure of numerical accuracy. Others have sought to link the filter operation to the computational mesh and have viewed C s as needing determination for correct implementation. Here tests with a high resolution of 224 × 224 × 200 grid points are found to confirm Mason and Callen’s view. These simulations are also used together with lower-resolution simulations to illustrate the degree of convergence achieved. Some erroneous features of the simulations are identified through this test.

For the case of buoyant convection, the buoyancy dependence of the subgrid model is further examined. Most available subgrid models allow for buoyancy fluxes changing the level of the subgrid energy but only allow stable buoyancy gradients to modify the subgrid length scale—a reduction in this case. In contrast to most applications, it has been suggested that for a fixed filter operation, the subgrid length scale should always have a buoyancy dependence and should increase, in a finite way, with unstable buoyant transfer. Here an examination of spectral behavior in high-resolution simulations supports such an approach and shows that the model with the buoyancy-dependent length scale is indeed consistent with a fixed filter operation. The more conventional models are shown to have less satisfactory behavior.

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R. L. Peace Jr., R. A. Brown, and H. G. Camnitz

Abstract

A generalized relationship is derived which relates an arbitrary horizontal wind field and the Doppler velocity field that results from its observation with a single horizontally directed radar. The relationship shows clearly how various elements of the horizontal wind field (velocity components and their space derivatives) combine to account for Doppler velocity variations with azimuth and range. The relationship also provides a basis for development of observational techniques for interpretation of the horizontal motion field from single pulse Doppler radar measurements.

Two sample techniques are presented for interpreting Doppler measurements in terms of horizontal motion in echoes located some distance from the radar. The underlying assumption upon which both techniques are based is the quasi-stationarity of the motion field relative to a reference frame moving with the observed echo. The practicality of one of these techniques is demonstrated for a model wind field by computer simulation of the Doppler velocity field that would be observed by a single radar with representative measurement accuracy, and computer reconstruction of the original motion field from these simulated Doppler measurements. Subject to the validity of the underlying assumption, the results demonstrate the feasibility of determining the horizontal motion field in a remote echo from the observations of a single pulse Doppler radar. Means of testing the basic assumption are presented.

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A. R. Brown, M. K. MacVean, and P. J. Mason

Abstract

Large eddy simulations sometimes use monotone advection schemes. Such schemes are dissipative, and the effective subgrid model then becomes the combined effect of the intended model and of the numerical dissipation. The impacts on simulation reliability are examined for the cases of dry convective and neutral planetary boundary layers. In general it is found that the results in the well-resolved flow interior are insensitive to the details of the advection scheme. However, unsatisfactory results may be obtained if numerical dissipation dominates where the flow becomes less well resolved as the surface is approached.

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Benjamin M. Herman, Samuel R. Browning, and John A. Reagan

Abstract

It can be shown, theoretically, that the polarization properties of laser light scattered by a volume of air containing aerosols include considerable information as to the size distribution of the aerosols. A theoretical inversion model, utilizing the above information, is developed, which uses the Stokes parameters of the angularly scattered laser light as input data. These input data are generated theoretically from assumed size distribution functions of the aerosols. Both “perfect” measurements and measurements into which random errors are introduced are employed. These data are then used in the inversion model to generate predicted size distribution functions. Numerical experiments are performed with 0, 1 and 2% random error in the observations, in order to determine what accuracy is required in the lidar measurements. Comparisons between the actual and predicted functions are then made in order to assess the accuracy of the model.

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Benjamin C. Trabing, Michael M. Bell, and Bonnie R. Brown

Abstract

Potential intensity theory predicts that the upper-tropospheric temperature acts as an important constraint on tropical cyclone (TC) intensity. The physical mechanisms through which the upper troposphere impacts TC intensity and structure have not been fully explored, however, due in part to limited observations and the complex interactions between clouds, radiation, and TC dynamics. In this study, idealized Weather Research and Forecasting Model ensembles initialized with a combination of three different tropopause temperatures and with no radiation, longwave radiation only, and full diurnal radiation are used to examine the physical mechanisms in the TC–upper-tropospheric temperature relationship on weather time scales. Simulated TC intensity and structure are strongly sensitive to colder tropopause temperatures using only longwave radiation, but are less sensitive using full radiation and no radiation. Colder tropopause temperatures result in deeper convection and increased ice mass aloft in all cases, but are more intense only when radiation was included. Deeper convection leads to increased local longwave cooling rates but reduced top-of-the-atmosphere outgoing longwave radiation, such that the total radiative heat sink is reduced from a Carnot engine perspective in stronger storms. We hypothesize that a balanced response in the secondary circulation described by the Eliassen equation arises from upper-troposphere radiative cooling anomalies that lead to stronger tangential winds. The results of this study further suggest that radiation and cloud–radiative feedbacks have important impacts on weather time scales.

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J. A. Weinman, J. T. Twitty, S. R. Browning, and B. M. Herman

Abstract

The intensity of sunlight multiply scattered in model atmospheres is derived from the equation of radiative transfer by an analytical small-angle approximation. The approximate analytical solutions are compared to rigorous numerical solutions of the same problem. Results obtained from an aerosol-laden model atmosphere are presented. Agreement between the rigorous and the approximate solutions is found to be within a few percent.

The analytical solution to the problem which considers an aerosol-laden atmosphere is then inverted to yield a phase function which describes a single scattering event at small angles. The effect of noisy data on the derived phase function is discussed.

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