Columnar Modelling of Nucleation Burst Evolution in the Convective Boundary Layer ? ﬁrst Results from a Feasibility Study Part I: Modelling Approach O

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Columnar modelling of nucleation burst evolution in the convective boundary layer ? ﬁrst results from a feasibility study Part I: Modelling approach O. Hellmuth

To cite this version: O. Hellmuth. Columnar modelling of nucleation burst evolution in the convective boundary layer ? ﬁrst results from a feasibility study Part I: Modelling approach. Atmospheric Chemistry and Physics, European Geosciences Union, 2006, 6 (12), pp.4175-4214.

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Columnar modelling of nucleation burst evolution in the convective boundary layer – ﬁrst results from a feasibility study Part I: Modelling approach

O. Hellmuth Leibniz Institute for Tropospheric Research, Modelling Department, Permoserstrasse 15, 04318 Leipzig, Germany Received: 1 August 2005 – Published in Atmos. Chem. Phys. Discuss.: 10 November 2005 Revised: 14 February 2006 – Accepted: 11 May 2006 – Published: 21 September 2006

Abstract. A high-order modelling approach to interpret essentially extended by Charlson, Lovelock, Andrea and “continental-type” particle formation bursts in the anthro- Warren2 (Charlson et al., 1987). In their postulated neg- pogenically influenced convective boundary layer (CBL) is ative feedback mechanism, abundant atmospheric sulphate proposed. The model considers third-order closure for plane- aerosols are hypothesised to play a key role in stabilising tary boundary layer turbulence, sulphur and ammonia chem- Earth’s climate. An assumed global warming, caused by ra- istry as well as aerosol dynamics. In Paper I of four papers, diative forcing of greenhouse gases, would lead to an in- previous observations of ultrafine particle evolution are re- crease of the sea surface temperature and, consequently, to viewed, model equations are derived, the model setup for a an increase of dimethyl sulphide emissions (DMS, (CH3)2S) conceptual study on binary and ternary homogeneous nucle- from marine phytoplankton (e.g., macroalgae) into the at- ation is defined and shortcomings of process parameterisa- mosphere. The gas phase oxidation of DMS produces sul- tion are discussed. In the subsequent Papers II, III and IV phur dioxide (SO2), which further oxidises with the hydroxyl simulation results, obtained within the framework of a con- radical (OH) to form sulphuric acid (H2SO4). According to ceptual study on the CBL evolution and new particle forma- the present level of process understanding, H2SO4 vapour tion (NPF), will be presented and compared with observa- and water (H2O) vapour are key precursor gases for the ho- tional findings. mogeneous heteromolecular (binary) nucleation of sulphate aerosols over the ocean. Due to condensation and coagulation processes, these microscopic particles can grow via interme- 1 Introduction diate, not yet fully understood stages from so-called thermodynamically stable clusters (TSCs) (Kulmala et al., 2000; At the end of the 1960’s, James Lovelock came up with an Kulmala, 2003) via ultrafine condensation nuclei (UCNs) to influential theory of homeostasis of a fictive planet Gaia, rep- non-sea-salt (nss) sulphate aerosols and, finally, to cloud con- resented by the conceptual “daisyworld” model (Lovelock, densation nuclei (CCNs). An increase of the CCN concen- 1993). Lovelock proposed a negative feedback mechanism tration leads to an enhancement of the number concentra- to explain the relative stability of Earth’s climate over geo- tion of cloud droplets. According to an effect proposed by logical times, which became generally known as “Gaia hy- Twomey (1974), a higher cloud droplet concentration causes pothesis”1. At the end of the 1980’s, the Gaia theory was an enhancement of the cloud reflectivity for a given liquid water content. An increased reflectivity of solar radiation Correspondence to: O. Hellmuth by clouds cools the atmosphere and counteracts the initial ([email protected]) warming. Via this route, a thermal stabilisation of Earth’s 1 In the Hellenistic mythology, “Gaia” is the name of the god- climate is accomplished. For the illustration of this theory hood of Earth. James Lovelock and the Nobel prize winner William the reader is referred, e.g., to Easter and Peters (1994) and Golding used this name as a synonym for a self-regulating geophys- Brasseur et al. (2003, pp. 142–143, Fig. 4.10). Apart from ical and biophysical mechanism on a global scale. According to the Gaia hypothesis, temperature, oxidation state, acidity as well as their climate impact, aerosol particles are also affecting hu- different physicochemical parameters of rocks and waters remain man health, visibility etc. constant at each time due to homeostatic interactions, maintained by massive feedback processes. These feedback processes are initi- gle creatures). ated by the “living world”, whereas the equilibrium conditions are 2According to the initials of the authors, this theory became changing dynamically with the evolution of Earth’s live (not of sin- known as CLAW hypothesis.

