Fritz Herbert

Scavenging of Airborne Particles by Collision with Water Drops - Model Studies on the Combined Effect of Essential Microdynamic Mechanisms

SummaryOut information from the limited amount of observations about the capture of airborne particles by cloud and precipitation drops reveals that this complex scavenging phenomenon depends on various, the dropletparticle collisions simultaneously driving micromechanisms. Theoretical models attempt to treat these processes in combination.A major task of the study reported here was to describe the responsible scavenging interactions as part of a microdynamical simulation model for particle and drop size spectra. Using for this purpose the theory of nonlinear stochastic collisions we show that key properties (such as size dependence, relaxation, etc.) associated with particle scavenging are predicted satisfactorily with meteorologically representative input data.ZusammenfassungDie aus Beobachtungen gewinnbare Information über das Einfangen von luftgetragenen Teilchen durch Wolken- und Niederschlagstropfen zeigt klar, daß dieses komplexe Auswaschphänomen von mehreren, gemeinsam für die Tropfen-Teilchenkoflisionen verantwortlichen Mikromechanismen abhängt. In theoretischen Modellen versucht man, diese Prozesse kombiniert zu behandeln.Kernpunkt der theoretischen Untersuchungen, die hier vorgestellt werden, war es, die verantwortlichen Wechselwirkungen in einem mikrodynamischen Simulationsmodell für Größenspektren von Teilchen und Tropfen zu beschreiben. Hierfür wurde die entsprechende Theorie nichtlinearer stochastischer Kollisionen verwendet. Die Rechnungen liefern, daß wichtige Eigenschaften (wie Größenabhängigkeit, Relaxation usw.), die mit dem Teilchenauswaschvorgang zusammenhängen, bei Annahme meteorologisch repräsentativer Eingabedaten zufriedenstellend bestimmt werden können.

Evaporation and Precipitation Surface Effects in Local Mass Continuity Laws of Moist Air

Abstract The local mass balance equations of cloudy air are formulated for a model system composed of dry air, water vapor, and four categories of water condensate particles, as typically adopted for numerical weather prediction and climate models. The choice of the barycentric velocity as reference motion provides the most convenient form of the total mass continuity equation. Mass transfer across the earth’s surface due to precipitation and evaporation causes a nonvanishing barycentric vertical velocity ws and is proportional to the local difference between evaporation rate and rain plus snow rate. Hence ws vanishes only in the special situation that evaporation and precipitation balance exactly. Alternative concepts related to different reference motions are reviewed. However, the choice of the barycentric velocity turns out to be advantageous for several reasons. The implication of the nonvanishing total mass transport across the earth’s surface is estimated from model simulations for two extreme weat...

Mathematical studies on the aerosol concentration in drops changing due to particle scavenging and redistribution by coagulation

SummaryPollution of cloud and precipitation water by airborne particles occuring primarily as a result of collision aerosol scavenging processes can within the drop size spectrum considerably be changed due to stochastic coagulation of polluted drops. The effect of mutaal drop coagulation shifts the size spectrum to larger drops. This is accompanied by a spectral redistribution of the drop water as well as the captured particle content. Describing particle scavenging and redistribution by drop coagulation in terms of a microdynamical model, we find how the degree of drop pollution varies with different cloud physical reference states.ZusammenfassungVerschmutzung von Wolken- und Niederschlagswasser durch luftgetragene Teilchen, die primär durch Auswaschen von Aerosol zustandekommt, kann innerhalb des Tropfengrößenspektrums aufgrund stochastischer Koagulation der verschmutzten Tropfen erheblich verändert werden. Der Effekt durch gegenseitiges Koagulieren (der Tröpfchen) verlagert das Größenspektrum zu größeren Tropfen. Dies führt zu einer spektralen Neuverteilung des Tropfenwassers sowie des eingeschlossenen Teilchengehalts. Das Auswaschen von Aerosol und die Neuverteilung durch Koagulation werden mit einem mikrodynamischen Modell beschrieben, mit dem der Verschmutzungsgrad der Tröpfchen und seine Veränderung bei verschiedenen wolkenphysikalischen Voraussetzungen berechnet wird.

Continuity equations as expressions for local balances of masses in cloudy air

The mathematical representation of the mass continuity equation and a boundary condition for thevertical velocity at the earth’s surface is re-examined in terms of its dependence on the frame ofreference velocity. Three of the most prominent meteorological examples are treated here: (a) thebarycentric velocity of a full cloudy air system, (b) the barycentric velocity of a mixture consisting ofdry air andwater vapour and (c) the velocity of dry air. Although evidently the physical foundation holdsindependently of the choice of a particular frame, the resulting equations differ in their mathematicalstructure: In examples (b) and (c) the diffusion flux divergence that appears in the corresponding massequation of continuity should not be omitted a priori. As to the lower boundary condition for the normalcomponent of velocity, special emphasis is placed on the net mass transfer across the earth’s surfaceresulting from precipitation and evaporation. It is shown that for a flat surface, the reference verticalvelocity vanishes only in case (c). Regarding cases (a) and (b), the vertical reference velocities aredetermined as functions of the precipitation and evaporation rates. They are nonzero, and it is shownthat they cannot generally be neglected.

