Water uptake and equilibrium sizes of aerosol particles at high relative humidities: Their dependence on the composition of the water-soluble material

Equilibrium water uptake and the sizes of atmospheric aerosol particles have for the first time been determined for high relative humidities, i.e., for humidities above 95 percent, as a function of the particles chemical composition. For that purpose a new treatment of the osmotic coefficient has been developed and experimentally confirmed. It is shown that the equilibrium water uptake and the equilibrium sizes of atmospheric aerosol particles at large relative humidities are significantly dependent on their chemical composition.List of symbols A proportionality factor - a _w activity of water in a solution - c _{p v} specific heat of water vapour at constant pressure - c _w specific heat of liquid water - f relative humidity - l _w specific heat of evaporation of water - M _i molar mass of solute speciesi - M _s mean molar mass of all the solute species in a solution - M _w molar mass of water - m ₀ mass of an aerosol particle in dry state - m _i mass of solute speciesi - m _s mass of solute - m _w mass of water taken up by an aerosol particle in equilibrium state - m total molality=number of mols of solute species in 1000 g of water - m _i molality of solute speciesi - m _k total molality of a pure electrolytek - O(m ²) remaining terms being of the second and of higher powers ofm - p ⁺ standard pressure - p total pressure of the gas phase - p pressure within a droplet - p ₁,p ₂,p ₃ coefficients in the expansion of M - p _1i, p_2i, p_3i specific parameters of ioni - p _s saturation vapour pressure - p _w water vapour pressure - R _w individual gas constant of water - r radius of a droplet - r ₀ equivalent volume radius of an aerosol particle in dry state - T temperature - T ₀ standard temperature - T ₁ temperature of the pure water drop in the osmometer - v _w specific volume of pure water - z _i valence of ioni - _i relativenumber concentration of ioni in a solution - correction term due to the adsorption of ions at liquid-solid interfaces - activity coefficient of solute speciesi in a solution, related to molalities - I bridge current - T temperature difference between solution and pure water drop in the osmometer - exponential mass increase coefficient - _w specific chemical potential of water vapour - _w specific chemical potential of water - ⁰ _w specific chemical potential of pure water vapour - ⁰ _w specific chemical potential of pure water - ₀ density of an aerosol particle in dry state - _w density of pure water - surface tension of a droplet - ₀ surface tension of pure water, i.e., at infinite dilution of the solute - osmotic coefficient - _k osmotic coefficient of a solution of a pure electrolytek - _k osmotic coefficient of a solution of a mixed solute - _M fugacity coefficient of water vapour - ^s _{i=1 i} z ² _i This work is part of a Ph.D. thesis carried out at the Meteorological Institute of the Johannes Gutenberg-Universität, Mainz.     

On the efficiency with which aerosol particles of radius larger than 0.1 μm are collected by simple ice crystal plates

A theoretical model is presented which allows computing the efficiency with which aerosol particles of radius 0.1r10 m are collected by simple ice crystal plates of radius 50a _c 640 m in air of various relative humidities, temperatures and pressures. Particle capture due to thermophoresis, diffusiophoresis and inertial impaction are considered. It is shown that the capture efficiency of an ice crystal in considerably affected by phoretic effects in the range 0.1r1 m. For aerosol particles ofr>1 m the efficiency is strongly controlled by the flow field around the crystal and the density of the aerosol material. Trajectory analysis also predicts that aerosol particles are preferentially captured by the ice crystal rim. Our theoretica results are found to agree satisfactorily with the laboratory studies presently available. Comparison shows that for the same pressure, temperature and relative humidity of the ambient air ice crystal plates are better aerosol particle scavengers than water drops.     

Humidity effects on gravitational settling and Brownian diffusion of atmospheric aerosol particles