Published by Copernicus GmbH on behalf of the European Geosciences Union. 4176 O. Hellmuth: Burst modelling

Recently, Kulmala et al. (2004c, Fig. 1) proposed a pos- cloud condensation nuclei”4 (Kulmala et al., 2000, p. 66). sible connection between the carbon balance of ecosys- Depending on the availability of pre-existing particles and/or tems and aerosol-cloud-climate interactions, also suspected condensable vapours, Kulmala et al. (2000) hypothesised two to play a significant role in climate stabilisation. Increas- possibilities for the growth of TSC particles to detectable ing carbon dioxide (CO2) concentrations accelerate photo- size: (a) When the concentration of pre-existing particles is synthesis, which in turn consumes more CO2. But forest low, TSCs can effectively grow by self-coagulation. (b) In the ecosystems also act as significant sources of atmospheric presence of a high source of available condensable vapours aerosols via enhanced photosynthesis-induced emissions of (e.g., organics, inorganics, NH3) TSCs can effectively grow non-methane biogenic volatile organic compounds (BVOCs) to detectable size and even to Aitken mode size by condensa- from vegetation3. Owing to the capability of BVOCs to get tion. By means of the advanced sectional aerosol dynamical activated into CCNs, biogenic aerosol formation may also model AEROFOR the authors concluded, that ternary nucle- serve as negative feedback mechanism in the climate system, ation is able to produce a high enough reservoir of TSCs, slowing down the global warming similar as proposed in the which can grow to detectable size and subsequently being CLAW hypothesis. able to explain observed particle formation bursts in the atmosphere. The occurrence of bursts was demonstrated to be connected to the activation of TSCs with extra condensable 2 Key mechanisms of new particle formation in the at- vapours (e.g., after advection over sources or mixing). Ion- mosphere induced nucleation was ruled out as a TSC formation path- way because of the typically low ionisation rate in the tro- Atmospheric NPF is known to widely and frequently occur in posphere. Later on, this concept has been advanced by Kul- Earth’s atmosphere. It is generally accepted to be an essen- mala (2003). The author refers to four main nucleation mech- tial process, which must be better understood and included anisms, which are suspected to be relevant for NPF in the in global and regional climate models (Kulmala, 2003). So atmosphere: far, the most comprehensive review of observations and phenomenological studies of NPF, atmospheric conditions under 1. Homogeneous nucleation involving binary mixtures which NPF has been observed, and empirical nucleation rates of H2O and H2SO4 (occurrence, e.g., in industrial etc. over the past decade, from a global retrospective and plumes); from different sensor platforms was performed by Kulmala et al. (2004d). The authors reviewed altogether 149 studies 2. Homogeneous ternary H2O/H2SO4/NH3 nucleation and collected 124 observational references of NPF events in (occurrence in the CBL); the atmosphere. The elucidation of the mechanism, by which new parti- 3. Ion-induced nucleation of binary, ternary or organic cles nucleate and grow in the atmosphere, is of key impor- vapours (occurrence in the upper troposphere (UT) and tance to better understand these effects, especially the role lower stratosphere); of aerosols in climate control (Kulmala et al., 2000; Kul- 4. Homogeneous nucleation involving iodide species (oc- mala, 2003). Kulmala et al. (2000) pointed out, that in some currence in coastal environments). cases observed NPF rates can be adequately explained by binary nucleation involving H2O and H2SO4. But in cer- The effectiveness of these mechanisms has been confirmed tain locations, e.g., within the marine boundary layer (MBL) in laboratory studies. However, the concentrations consid- and at continental sites, the observed nucleation rates exceed ered in these studies were typically far above those observed those predicted by the binary scheme. In such cases, ambi- in the atmosphere. Moreover, the concentrations of