The coalescence of water drops II. The coalescence problem and coalescence efficiency

By the use of the model of approaching drops (Arbel and Levin, 1977) the coalescence efficiencies of drops are computed. It is found that for interactions of drops at their terminal velocities the coalescence depends both on the size of the large drop and on the size ratio of the interacting drops in agreement with the experimental results of Whelpdale and List (1971) and Levin and Machnes (1977).The results were found to be sensitive to the assumption of the drops deformation and to the critical separation distance. This distance is defined as the distance at which the drops begin to merge. The variations of the coalescence efficiency with these parameters is discussed.Appendix: List of symbols D distance between the deformed surfaces of the drops - D _o initial value ofD - D _s stop distance, the distance at which the impact velocity vanishes - D _c critical coalescence distance - E collection efficiency - E ₁ collision efficiency - E ₂ coalescence efficiency - E _2R coalescence efficiency for collisions with stationary targets - F _c centrifugal force - p ratio of the radii of the interacting drops - r _o initial distance between drops' centers - R _L radius of larger drop - R _s radius of smaller drop - R _D radius of deformation - v approach velocity of two deformed surfaces - v _o initial value ofv - V _i impact velocity (given negative sign when drops approach each other) - V _c critical impact velocity - W _i velocity of the smaller drop at infinity for it to reachD _o with velocityv _o - x _i impact distance, the distance between the trajectories of the two drops - x _c critical impact distance for coalescence -  average critical impact distance for coalescence - X _c critical impact distance for collisions - coefficient of deformation given in equation 1 - _i impact angle defined byWhelpdale andList (1971) given also inArbel andLevin (1977) - coefficient of deformation given in equation 2 - viscosity of air - _i impact angle used inArbel andLevin (1977) and here - _c critical angle for coalescence - average critical angle for coalescence On sabbatical leave (1976–77) from the Department of Geophysics and Planetary Sciences, Tel Aviv University, Ramat Aviv, Israel.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

The coalescence of water drops I. A theoretical model of approaching drops

The approach of two water drops in the absence of air flow around them is theoretically investigated. By assuming deformation criteria it is possible to solve the equation of motion of the drops under the influence of a variety of forces. These forces include the viscous force exerted by the air between the two deformed surfaces, the London-Van Der Waals forces and the force of gravity. It is found that the viscous forces dominate over the whole distance of the interaction. The equations have analytical solutions when a head-on approach is considered and when the deformation of the drops is assumed constant during the interaction. The equations were solved numerically for other deformation criteria and for non head-on approaches.The results of the present model are used in the following paper to compute the coalescence efficiencies of water drops. The model is primarily applicable to situations in which the large drop is stationary and the small one approaches it from below. However, it could also be used for interaction between freely falling drops as long as their relative velocities exceed about 13 cm/sec.Appendix: List of symbols C constant of the motion - D distance between the deformed surfaces of the drops - D _o initial value ofD - D _m the value at which the viscous force is maximum - D _N normalized distance - D _s the distance at which the velocity of approach vanishes - F _c centrifugal force - F _g force due to gravity - F _N normalized viscous force - F _LV force due to London-Van der Waals effect - F _R radial component of the force - F _V viscous force - F _t tangential component of the force - g acceleration due to gravity - M _L mass of large drop - m _s mass of small drop - p ratio of radii of interacting drops - R radius of an arbitrary drop - r distance between the centers of mass of the two drops - R _D radius of deformation - R _L radius of larger drop - R _s radius of smaller drop - t time - u defined in equation 20 — has the meaning of kinetic energy - v relative velocity of the deformed surfaces - v ₀ initial value ofv - V ₀ initial relative velocity of the centers of the drops - V _c critical impact velocity - V _i impact velocity - V _N ,v _n normalized velocity - V _t tangential component of the velocity - W _i velocity of the small drop at infinity for it to reach the pointD ₀ at velocityV ₀ - x instantaneous impact distance -  average critical impact distance for coalescence - x ₀ initial value of the impact distance - x _c critical impact distance for coalescence - coefficient of deformation - _i impact angle according toWhelpdale andList (1971) - coefficient of deformation - viscosity - surface tension - F _s sum of forces acting on the small drop - F _L sum of forces acting on the large drop - time constant - _R Rayleigh's oscillation period On sabbatical leave (1976–77) from the Department of Geophysics and Planetary Sciences, Tel Aviv University, Ramat Aviv, Israel.The National Center for Atmospheric Research is sponsored by the National Science Foundation.     

