Let Ω be some domain in C symmetric with respect to the real axis and such that Ω ∩ R = ∅ and the intersections of Ω with the upper and lower open half-planes are simply connected. We study the class of piecewise meromorphic R-symmetric operator functions G in Ω \ R such that for any subdomain Ω ′ of Ω with Ω ′ ⊂ Ω, G restricted to Ω ′ can be written as a sum of a definitizable and a (in Ω ′ ) holomorphic operator function. As in the case of a definitizable operator function, for such a function G we define intervals ∆ ⊂ R∩Ω of positive and negative type as well as some "local" inner products associated with intervals ∆ ⊂ R ∩ Ω.Representations of G with the help of linear operators and relations are studied, and it is proved that there is a representing locally definitizable selfadjoint relation A in a Krein space which locally exactly reflects the sign properties of G: The ranks of positivity and negativity of the spectral subspaces of A coincide with the numbers of positive and negative squares of the "local" inner products corresponding to G.

SUMMARYThe forces on two moving spheres are found from the Stokes equations of motion, full allowance being made for the mutual interference of the flows round the spheres, and the relative trajectories when they are falling under gravity are calculated so that the collision efficiency E can be found. The values of E are determined for drops of radius up to 30 p colliding with droplets of any smaller size. The results differ widely from those found by Pearcey and Hill (1957) and it is suggested that the reason for the discrepancy is that the motion of the spheres when they are close together is of great importance in determining E.The results also differ from those found by Schotland and Kaplin (1956). Their experiments did not model the ratio of the density of the spheres to that of the medium through which they are falling and since it is shown that this factor is of importance, their results are not considered applicable to water drops falling in air.

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Any population of cloud droplets forming on polydisperse condensation nuclei is thermodynamically unstable. There is no value of the supersaturation for which the growth rate of all the droplets is zero, so that if some droplets are in equilibrium, then some must have positive and some negative growth rates. Droplets with positive growth rates will continue to grow at the expense of those with negative growth rates. This effect has been termed the ripening process, and has been postulated to be a potential mechanism to explain broad droplet size distributions in stratiform clouds. In this paper multiple parcel trajectories are used, derived using a simple representation of the turbulent dynamics, to examine the time evolution of the droplet size distribution in a nonentraining stratiform cloud. It is shown that the magnitude of the effect is critically dependent upon the mean parcel in-cloud residence time. The simulations suggest that, for a stratiform clouds of h ϭ 400 m thickness, and a vertical wind standard deviation of w ϭ 0.6 m s Ϫ1 (typical for stratocumulus clouds in a fairly vigorous, well-mixed boundary layer), the ripening effect is negligible, in that the droplet size distribution changes little with time. However, clouds with low w ϭ 0.2 m s Ϫ1 (typical of weaker stratus clouds) show a marked spectral ripening effect over a period of several hours. Ripening is observed in the numerical model in both clean and polluted aerosol distributions. Autoconversion rates calculated from the droplet size distributions increase markedly with time as ripening takes place. It is suggested that to accurately model droplet size distributions in stratus cloud, it may be necessary to take into account the distribution of in-cloud parcel residence time.

From theoretical, numerical and experimental studies of small inertial particles with density equal to β (>1) times that of the fluid, it is shown that such particles are ‘centrifuged’ out of vortices and eddies in turbulence. Thus, in the presence of gravitational acceleration g , their average sedimentation velocity V T in a size range just below a critical radius a cr is increased significantly by up to about 80%. We show that in fully developed turbulence, a cr is determined by the circulation Γ k of the smallest Kolmogorov micro-scale eddies, but is approximately independent of the rate of turbulent energy dissipation ϵ , because Γ k is about equal to the kinematic viscosity ν . It is shown that a cr varies approximately like and is about 20 μm (±2 μm) for water droplets in most types of cloud. New calculations are presented to show how this phenomena causes higher collision rates between these ‘large’ droplets and those that are smaller than a cr , leading to rapid growth rates of droplets above this critical radius. Calculations of the resulting droplet size spectra in cloud turbulence are in good agreement with experimental data. The analysis, which explains why cloud droplets can grow rapidly from 20 to 80 μm irrespective of the level of cloud turbulence is also applicable where a cr ∼1 mm for typical sand/mud particles. This mechanism, associated with unequal droplet/particle sizes is not dependant on higher particle concentration around vortices and the results differ quantitatively and physically from theories based on this hypothesis.