the nu- ent H2SO4 concentrations are typically lower than those re- cleating vapours in laboratory experiments are regularly so quired for binary nucleation, but are sufficient for ternary nu- high, that these vapours are also responsible for conden- cleation involving H2O, H2SO4 and ammonia (NH3). The sation growth. In opposite to this, the nucleation of atmo- authors presented a hypothesis, “which – in principle – en- spheric particles can be kinetically limited by TSCs, which ables us to explain all observed particle production bursts are formed during intermediate steps of particle nucleation in the atmosphere: (1) In the atmosphere, nucleation is oc- (Kulmala, 2003). As the author pointed out, homogeneous or curring almost everywhere, at least in the daytime. The con- ion-induced ternary nucleation mechanisms can explain the ditions in the free troposphere (cooler temperature, fewer nucleation of new atmospheric particles (diameter <2 nm) pre-existing aerosols) will favour nucleation. (2) Nucleation in many circumstances, but the observed particle growth maintains a reservoir of thermodynamically stable clusters can not be explained by condensation of H2SO4 alone. Kul- (TSCs), which are too small to be detected. (3) Under cer- mala (2003, see references therein) concluded, that nucle- tain conditions TSCs grow to detectable sizes and further to 4TSCs are newly formed particles, whose sizes are greater than 3BVOCs are important precursors for atmospheric NPF, e.g., in a critical cluster (about 1 nm), but are smaller than the detectable forest regions. size of ∼3 nm.

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4177 ation and growth are probably decoupled under most atmo- a threshold size. The second, subsequent process involves spheric conditions and other vapours, such as organic acids, organic condensation during activation. For organic aerosol are involved in the growth process. According to this con- formation to occur, two preconditions must be fulfilled: (1) 6 −3 cept, non-soluble organic vapour (n-nonane) starts to con- The H2SO4 vapour concentrations must be ≥2.5×10 cm , dense on newly nucleated water clusters. After formation, otherwise the clusters are scavenged before they can be ac- the particles can grow, e.g., by heterogeneous nucleation of tivated. (2) The condensable organic vapour concentration organic insoluble vapours on ternary clusters, activation of must be greater than ∼107 cm−3. Higher concentrations of the clusters for condensation of soluble organic vapours and H2SO4 will result in activation at lower organic vapour con- chemical reactions at the surfaces of small clusters. Follow- centrations. The proposed mechanism of cluster activation by ing these initial steps, multicomponent condensation of or- organic vapours suggests, that small increases in the avail- ganic and inorganic vapours is suspected to occur. Conden- ability or production of organic vapours will lead to large sation growth and coagulation loss are competing processes, increases in the aerosol population. The authors refer to an i.e., the more effective the condensation growth, the larger increase in the production of condensable organic vapours, the fraction of nucleated particles, that can survive. Among e.g., resulting from increasing oxidant concentration (e.g., the serious problems, which are deserved to be solved in fu- ozone) associated with globally increasing pollution levels. ture, the author stated the need for an interdisciplinary ap- A description of state-of-the-art growth hypotheses and proach involving laboratory studies, continuous field obser- their theoretical background is given in Kulmala et al. vations, new theoretical approaches and dynamical models. (2004b). Based on measurements of particle size distribu- Based on the traditional Kohler¨ theory to describe the for- tions and air ion size distributions, the authors provided ob- mation of cloud droplets due to spontaneous condensation servational evidences, that organic substances soluble into of water vapour (Kohler¨ , 1936), Kulmala et al. (2004a) pro- TSCs (either ion clusters or neutral clusters) are