Characteristics of cloud drop spectra in tropical warm clouds

Characteristics of cloud drop spectra were studied using 400 samples obtained from 120 warm cumulus clouds formed during the summer monsoon season.The total concentration of cloud drops (N _T) varied from 384 to 884 cm^–3 and the maximum concentration was observed in the layer below the cloud-top. The width of the drop spectrum was broader in the cloud-base region and in the region below the cloud-top. The spectrum was multimodal at all levels except in the cloud-top region where it was unimodal. The concentration of drops with diameter greater than 50 m (N _L) varied from 0.0 to 0.674 cm^–3.N _L was larger in the cloud-base region.N _L decreased with height up to the middle level and thereafter showed an increase. In the cloud-top region no large drops were present. The computed values of the liquid water varied between 0.132 and 0.536 g m^–3 and the mean volume diameter (MVD) varied between 8.1 and 12.0 m. The LWC and MVD showed a decrease with height except in the middle region of the cloud where the values were higher than the adjacent levels. The dispersion of the cloud drops was lower (0.65) in the cloud-top region and higher (1.01) in the cloud-base region.The observed cloud microphysical characteristics were attributed to vertical mixing in clouds induced by the cloud-top gravity oscillations (buoyancy oscillations) generated by the intensification of turbulent eddies due to the buoyant production of energy by the microscale-fractional-condensation (MFC) in turbulent eddies.     

Explosive source and its influence on ground motion dynamics

Summary In adjusting measured values in sets A(r*), v(r*) and f(r*) by means of a power function in the form of P=Kr^* a region of discontinuity of the approximating curves was found at the distance r*11.5 m kg ^–1/3. It is assumed that this discontinuity was caused by the varying character of the source of seismic waves. For scaled distances r*>11.5 m kg ^–1/3 the explosion was considered to be a spherical source from the point of view of the charge geometry and of the distance of the pick-up from the centre of the charge, whereas if r*<11.5 m kg ^–1/3 the explosion in the borehole had the character of a cylindrical source. The difference of the two types of sources was reflected in the exponent with both the functions A(r*) and v(r*), so that for r*>11.5 m kg ^–1/3 –4.0 and–2.4, and for r*<11.5 m kg ^–1/3 –2.5 and–1.5. For the same intervals of scaled distance in the set f(r*)1.4 and1.2.     

On reflection and diffraction of pressure waves from floating ice

Summary The diffraction of a pressure wave arising from the edge of the floating ice due to an incident wave of the formH[t–(xcos/)+ (ysin/)], whereH(t) is the Heaviside step function, has been studied. The ice sheet has been taken to be moving with a velocity much less than the velocity of sound in the liquid. The problem has been solved using an approximate method of solving Wiener Hopf problems. Finally, the pressure field at a large distance from the edge has been derived in a closed form evaluating the integrals by the saddle point method.     

Tectonosedimentary Evolution of the Cheb Basin (NW Bohemia,Czech Republic) between Late Oligocene and Pliocene: A Preliminary Note