Similarity Hypotheses for the Atmospheric Surface Layer Expressed byNon-Dimensional Characteristic Invariants – A Review

In this paper, similarity hypotheses for the atmospheric surface layer (ASL) are reviewed using nondimensional characteristic invariants, referred to as -numbers. The basic idea of this dimensional -invariants analysis (sometimes also called Buckingham’s -theorem) is described in a mathematically generalized formalism. To illustrate the task of this powerful method and how it can be applied to deduce a variety of reasonable solutions by the formalized procedure of non-dimensionalization, various instances are represented that are relevant to the turbulence transfer across the ASL and prevailing structure of ASL turbulence. Within the framework of our review we consider both (a) MoninObukhov scaling for forced-convective conditions, and (b) Prandtl-Obukhov-Priestley scaling for free-convective conditions. It is shown that in the various instances of Monin-Obukhov scaling generally two -numbers occur that result in corresponding similarity functions. In contrast to that, Prandtl-Obukhov-Priestley scaling will lead to only one number in each case usually considered as a non-dimensional universal constant. Since an explicit mathematical relationship for the similarity functions cannot be obtained from a dimensional invariants analysis, elementary laws of -invariants have to be pointed out using empirical or/and theoretical findings. To evaluate empirical similarity functions usually considered within the framework flux-profile relationships, so-called integral similarity functions for momentum and sensible heat are presented and assessed on the basis of the friction velocity and the vertical component of the eddy flux densities of sensible and latent heat directly measured during the GREIV I 1974 field campaign.

Thermodynamic non-equilibrium theory of the condensational growth of atmospheric water drops

SummaryThe application of the theory of non-equilibrium thermodynamics to phenomena of cloud micro-physics has been examined for the example of mass growth of atmospheric water drops due to vapour diffusion and condensation. A materially and energetically closed heterogeneous system composed of a drop phase and a surrounding dry air-water vapour mixture is assumed as appropriate basic model in order to treat the concomitant theoretical aspects (comprising description of the individual growth rate, variation of moist air temperature, and drop surface conditions) in dependency on the central criteria of irreversibility. Owing to this, a main object of our theory is the thorough derivation of the budget equations of thermal energy and entropy representing, respectively, the first and second law of thermodynamics. Physically, these principles, associated with the peculiar thermodynamic behaviour of coexisting atmospheric drops and vapour, embody a suitable theoretical frame for the line of reasoning. The dominant position is owned by the production rate of entropy, a bilinear form of thermodynamic forces and fluxes. The occasion arises to postulate adequate non-equilibrium laws for the irreversible transport of matter and heat. With regard to the entropy rate of change and the thermodynamic situation in the drop-moist air model, one is left with the option to consider several alternative postulates for the fluxes and, hence, several equivalent parameterizations of the growth rate of drops. Four such approaches in accord with the thermodynamic context are discussed. As each of them depends on the surface temperatures of drops, it is expedient to complete the growth equations by a separate treatment of these micro-state variables. Practical scaling arguments for these internal thermodynamic parameters reveal that a suitable reduced form of the energy budget for the isolated drop-moist air-model system can be assumed. As a consequence, the droplet surface temperature becomes a diagnostic parameter which can be eliminated from the growth equation.

An internal energy theorem for the atmosphere and its association with turbulent (potential) temperature variances

An extended treatment of the internal energy as a thermodynamic potential in the energy-entropy picture is presented in connection with atmospheric systems under turbulent conditions. One finds that a turbulent thermodynamically autonomous energy quantity, the turbulent internal energy (TIE) which belongs to the mean state internal energy depending on averaged state variables, can be derived. The elementary relevance of TIE becomes expediently represented by means of a second-order Taylor approach; it explains TIE as the energetic response to the eddy variances of the independent thermodynamic variables of state (i.e. temperature and mass density or potential temperature). The difference of the novel concept of TIE to existing available energy formulations is discussed.

Parameterization of the CCN-humidity spectrum in dependency on nucleation conditions and aerosol size distribution

SummaryAtmospheric CCN-humidity spectra (describing the CCN-number concentration as function of supersaturation) are derived as the integral over given particle size distributions. In that concept the finite boundary, representing the limiting activated particle size, results from the critical values of the Köhler-curve. As utilization of this general outcome different representative aerosol size distributions of the power law type as well as the log-normal type are chosen for case studies which are compared to empirical results. The dependency on temperature of the limiting activated particle size is shown to provide a non-negligible influence on the number of activated particles.

On the relationship between turbulent internal energy and available energy in local formulation

The interdependence between the turbulent internal energy (as it was introduced in Herbert and Kucharski [1998]) and the atmospheric shallow fluid available energy is analytically investigated in conformity with the corresponding local energy balance differential equations that are imposed as central physical criteria. It is shown that the available and turbulent internal energy forms do well combine together under the assumption of a further thermal energy function coupled with appropriate substantial simplifications.

On different moisture variables in the constitutive equation for condensation growth of cloud drops

The irreversible H2O-transfer between a drop and moist air is described on the basis of the diffusion model for condensational drop growth. Depending on the chosen form of the flux-gradient relationship to approach the irreversible vapour diffusion flux, distinct equations for the growth rate follow, all of which are equally in accord with thermodynamics. They differ in the formulation of the driving force and in the choice of the relevant moisture variable as well as in the correction term due to thermal effects. Considering all assumptions entering the approaches of the condensational drop growth rate, arguments are provided that the most relevant form of the growth equation should be that depending linearly on the mass fraction of water vapour in air as representative moisture variable.