mainly posed an analogous attempt to describe the activation of inor- responsible for the growth of clusters and nanometer-size ganic stable nanoclusters into aerosol particles in a supersat- aerosol particles. Although no observational evidences were urated organic vapour, which initiates spontaneous and rapid found, that support the dominance of charged-enhanced par- growth of clusters5. The authors pointed out, that the existing ticle growth associated with ion-mediated nucleation, ini- evidences, supporting the prevalence of condensed organic tial growth (around 1–1.5 nm) may be dominated by ion- species in new 3–5 nm particles, can not be explained by ho- mediated condensation. It is hypothesised, that later those mogeneous nucleation of organic vapours under atmospheric charged clusters are neutralised by recombination and that conditions. The supersaturation of organic vapours was esti- they continue their growth by vapour condensation on neutral mated to be well below that required for homogeneous nucle- clusters. The observations revealed a remarkably steady ion ation to occur. The Nano-Kohler¨ theory “describes a thermo- distribution between about 0.5 and 1.5 nm, which appears to dynamic equilibrium between a nanometer-size cluster, wa- be present all the time during both NPF event and non-event ter and an organic compound, which is fully soluble in wa- days. This finding supports the hypothesis of Kulmala et al. ter, i.e., it does not form a separate solid phase, but is totally (2000) on the frequent abundance of nanometer-size thermo- dissolved into the solution. Furthermore, organic and inor- dynamically stable neutral clusters, which can grow to de- ganic compounds present together in growing clusters form tectable size at sufficient high concentrations of condensable a single aqueous solution. It is also assumed, that gaseous vapours. The study of Kulmala et al. (2004b) strongly sup- sulphuric acid condenses irreversibly into the cluster along ports the idea, that vapours responsible for the main growth with ammonia, so that a molar ratio of 1:1 is maintained of small clusters/particles are different from those driving the between the concentrations of these two compounds in the nucleation, i.e., nucleation and subsequent growth of nucle- solution”(Kulmala et al., 2004a, pp. 2–3 of 7). Using an ad- ated clusters are likely to be uncoupled from each other. The vanced sectional aerosol dynamics box model, the authors suggested abundance of TSCs in connection with less fre- evaluated the new theory in order to predict the yield of quent occurrence of NPF events indicates, that atmospheric aerosol particles as a function of organic vapour density. It NPF may be limited by initial steps of growth rather than by was demonstrated, that for a pseudo-constant reservoir of nucleation. TSCs the proposed mechanism leads to a rapid activation More recently, Kulmala et al. (2005) confirmed the hypoth- of the clusters by organic vapours. The model simulations esis on the existence of neutral TSCs both experimentally of Kulmala et al. (2004a) show, that prior to the activation, and theoretically. With respect to their modelling approach, the cluster growth is driven by H2SO4 condensation. The au- the best agreement with the observations was obtained by thors proposed the following dual step process to interpret using a barrierless (kinetic) ternary nucleation mechanism their modelling results: One process considers the conden- together with a size dependent growth rate, supporting the sation growth of nuclei by H2SO4 and organic vapour to Nano-Kohler¨ theory. The model was able to predict both the existence of nucleation events and, with reasonable accu- 5The authors introduced the annotation “Nano-Kohler¨ theory” racy, the concentrations of neutral and charged particles of to emphasise the analogy to Kohler’s¨ traditional approach. all sizes. www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4178 O. Hellmuth: Burst modelling

3 On the role of multiscale transport processes in new mixing of vapour and aerosol loaded surface layer with particle formation clean residual layer air)?