A working model of tectono-sedimentary evolution is proposed for the Cheb Basin, a polyhistory sedimentary basin formed between the late Oligocene and Pliocene by reactivation of basement fracture systems in the northwestern part of the Bohemian Massif. The basin is located at the intersection of the Ohe (Eger) Graben structural domain, characterized by dominance of NE-striking graben systems in present-day geology, and the NW-striking Cheb-Domalice Graben, a major strike-slip – dominated structure in Western Bohemia. The first significant depositional episode in the Cheb Basin coincides with the deposition of late Oligocene-Miocene clastics in the whole extensional system of the Ohe Graben, controlled by E-W – trending depocenters. The main structural feature of the Cheb Basin region at that time was a palaeohigh caused by a NW- trending accommodation zone separating minor E-W – trending depocentres. The second, late Pliocene, episode of sedimentation occurred under a very different kinematic regime than the Oligo-Miocene rift basin evolution. During this time, the present-day structure of the Cheb Basin and the Cheb-Domalice Graben formed as a consequence of sinistral displacement on the Mariánské Lázn Fault Zone. Reactivation of this strike-slip fault zone led to the formation of a horsetail splay of oblique-extensional faults at the northern termination of the Mariánské Lázn Fault Zone, which contained the present-day Cheb Basin.     

Cyclic variations of the initial height of the sporadicE-layer at middle geographical latitudes

Summary The variations of the initialh E _s height are investigated in the solar cycle 1957–1968, deriving the regressive dependency:h E _s =121.4–6·10^–2 R referring to the median monthly values at a solar zenith angle =75°. The similar variations ofh E _s (R) during the day and night are interpreted as a domination of the sporadic layer formation from a redistribution of the day-time ionization and secondary participation of nightly ionizing sources. The analogous cyclich E _s andh E variations confirm this conclusion while the seasonal variations in the state of the sporadic layer show outlined dynamical effects. The comparatively not big cyclic variation in the spatial state of theE-region are considered to confirm the predominating ionizing action of the ultraviolet range (933–1038 Å) in the lower part of theE-region, while the soft X-radiation influences mainly the near maximum part of this region.     

On the frequency distribution of interoccurrence times of earthquakes

The distributions of discrete frequency, N, versus interoccurrence time, t (in days), of M 7 earthquakes in the Taiwan region during the 1900–1994 period, M 6 earthquakes in the north-south seismic belt of China during the 1900–1990 period, and M 5.5 earthquakes in Southern California, U.S.A., during the 1914–1995 period are studied through two statistical models (gamma function and exponential function). Results show that both the exponential function and gamma function can describe the distributions. However, the former is more appropriate than the latter. This indicates that the three time series of earthquakes have a significant component of Poisson processes, even though the tectonic conditions, the fault distributions and the size of the three seismic regions are different.     

Determination of characteristics of steam reservoirs by Radon-222 measurements in geothermal fluids

The authors conducted a Rn²²² survey in wells of the Larderello geothermal field (Italy) and observed considerable variations in concentrations. Simple models show that flow-rate plays an important part in the Rn²²² content of each well, as it directly affects the fluid transit time in the reservoirs. Rn²²² has been sampled from two wells of the Serrazzano area during flow-rate drawdown tests. The apparent volume of the steam reservoir of each of these two wells has been estimated from the Rn²²² concentration versus flow-rate curves.List of symbols Q Flow-rate (kg h^–1) - Decay constant of Rn²²² (=7.553×10^–3 h^–1) - Porosity of the reservoir (volume of fluid/volume of rock) - ₁ Density of the fluid in the reservoir (kg m^–3) - ₂ Density of the rock in the reservoir (kg m^–3) - M Stationary mass of fluid filling the reservoir (kg). - E Emanating power of the rock in the reservoir (nCi kg _rock ^–1 h^–1). - P Production rate of Rn²²² in the reservoir: number of atoms of Rn²²² (divided by 1.764×10⁷) transferred by the rock to the mass unit of fluid per unit time (nCi kg _fluid ^–1 h^–1). - N Specific concentration of Rn²²² in the fluid (nCi kg^–1) - Characteristic time of the steam reservoir at maximum flow-rate (=M/Q)     

Effects of collision-induced breakup on drop size distributions in steady state rainshafts