The dependency on relative humidity of the settling velocity of aerosol particles in stagnant air and of the diffusion coefficient due to Brownian motion of aerosol particles was computed for six aerosol types and different particles sizes in dry state. The computations are based (1) on mean bulk densities of dry aerosol particles obtained from measurements or from the knowledge of the chemical composition of the particles, (2) on micro-balance measurements of the water uptake per unit mass of dry aerosol substance versus water activity at thermodynamic equilibrium, and (3) on measurements of the equilibrium water activity of aqueous sea salt solutions. The results show a significant dependence of the settling velocity and Brownian diffusion of aerosol particles on relative humidity and on the particle's chemical composition.Nomenclature A surface parameter of a particle - B surface parameter of a particle - c _L velocity of sound in moist air - C 1+Kn[A+Qexp(–B/Kn]=slip correction - D diffusion coefficient of a particle - D ₁ D(=1)=diffusion coefficient of a spherical particle - f P _w /P _we (T,P)=relative humidity (f=0 dry air,f=1 saturated air) - g acceleration due to gravity - g |g| - k 1.3804×10^–16 erg/°K=Boltzmann constant - Kn _L /r=Knudsen number of a particle - Kn _{0 0L} /r ₀=Knudsen number of a dry particle - m 4r ³/3=mass of a particle - m _L 4r ³ _L /3=mass of the moist air displaced by a particle - M mobility of a particle - M ₀ molar mass of dry air - M _w molar mass of water - Ma |u–u _L |/c _L =Mach number of the particles motion relative to the ambient air - n particle number per unit volume of air - P P ₀+P _w =pressure of the moist air - P ₀ partial pressure of the dry air - P _w partial pressure of the water vapour - P _we P _we (T,P)=equilibrium partial water vapour pressure over a plane surface of water saturated with air - Q surface parameter of a particle - r equivalent radius of a particle (radius of a sphere with the particles volume) - r ₀ equivalent radius of a particle in dry state - R 1+0.13Re ^0.85=inertia correction - R ₀ specific gas constant of dry air - R _w specific gas constant of water - Re 2r _L u–u _L / _L =Reynolds number of the particles motion relative to the ambient air - t time - T absolute temperature - u velocity of a particle - u (amount of the) settling velocity of a particle in stagnant air - u ₁ u(=1)=(amount of the) settling velocity of a spherical particle in stagnant air - u _L velocity of the ambient moist air (far enough from the particle where the flow pattern remains undistorted) - W drag coefficient of a particles equivalent sphere - empirical parameter in equation (3.1) - dynamic viscosity of a particles liquid cover - _L dynamic viscosity of moist air - _0L dynamic viscosity of dry air (at the same pressure and temperature like the moist air) - celsius temperature - dynamic shape factor of a particle (=1 for a sphere) - ₀ dynamic shape factor of a dry particle - _L mean free path of the molecules in moist air - _0L mean free path of the molecules in dry air (at the same pressure and temperature like the moist air) - _Po mean free path of the molecules in dry air at the pressureP ₀ of the dry air and the temperature given - factor of solid to liquid change-over (=1 for a solid particle) - mean bulk density of a particle - _L density of the moist air - _0L density of the dry air at the same pressure and temperature like the moist air - ₀ mean bulk density of a dry particle - ₀ mean diameter of the molecules of dry air - _w diameter of water molecules - relaxation time of a particle - gradient operation - 3.141593     

On the origin of the fourth zonal geopotential harmonic

Summary It has been proved that the fourth geopotential (Stokes) parameter J₄ of the actual Earth can be explained by its long-term rotational distortions. However, this is not the case of the sixth zonal prameter J₆; its origin should be explained geophysically in another way.Dedicated to the Memory of Professor Karel P     

Solution to the Stokes Boundary-Value Problem on an Ellipsoid of Revolution

We have constructed Green's function to Stokes's boundary-value problem with the gravity data distributed over an ellipsoid of revolution. We show that the problem has a unique solution provided that the first eccentricity e₀ of the ellipsoid of revolution is less than 0·65041. The ellipsoidal Stokes function describing the effect of ellipticity of the boundary is expressed in the E-approximation as a finite sum of elementary functions which describe analytically the behaviour of the ellipsoidal Stokes function at the singular point = 0. We prove that the degree of singularity of the ellipsoidal Stokes function in the vicinity of its singular point is the same as that of the spherical Stokes function.     