Among others, the question how multiscale transport pro- Previous eddy co-variance particle flux measurements above cesses influence NPF is not yet answered and a subject of forests, such as performed by Buzorius et al. (2001, Figs. 4, ongoing research. A review of scales and the potential of at- 6, 8), Nilsson et al. (2001b, Fig. 9) and Held et al. (2004, mospheric mixing processes to enhance the binary nucleation Fig. 3), yield net deposition of particles, i.e., downward di- rate was performed by Nilsson and Kulmala (1998). On the rected particle fluxes, but these measurements were restricted base of the classical concept of mixing-induced supersatu- to altitudes nearby the canopy layer. To date, the net effect of ration – to our knowledge at first proposed by James Hut- forest stands on particle mass is not yet determined (Held ton in 1784 (Bohren and Albrecht, 1998, see p. 322–324) et al., 2004). – Nilsson and Kulmala (1998) proposed a parameterisation Turbulence-related investigations of NPF are subject of for the mixing-enhanced nucleation rate. The influence of at- ongoing research, e.g., on the European as well as on mospheric waves, such as Kelvin-Helmholtz instabilities, on the process scale within the framework of the QUEST NPF was investigated by Bigg (1997), Nyeki et al. (1999) and project (Quantification of Aerosol Nucleation in the Eu- Nilsson et al. (2000a). The effects of synoptic weather, plan- ropean Boundary Layer, http://venda.uku.fi/quest/, http:// etary boundary layer (PBL) evolution, e.g., adiabatic cool- www.itm.su.se/research/project.php?id=84). A comprehen- ing, turbulence, entrainment and convection, respectively, on sive compilation of previous and current results of the aerosol formation were analysed: QUEST project can be found in an ACP Special Issue edited by Hameri¨ and Laaksonen (2004)(http://www.copernicus. – In the MBL by Russell et al. (1998), Coe et al. (2000), org/EGU/acp/acp/special issue9.html). Pirjola et al. (2000) and O’Dowd et al. (2002); Present day modelling studies to explain NPF events are often based on box models, e.g., applied within a Lagrangian – In the CBL by Nilsson et al. (2000b), Aalto et al. (2001), framework: Buzorius et al. (2001), Nilsson et al. (2001a,b), Boy and Kulmala (2002), Buzorius et al. (2003), Stratmann 1. Analytical and semi-analytical burst models, intended et al. (2003), Uhrner et al. (2003), Boy et al. (2004) and to be used for the parameterisation of SGS bursts in Siebert et al. (2004); large-scale global transport models (Clement and Ford, 1999a,b; Katoshevski et al., 1999; Clement et al., 2001; – In the free troposphere (FT) and UT by Schroder¨ and Dal Maso et al., 2002; Kerminen and Kulmala, 2002); Strom¨ (1997), Clarke et al. (1999), de Reus et al. (1999), Hermann et al. (2003) and Khosrawi and Konopka 2. Multimodal moment models (Kulmala et al., 1995; (2003). Whitby and McMurry, 1997; Wilck and Stratmann, The influence of small-scale and subgridscale (abbreviated 1997; Pirjola and Kulmala, 1998; Wilck, 1998); as SGS) fluctuations, respectively, on the mean-state nucle- 3. Sectional models (Raes and Janssens, 1986; Kerminen ation rate and the parameterisation of turbulence-enhanced and Wexler, 1997; Pirjola, 1999; Birmili et al., 2000; nucleation was subject of investigations performed by Easter Korhonen et al., 2004; Gaydos et al., 2005). and Peters (1994), Lesniewski and Friedlander (1995), An- dronache et al. (1997), Jaenisch et al. (1998a,b), Clement and Such models were demonstrated to successfully de- Ford (1999b), Elperin et al. (2000), Hellmuth and Helmert scribe NPF events, when transport processes can be ne- (2002), Schroder¨ et al. (2002), Buzorius et al. (2003), Hou- glected. However, from in situ measurements Stratmann et al. siadas et al. (2004), Lauros et al. (2004), Shaw (2004) and (2003) and Siebert et al. (2004) provided evidences for a di- Lauros et al. (2006). rect link between turbulence intensity near the CBL inver- Summing up previous works, further investigations are de- sion and ground-observed NPF bursts on event days. When served to answer the following questions: CBL turbulence is suspected to be important, occurrence and evolution of NPF bursts are not yet satisfying modelled (Bir- 1. How strong can SGS fluctuations enhance the nucle- mili et al., 2003; Stratmann et al., 2003; Uhrner et al., 2003; ation rate during intensive mixing periods? Wehner and Wiedensohler, 2003). Independent from the de- 2. Can NPF be triggered by upward moving air parcels gree of sophistication, zero-dimensional models are a priori across atmospheric layers with large temperature gra- not able to explicitly describe transport processes, neither dients? gridscale nor SGS ones. At most, such models can implic- itly consider transport effects by more or less sophisticated 3. Where does NPF occur in the PBL (within the surface artificial tendency terms (e.g., for entrainment) or by empiri- layer or at levels above, followed by downward trans- cally adjusted tuning parameters. For example, Uhrner et al. port after breakup of the nocturnal residual layer and (2003, Figs. 1 and 5) derived a semi-empirical prefactor for