New equations and techniques for dealing with drop breakups are developed and applied to the modelling of the evolution of raindrop spectra in rainshafts. Breakup experiments byMcTaggart-Cowan andList (1975) served as data base.No matter what the original size distribution, the spectrum evolution will always lead to a Marshall-Palmer type equilibrium di tributionN=N ₀e^–D, with =constant andN ₀ proportional to the rainfall rateR. (D stands for raindrop diameter.) ForR29 mm h^–1 and an original Marshall-Palmer distribution, the required fall height to reach equilibrium is 2 km.The equilibrium distributions are characterized by linear relationships betweenR, the radar reflectivity factorZ, the liquid water content LWC and theN ₀ of the Marshall-Palmer distribution. Possible explanations for the discrepancy with observations are given.The fact that the all-water processes cannot produce drops withD2.5 mm (as confirmed by observations) leads to the conclusion that observed large raindrops withD5 mm represent melted hailstones and have not yet reached an equilibrium distribution. These latter conclusions were reached within the original assumption of videspread, steady state precipitation.     

AuroralH-emission and interplanetary neutral hydrogen

Summary Auroral spectra taken from within the auroral zone reveal H-emission as not limited to the magnetic zenith but as extending over the sky. The surface brightness forH andH may tentatively be estimated to a few airglows. The assumption is made that some of this emission may extend to the very low gm latitudes and be practically universal and the possible mode of excitation of neutral hydrogen in space is speculated upon. At the earth's orbit the density number of the cosmic, infalling, hydrogen accreted from beyond the solar system towards the sun is about 10² neutral atoms cm^–3 and coronal plasma contributions cannot encrease this number by more than a factor of two. If an exceptionally violent solar disturbance ejects stream-electrons attaining velocities-of 2,200 km sec^–1 corresponding to the threshold energy forH excitation of 12.8 eV and if these electrons, at the earth's orbit, have a density number of 10³ cm^–3, then a shell about 100,000 km thick may contributeH emission having a surface brightness of about 0.1 airglow this being the higher probable value. Such anH-emission could be detectable from all latitudes during, or, most probably, just before the occurence of an auroral storm.     

Ion composition measurements and modelling at altitudes from 140 to 350 km using EISCAT measurements

This work aims at processing the data of CP1 and CP2 programs of EISCAT ionospheric radar from 1987 to 1994 using the –full profile method which allows to solve the –temperature-composition ambiguity problem in the lower F region. The program of data analysis was developed in the CEPHAG in 1995–1996. To improve this program, we implemented another analytical function to model the ion composition profile. This new function better reflects the real profile of the composition. Secondly, we chose the best method to select the initial conditions for the –full profile procedure. A statistical analysis of the results was made to obtain the averages of various parameters: electron concentration and temperature, ion temperature, composition and bulk velocity. The aim is to obtain models of the parameter behaviour defining the ion composition profiles: z₅₀ (transition altitude between atomic and molecular ions) and dz (width of the profile), for various seasons and for high and low solar activities. These models are then compared to other models. To explain the principal features of parameters z₅₀ and dz, we made an analysis of the processes leading to composition changes and related them to production and electron density profile. A new experimental model of ion composition is now available.     

Statistical analysis of earthquake activity at Etna volcano (March 1981 eruption)

Seismic activity that preceded, accompanied, and followed the 17–23 March 1981 Etnean eruption has been statistically analyzed.On the grounds of both time evolution of seismicity and catalogue completeness, three time intervals have been defined (12 February–2 March, 12–17 March, 19–31 March) and for each of these periods both the b coefficient of theGutenberg-Richter's (1956) relationship and the E parameter of the cluster size (Shlien andToksoz, 1970) have been calculated.No significant variations were observed between the first and third periods, while lower values of bothb andE coefficients were found in the second one. These findings might indicate that changes in the seismicity features occur just before the eruption start.Small but fast variations in the stress field acting on the volcano might originate this type of seismic activity, while the importance of the tectonic control on volcanic phenomena seems to be confirmed.     