Friction constitutive law with rate and state dependences

A general formula for the Dieterich-Ruina friction constitutive law with rate and state (n-state variables,n=1, 2,...) dependences has been obtained and discussed under the assumption that the slip acceleration a varies ion a linearly with the slip displacement , namelya = a ₀ + (-₀). Wherea ₀, ₀ are initial constants, is the acceleration rate and constant.a ₀ and may be arbitrary constants (positive, negative or zero).The extreme value of frictional resistance and the existence condition of the extreme value, which are very important and govern to some degree the motion process of a frictionally slipping mechanical system, have been analyzed. A critical value _c which is the measure of the velocity weakening and velocity strengthening of the mechanical system, and its properties and the relationship to the extreme problem have been studied. Again, according to the critical value _c, the concepts of light or strong velocity weakening (or strengthening) are introduced.A possibly new phenomenon that frictional resistance may vary in some kind of decayed oscillation is found. Finally, the condition for the smallest frictional resistance for a slipping mechanical system with nonuniform acceleration has been obtained.     

Size distribution models of fog and cloud droplets and their volume extinction coefficients at visible and infrared wavelengths

Based on the average variability of the skewness with respect to the droplet mode radius, a wide set of mean size-distribution models is presented in terms of the modified gamma function for fog and stratified cloud droplets. These models appear appropriate for giving reliable size-distribution curves relative to the various formation stages of the droplet population both in fogs and in stratus and stratocumulus clouds.The corresponding volume extinction coefficient has been computed at various wavelengths from 0.4 to 17 m using Van de Hulst's (1957) approximation multiplied by Deirmendjian's (1960) correction factors. This set of theoretical extinction data has been compared with experimental extinction measurements performed in atmospheres characterized by a marked thermal inversion for describing the evolutionary features of the water droplet size distribution within the whole ground layer.     

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.     

A numerical study of the heat transfer through a fluid layer with recirculating flow between concentric and eccentric spheres

A numerical study has been made of the heat transfer through a fluid layer with recirculating flow. The outer fluid surface was assumed to be spherical, while the inner surface consisted of a sphere concentrically or eccentrically located with respect to the outer spherical surface. The recirculating flow was assumed to be driven by a gas flow creating stress on the fluid's outer surface so that creeping (low Reynolds number) flow developed in its interior. The present study solves the Stokes equation of motion and the convective diffusion equation in bispherical coordinates and presents the streamline and isotherm patterns.Nomenclature a _i inner sphere radius - a _d outer sphere radius - A ₁ defined by equation (5) - A ₂ defined by equation (6) - B ₁ defined by equation (7) - B ₂ defined by equation (8) - c dimensional factor for bispherical coordinates - C constant in equation (4) - d narrowest distance between the two eccentric spheres - E ² operator defined by equation (1) in spherical coordinates and by equation (21) in bispherical coordinates - G modified vorticity, defined in equation (22) - G ^* non-dimensional modified vorticity, defined in equation (28) - h metric coefficient of bispherical coordinate system, defined in equation (18) - k _w thermal conductivity of water - K ₁ defined by equation (9) - K ₂ defined by equation (10) - N _Re Reynolds number=2a _dU/gn - N _Pe,h Peclet number=2a _dU/ - n integer counter - q heat flux - r radius - r ^* non-dimensional radius=r/a _d - S surface area - t time - t ^* non-dimensional time=t/a _d ² - T temperature - T _o temperature at inner sphere surface - T _a temperature at outer sphere surface - T ^* non-dimensional temperature;=(T–T _o)/(T_a–T_o) - u velocity - u _r radial velocity in spherical coordinates - u angular velocity in spherical coordinates - u radial velocity in bispherical coordinates - u angular velocity in bispherical coordinates - U free stream velocity - u _r ^* =u _r/U - u ^* =u /U - u ^* =u /U - u ^* =u /U Greek symbols a ₁ small displacement - vorticity, defined in equation (17) - ^* non-dimensional vorticity, defined in equation (27) - radial bispherical coordinates - _o bispherical coordinate of inner sphere - _a bispherical coordinate of outer sphere - angular coordinate in spherical coordinates - thermal diffusivity - _w thermal diffusivity of water - kinematic viscosity - angular bispherical coordinate - spherical coordinate - streamfunction - non-dimensional streamfunction for spherical coordinates, = /(U a _d ² ) - ^* non-dimensional streamfunction for bispherical coordinates, defined in equation (26)     

Measurements of the effective attachment coefficient between RaA ions and condensation nuclei

Summary To clarify the interaction between RaA ions and condensation nuclei, simultaneous measurements of the concentration of RaA ions (n _A), radon-222, and condensation nuclei (Z) were carried out at several stations. In some occasions, the diffusion coefficients of nuclei (D) were also measured. It was found that the correlation among them may be well expressed by the simple formula;q _A = _A n_A Z. The correlation between _A andD (or radius of nuclei) was also obtained.The effective attachment coefficient of RaA ions was compared with that of small ordinary ions, and it was presumed that about one fourth of RaA atoms formed on the decay of radon-222 is positively charged, and the rest is neutral.     