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4179 the binary nucleation rate to correct for the influence of ver- ables are predictive, and second-order moments are parame- tical exchange processes in their box model study. The loga- terised. In a second-order closure scheme first-moment and rithm of that prefactor varied from −3 to 17.2 depending on second-moment variables are predictive, and third-moment the local temperature gradient in the Prandtl layer, whereas variables are parameterised. Finally, in a third-order closure largest values were obtained for very unstable conditions. scheme all first-moment, second-moment and third-moment To overcome present shortcomings in NPF burst mod- variables are determined via prognostic equations, while elling, Boy et al. (2003a) proposed an one-dimensional the fourth-moment variables are parameterised in terms of boundary layer model with aerosol dynamics and a second- lower-moment variables. With respect to the treatment of order turbulence closure (BLMARC, Boundary, layer, mix- the “parameterisation problem”, the reader is also referred ing, aerosols, radiation and clouds) including binary and to Stull’s ostensive perception of this tricky subject, which ternary nucleation. The underlying assumption of horizontal culminates in a quotation of Donaldson (1973): “There are homogeneity is justified by the fact, that NPF often quasi- more models for closure of the equations of the motion at the simultaneously occurs over distances ranging from approxi- second-order correlation level than there are principal inves- mately 50 km to the synoptic scale with a horizontal exten- tigators working on the problem” (Stull, 1997, p. 201). Con- sion of more than 1000 km (Nilsson et al., 2001a; Birmili sidering the important role of human interpretation and cre- et al., 2003; Komppula et al., 2003a; Plauskaite et al., 2003; ativity in the construction of approximations, the parameteri- Stratmann et al., 2003; Wehner et al., 2003; Vana et al., 2004; sation can be to some degree located at an intermediate stage Gaydos et al., 2005). between science and art. Hence, “[...] parameterisation will As a contribution to the ongoing discussion on the role of rarely be perfect. The hope is that it will be adequate” (Stull, turbulence during the evolution of “continental-type” NPF 1997, p. 201). bursts in anthropogenically influenced regions, a columnar So far, previous attempts to extend third-order closure to modelling approach is proposed here. Compared to former aerosol dynamics in the PBL are not known. In this paper, the studies, CBL dynamics, chemistry reactions and aerosol dy- approach is motivated, model formalism and assumptions are namics will be treated in a self-consistent manner by apply- described. In subsequent papers, a conceptual study on mete- ing a high-order closure to PBL dynamics, appropriate sul- orological and physicochemical conditions, that favour NPF phur and ammonia chemistry as well as aerosol dynamics. in the anthropogenically influenced CBL, will be performed. A comprehensive explanation of the annotation applied in turbulence closure techniques can be found, e.g., in Stull (1997, Chapter 6, p. 197–250). In general, the closure prob- 4 Characterisation of “continental-type” NPF bursts lem is a direct consequence of Reynolds’ flow decomposition and averaging approach (Stull, 1997, p. 33–42). The closure In each case over a period of 1.5 years, Birmili and Wieden- problem results from the fact, that “the number of unknowns sohler (2000) observed NPF events in the CBL on approxi- in the set of equations for turbulent flow is larger than the mately 20% of all days, Stanier et al. (2003, Pittsburgh re- number of equations” (Stull, 1997, p. 197). The introduction gion, Pennsylvania) on over 30% of the days, most frequent of additional diagnostic or prognostic equations to determine in fall and spring and least frequent in winter. NPF was ob- these unknown variables results in the appearance of even served to be favoured on sunny days with below average more new unknowns. Consequently, the total statistical de- PM2.5 concentrations. NPF events were found to be fairly scription of a turbulent flow requires an infinite set of equa- correlated with the product of ultraviolet (UV) intensity and tions. In opposite to this, for a finite set of governing equa- SO2 dioxide concentration and to be dependent on the effec- tions the description of turbulence is not closed. This fact tive area available for condensation, indicating that H2SO4 is is commonly known as the “closure problem”. As demon- a component of new particles. strated by Stull (1997, Tables 6–1, p. 198), the prognostic Makel¨ a¨ et al. (1997) performed an early study on NPF occur- equation