A new aspect of condensation theory

This paper examines the effects of the mixing of dry air into a cloud top from the point of view of the droplet spectra. It is shown theoretically that the resulting cycling of the air up and down in the cloud, as seems to be the essential mechanism by which cumuli have been diluted to their observed liquid water mixing ratio, can double the largest drop radius and generate cloud parcels containing drops of all sizes up to this maximum. These changes in the droplet distribution with size occur by a process which is not greatly influenced by the cloud condensation nuclei or the details of droplet growth since maritime like spectra can develop in continental type cumuli. It shows that large numbers of cloud condensation nuclei should not have much effect in inhibiting the rainforming process by reducing coalescence growth. On the contrary, the controlling parameters which determine precipitation efficiency and times seem to be those which control the mixing.     

Source parameters of mining-induced seismic events: An evaluation of homogeneous and inhomogeneous faulting models for assessing damage potential

Source parameter estimates based on the homogeneous and inhomogeneous source models have been examined for an anomalous sequence of seven mine-induced events located between 640 and 825 m depth at Strathcona mine, Ontario, and having magnitudes ranging betweenm _N 0.8 and 2.7. The derived Brune static stress drops were found to be similar to those observed for natural earthquakes (30 bars), whereas dynamic stress drops were found to range up to 250–300 bars. Source radii derived from Madariaga's model better fit documented evidence of underground damage. These values of source radii were similar to those observed for the inhomogeneous model. The displacement at the source, based on the observed attenuation relationship, was about 60 mm for three magnitude 2.7 events. This is in agreement with slip values calculated using peak velocities and assuming the asperity as a Brune source within itself (72 mm). By using Madariaga's model for the asperity, the slip was over 3 times larger than observed. Peak velocity and acceleration scaling relations with magnitude were investigated by incorporating available South African data, appropriately reduced to Canadian geophysical conditions. The dynamic stress drop scaled as the square root of the seismic moment, similar to reported results in the literature for crustal earthquakes. This behavior suggests that the size of the asperities responsible for the peak ground motion, with respect to the overall source size, follow distributions that may be similar over a wide range of magnitudes. Measurements of source rupture complexity (ranging from 2 to 4) were found to agree with estimates of overall source to asperity radii, suggesting, together with the observed low rupture velocities (0.3 to 0.6 ), that the sources were somewhat complex. Validation of source model appropriateness was achieved by direct comparison of the predicted ground motion level to observed underground damage in Creighton mine, located within the same regional stress and geological regime as Strathcona mine. Close to the source (<100 m), corresponding to relatively higher damage levels, a good agreement was found between the predicted peak particle velocities for the inhomogeneous model and velocities derived based on established geomechanical relationships. The similarity between asperity radii and the regions of the highest observed damage provided additional support for the use of the inhomogeneous source model in the assessment of damage potential.     

Primary ionization coefficients for dry and moist air; secondary ionization coefficients for a water surface

Summary A comparison has been made between the Townsend primary ionization coefficient, , for dry air and for air with humidities typical of those in the atmosphere. is defined as the number of new electrons produced by an electron per centimeter of drift in a field. A range of field/pressure ratios,E/p ₀, of 40 to 100 V (cm torr)^–1 was employed. The variation of with humidity is very small.Over the range ofE/p ₀ from 50 to 100 V (cm torr)^–1, the secondary ionization coefficient, , of a water surface has been found from sparking potential data to be typically 2×10^–4. represents the fraction of primary ionizing collisions that ultimately result in the production of additional electrons at the surface.     

Multiple scattering of seismic waves in fractured media: Velocity and effective attenuation of the coherent components ofP waves andS waves