Source function of stress waves of a spherical explosive source

Summary The source functions of the stress wave patterns at an elastic source of these waves are analysed. The comparison of the properties of the functions with the stress wave records obtained earlier showed that their parameters do not satisfy, to a greater or lesser extent, the stress wave patterns in the neighbourhood of explosive sources. For this reason a new source function (1) was defined, which fully approximates the observed stress wave patterns in gravel sandy soil. The coefficientsP ₀, , , and were experimentally determined as functions of the distance from the source, its size and the radius of the elastic source in the medium considered. The properties of source function (1) are demonstrated on an example.Paper presented at the XIIIth General Assembly of the European Seismological Commission, Braov (Romania), 28 August to 6 September, 1972.     

Measurement of lower-stratospheric aerosol by a balloon-borne photometer

Height distribution of the stratospheric aerosol extinction coefficient was measured in the altitude range 10 to 20 km by a balloon-borne multi-color sunphotometer in May 1978. It is demonstrated that detailed structures of the distribution of stratospheric aerosol can be remotely measured by the solar occultation method as well as by lidar andin situ particle counter observations. In the aerosol layer appearing at 18 km altitude the extinction coefficient at 800–1000 nm wavelength reached to 3×10^–7 m^–1, which was reasonable compared with lidar observations. Wavelength dependence of the aerosol optical depth was crudely estimated to be proportional to ^–1.5.     

Reservoir lifetime and heat recovery factor in geothermal aquifers used for urban heating

Simple models are discussed to evaluate reservoir lifetime and heat recovery factor in geothermal aquifers used for urban heating. By comparing various single well and doublet production schemes, it is shown that reinjection of heat depleted water greatly enhances heat recovery and reservoir lifetime, and can be optimized for maximum heat production. It is concluded that geothermal aquifer production should be unitized, as is already done in oil and gas reservoirs.Nomenclature a distance between doublets in multi-doublet patterns, meters - A area of aquifer at base temperature, m² drainage area of individual doublets in multidoublet patterns, m² - D distance between doublet wells, meters - h aquifer thickness, meters - H water head, meters - Q production rate, m³/sec. - r _e aquifer radius, meters - r _w well radius, meters - R _g heat recovery factor, fraction - S water level drawdown, meters - t producing time, sec. - T aquifer transmissivity, m²/sec. - v stream-channel water velocity, m/sec. - actual temperature change, °C - theoretical temperature change, °C - water temperature, °C - heat conductivity, W/m/°C - _r rock heat conductivity, W/m/°C - _aC_a aquifer heat capacity, J/m³/°C - _aC_r rock heat capacity, J/m³/°C - _WC_W water heat capacity, J/m³/°C - aquifer porosity, fraction     

Earthquake grouping criteria for spatially heterogeneous seismicity

An advanced method for estimating the earthquake grouping parameters R_cr and T_cr is proposed in order to identify interrelated seismic events. The method pursues continuity with the previous algorithm suggested in (Mirzoev, 1980; 1988a; 1988b; 1992; Mirzoev and Azizova, 1983; 1984) but uses a more realistic spatial model of the background seismicity. All the calculations in the method can be fully formalized and a preliminary expert estimation of the parameters is not required. The method provides stable estimates of the critical radius R_cr and time T_cr of grouping. Group earthquakes make up 50 to 75% of their total number.     

On the stationary charge distribution on aerosol particles in a bipolar ionic atmosphere

Summary By the «limiting sphere» method the combination coefficients for gaseous ions and aerosol particles were calculated, allowing for the jump in ion concentration at the surface of the particles. Hence the stationary charge distribution on aerosol particles in a symmetrical bipolar ionic atmosphere was determined. The use of the Boltzmann equation for this purpose proposed by some authors is theoretically wrong asthis equation applies to equilibrium rather than to stationary states. In practice, the Boltzmann equation can be used for particles with radius 3·10^–5 cm (under atmospheric pressure). Within this range the image forces and the jump in ion concentration may be neglected. The conditions of the applicability of the steady diffusion equations to the theory of the stationary charge distribution in aerosols are discussed.     