for any mean variable α (first statistical moment) ring in boreal forests. Evaluating number size distributions of includes at least one double correlation term α0β0 (second ambient submicron and ultrafine aerosol particles for three statistical moment). The forecast equation for this second- quarters of the year 1996, measured at the Hyytial¨ a¨ station in moment turbulence term contains additional triple correla- Southern Finland, the authors clearly demonstrated the ex- tions terms α0β0γ 0 (third statistical moments). Subsequently, istence of a well-defined mode of ultrafine particles. With the governing equations for the triple correlations contain reference to the literature, the authors noted: “The origin of fourth-moment quantities α0β0γ 0δ0 and so on. For practical the particles observed in this study may be connected with reasons only a finite number of equations can be solved, and biogenic activity [...]. On the other hand, since the for- the remaining unknowns have to be parameterised in terms mation events are observed as early as February, we can of known variables: “Such closure approximations or clo- not directly explain all the events by photosynthesis or re- sure assumptions are named by the highest order prognostic lated activity. There should be no photosynthesis going on equations that are retained” (Stull, 1997, p. 199). For exam- in February–March in Finland. However, the role of the bio- ple, in a first-order closure scheme only first-moment vari- genic sources, whether from plant processes or soil microbial www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4180 O. Hellmuth: Burst modelling processes, or reactions on the surfaces of the plants deserve acid–ammonia). After nucleation, growth into observ- further research” (Makel¨ a¨ et al., 1997, p. 1222). able sizes (≥3 nm) is required before new particles ap- Kulmala et al. (1998b) analysed the growth of nucleation pear. The major part of this growth is probably due mode particles observed at a boreal forest site in Southern to condensation of organic vapours” (Kulmala et al., Finland. During an one-year period the authors observed sig- 2001b, p. 325). nificant NPF at the Hyytial¨ a¨ measurement site on more than 50 days. Using an integral aerosol dynamics model, the au- Anyway, Kulmala et al. (2001b) emphasised the preliminary thors selected 10 days of the data, representing NPF event character of their conclusions, i.e., the lack of direct proof days with a clear subsequent growth, for a closer examina- of this phenomenon because the composition of 1–5 nm size tion. Among others, the model predicts the gas phase concen- particles is extremely difficult to determine using the present tration of the condensing species as a function of time. From state-of-the-art instrumentation. a sensitivity study the authors concluded, that the molecu- In a forest region Boy et al. (2003b) found, that SO2 is not lar properties of the condensing compound have only a small correlated or even anti-correlated with the number concen- effect on the daily maximum concentration of the condens- tration of small particles. As SO2 is mainly of anthropogenic ing species. Investigating the sensitivity of the model re- origin, it is mostly associated with high number concentra- sults against variations in the source term of the condens- tions of pre-existing aerosols, representing polluted condi- able species and against variations in temperature and rel- tions. Previous findings from laboratory studies revealed a ative humidity, the authors concluded, “that the maximum direct correlation between increasing SO2 concentrations and concentration of the condensing molecules is in the order of measured number concentrations of newly formed particles 107−109 cm−3, while their saturation vapour density is be- in an ozone/α-pinene reaction system (Boy et al., 2003b). low 105 cm−3” (Kulmala et al., 1998b, p. 461). However, the Held et al. (2004, BEWA field campaign in summer 2001 and condensable vapour causing the observed growth could not 2002) observed NPF on approximately 22% of all days at an yet be identified in the study of Kulmala et al. (1998b). ecosystem research site in the Bavarian Fichtelgebirge. Dur- In their overview of the EU funded project BIOFOR (Bio- ing a 15 month field campaign Gaydos et al. (2005, Pitts- genic aerosol formation in the boreal forest), performed at burgh Air Quality Study (PAQS)) observed regional NPF on the measurement site Hyytial¨ a¨ in Southern Finland, Kulmala approximately 33% of all days. et al. (2001b) stated, that the exact formation route for 3 nm During 21 months (April 2000–December 2001) of contin- particles remains unclear. However, the main results could be uous measurements of aerosol particle number size distribu- summarised as follows: tions in a subarctic area in Northern Finland Komppula et al. – NPF was always connected with arctic and polar air ad- (2003a, Pallas station in Western Lapland and Varri¨ o¨ station vecting over the site, leading to the formation of a stable in Eastern Lapland, distance 250 km) observed over 90 NPF nocturnal boundary layer, followed by a rapid forma- events, corresponding to >14% of all days. All events were tion and growth of a turbulent convective mixed layer, associated with marine/polar air masses originating from the closely followed by NPF. Northern Atlantic or the Arctic Ocean. 