We show that the multiple scattering by small fractures of seismic waves with wavelengths long compared to the fracture size and fracture spacing is indistinguishable from multiple-scattering effects produced by regular porosity, except for an orientation factor due to fracture alignment. The fractures reduce theP-wave andS-wave velocities and produce an effective attenuation of the coherent component of the seismic waves. The attenuation corresponds to 1000/Q of about unity for a Gaussian spectrum of fractures, and it varies with frequencyf asf ³. For a Kolmogorov spectrum of fractures of spectral index the attenuation is an order of magnitude or so larger and varies with frequency asf ^3-v The precise degree of attenuation depends upon the matrix properties, the fracture porosity, the degree of fracture anisotropy, the type of fluid filling the fractures, and the incidence angle of the wave.For fracture porosities less than about 15% theP-wave andS-wave velocities are decreased by the order of 5–10% with a lesser dependence on the type of fluid filling the fractures (gas, oil, or brine) and with a dependence on both the degree of anisotropy and the incident angle made by the wave. The tendency of fractures to occur perpendicularly to bedding suggests that the best way to measure seismically fractured rock behavior in situ is by using the travel-time delay and reflection amplitude. As both the offset and the azimuth of receivers vary from a shot, the travel-time delay and reflection amplitude should both show an elliptical pattern of behavior—the travel-time delay in response to the varying seismic speed, and the reflection amplitude in response to angular variations in the multiple scattering. Observations of attenuation at several frequencies should permit (a) determination of the spectrum of fractures (Gaussian versus Kolmogorovian) and (b) determination of the contribution of viscous damping to the effective attenuation.     

The effects of sliding velocity on the frictional and physical properties of heated fault gouge

The frictional properties of a crushed granite gouge and of gouges rich in montmorillonite, illite, and serpentine minerals have been investigated at temperatures as high as 600°C, confining pressures as high as 2.5 kbar, a pore pressure of 30 bar, and sliding velocities of 4.8 and 4.8×10^–2 m/sec. The gouges showed nearly identical strength behaviors at the two sliding velocities; all four gouges, however, showed a greater tendency to stick-slip movement and somewhat higher stress drops in the experiments at 4.8×10^–2 m/sec. Varying the sliding velocity also had an effect on the mineral assemblages and deformation textures developed in the heated gouges. The principal mineralogical difference was that at 400°C and 1 kbar confining pressure a serpentine breakdown reaction occurred in the experiments at 4.8×10^–2 m/sec but not in those at 4.8 m/sec. The textures developed in the gouge layers were in part functions of the gouge type and the temperature, but changes in the sliding velocity affected, among other features, the degree of mineral deformation and the orientation of some fractures.     

Time pattern of large earthquake sequences in Europe in 1976–1979

Summary The time development of three large sequences, Friuli 1976, Thessaloniki 1978 and Monte Negro-Albania 1979, was investigated by analysing the characteristics of the time distribution of magnitudes of the events, M(t), of average magnitudes, (t), and of the number of events, n(t), as well as the strain release curves and the magnitude-frequency relations, N(M). The existence of time gaps and of an anomalous decay of the numbers of shocks before major events, as well as the oscillation of ( const b^–1) may be considered as precursory phenomena. The clustering of events has been tested by the Poisson and negative binomial laws.Communication presented at the XVIIth General Assembly of the European Seismological Commission in Budapest, 21–29 August, 1980 under the title Properties of Seismic Regime and Recent Earthquake Sequences in Europe.     

Unified approach to amplitude attenuation and coda excitation in the randomly inhomogeneous lithosphere

A unified model is proposed for explaining the frequency dependent amplitude attenuation and the coda wave excitation on the basis of the single scattering process in the randomly inhomogeneous lithosphere. Adopting Birch's law and a direct proportion between density and wave velocity, we statistically describe the inhomogeneous medium by one random function characterized by the von Karman autocorrelation function. We calculate the amplitude attenuation from the solid angle integral of scattered wave energy on the basis of the Born approxiimation after subtracting the travel-time fluctuation effect caused by slowly varying velocity inhomogeneities. This subtraction is equivalent to neglect energy loss by scattering within a cone around the forward direction. The random inhomogeneity of the von Karman autocorrelation function of order 0.35 with the mean square fractional fluctuation of 7.2×10^–3 1.3×10^–2 and the correlation distance of 2.15.1 km well explains observed backward scattering coefficientg and the ratioQ _P ^–1 /Q _S ^–1 , and observed and partially conjecturedQ _S ^–1 for frequencies between 0.5 Hz and 30 Hz.