Actinometric determination of turbidity parameters in India

Summary A study of the wavelength exponent of aerosol scattering in the Ångström relation for extinction by aerosol has been made from the ground-based measurments of direct solar radiation using Ångström pyrheliometer with and without Schott filters. It has been observed that in India, mainly for the middle part of the year this exponent is zero or even negative which means that the aerosol scattering is nearly neutral which is in marked contrast with the condition prevailing in middle latitudes. It is evident from the -values that the aerosol size distribution in India is far different from that prevailing in middle latitudes. At four representative stations in India, the values of the wavelength exponent and the atmospheric turbidity coefficient have been determined using the method introduced byÅngström [1,2]) and are discussed here.     

Nonlinear Magnetoconvection and the Geostrophic Flow - Subcritical Solutions

The magnetoconvection problem under the magnetostrophic approximation is investigated as the nonlinear regime is entered. The model consists of a fluid filled sphere, internally heated, and rapidly rotating in the presence of a prescribed, axisymmetric, toroidal magnetic field. For simplicity only a dipole parity and a single azimuthal wavenumber (m = 2) is considered here. The leading order nonlinearity at small amplitude is the geostrophic flow U _g which is introduced to the previously linear model (Walker and Barenghi, 1997a, b). Walker and Barenghi (1997c) considered parameter space above critical and found that U _g acts as an equilibration mechanism for moderately supercritical solutions. However, for solutions well above critical a Taylor state is approached and the system can no longer equilibrate. More importantly though, in the context of this paper, is that subcritical solutions were found. Here subcritical solutions are considered in more detail. It was found that, at is strongly dependent on . ( is the critical value of the modified Rayleigh number is a measure of the maximum amplitude of the generated geostrophic flow while , the Elsasser number, defines the strength of the prescribed toroidal field.) R_m at proves to be the key measure in determining how far into the subcritical regime the system can advance.     

Turbulent “Dynamo” in the Passive Scalar Problem

A turbulent magnetic dynamo can be considered as the evolution of a vector field in a turbulent fluid flow. The problem of evolution of scalar fields (e.g., number density of small particles) in a turbulent fluid flow is similar to the turbulent magnetic dynamo. The dynamo instability results in generation of magnetic field. The most important effect which can cause a generation of mean magnetic field in a turbulent fluid flow is the -effect: = – (1/3) u · ( × u), where u is the turbulent velocity field with the correlation time . A similar instability in the passive scalar problem results in formation of large-scale inhomogeneous structures in a spatial distribution of particles due to the -effect: = u_p ( · u_p), where u _p is the random velocity field of the particles which they acquire in a turbulent fluid velocity field. The effect is caused by inertia of particles which results in divergent velocity field of the particles. This results in additional turbulent nondiffusive flux of particles. The mean-field dynamics of inertial particles are studied by considering the stability of the equilibrium solution of the derived evolution equation for the mean number density of the particles in the limit of large Péclet numbers. The resulting equation is reduced to an eigenvalue problem for a Schrödinger equation with a variable mass, and a modified Rayleigh-Ritz variational method is used to estimate the lowest eigenvalue (corresponding to the growth rate of the instability). This estimate is in good agreement with obtained numerical solution of the Schrödinger equation. Similar effects arise during turbulent transport of gaseous admixtures (or light noninertial particles) in a low-Mach-number compressible fluid flow. The discussed effects are important in planetary and atmospheric physics (cloud formation, pollutant dynamics, preferential concentration of particles in protoplanetary disks and also planetesimals in them).     

Rayleigh waves in an inhomogeneous medium

Summary Dispersion in Rayleigh waves is discussed for semi-infinite media with = ₁(1 ± cos s z) and = ₁(1 ± cosh s z), being the rigidity of the medium. A few workers tried with the above Fourier type of model but failed to find the dispersive nature. Because they neglected s due to the complexity of the calculation they arrived at a non dispersive frequency equation. This difficulty is removed in this paper and a dispersive frequency equation is obtained which shows both direct and inverse dispersion. The second model leads to non-convergent solution forz but shows many interesting results which are also discussed.