55% of NPF events were observed on the same day at both sites, while 45% were – NPF seems to occur in the mixed layer or entrainment observed at only one of the stations. Most of the differences zone. could be related to the influence of air masses of different origin or rain/fog at the respective other station. Among the – Certain thresholds of high enough H SO and NH con- 2 4 3 NPF events observed only in Eastern Lapland, experiencing centrations are probably needed, as the number of newly air masses from Kola Peninsula industrial area, about 80% formed particles was correlated with the product of the were associated with high SO concentrations. In Western H SO production and the NH concentration. A cor- 2 2 4 3 Lapland, no increase in SO concentrations was observed in responding correlation with the oxidation products of 2 connection with NPF events. The authors concluded, that in terpenes was not found. plumes originating from the Kola industrial area, new par- – The condensation sink, i.e., the effective area of pre- ticles are formed by nucleation involving H2SO4 from SO2 existing particles, is probably of importance, as no nu- oxidation. About one third of the NPF events, observed in cleation was observed at high values of the condensa- Eastern Lapland, was found to be affected or caused by the tion sink. Kola Peninsula industry. Evaluating continuous measurements, performed at the Pal- – Inorganic compounds and hygroscopic organic com- las Global Atmosphere Watch (GAW) station area, which pounds were found to contribute both to the particle started on 13 April 2000 and ended on 16 February 2002, growth during daytime, while at night-time organic Komppula et al. (2003b) identified in total 65 NPF events, compounds dominated. corresponding to 10% of all days, i.e., not as frequent and not – The authors concluded: “The most probable forma- as intense as seen in the measurements in Southern Finland tion mechanism is ternary nucleation (water–sulphuric (>50 events per year in Hyytial¨ a,¨ see references therein). The

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4181 largest number of events occurred in April and May. The synoptic analysis revealed NPF events favourably occurring in polar or arctic air masses. Solar radiation was observed to be one key factor needed for NPF. Additionally, a link between NPF and vertical movement of air masses caused by turbulence was observed. From the analysis of an eight-year dataset of aerosol size distributions observed at the boreal forest measurement site in Hyytial¨ a¨ (30 January 1996–31 December 2003, total of 2892 days) Dal Maso et al. (2005) found, that NPF events hap- pened in the boreal forest boundary layer on at least 24% of Fig. 1. Typical UCN evolution in the CSL during a NPF burst. all days. The average formation rate of particles larger than 3 nm was 0.8 cm−3 s−1, with enhanced rates during spring and autumn. The created event database is valuable for the 5 Modelling approach elucidation of the mechanism leading to atmospheric NPF. Recently, Hyvonen¨ et al. (2005, see references therein) re- 5.1 Rationale of non-local and high-order modelling ported on an occurrence of NPF events in boreal forest environments of around 50–100 times a year, corresponding to The modelling approach is motivated by the following facts: 14–27% of all days. A detailed evaluation of their findings 1. Local closure (known, e.g., as K-, small-eddy, or down- will be given in Paper IV. gradient theory) is generally accepted to be only valid, In general, NPF events are characterised by a strong increase if the characteristic scale of turbulent motions is very in the concentration of nucleation mode particles (diame- small compared to the scale of the mean flow. This is 4 −3 ter <10 nm, particle number concentration >10 cm ), a ensured for stable and neutral conditions. subsequent shift in the mean size of the nucleated particles and the gradual disappearance of particles over several hours 2. During convective conditions, the dominant eddy length (Birmili and Wiedensohler, 2000; Birmili, 2001). Figure 1 scale often exceeds the CBL depth, hence: shows a generalised pattern of the diurnal evolution of the UCN number concentration during a typical NPF event, ob- (a) Turbulent motions are not completely SGS ones in servable in the convective surface layer (CSL). The typical grid layers. UCN evolution, shown in that figure, is derived from a num- (b) Vertical gradients in the well-mixed layer are usu- ber of previous observations, published, e. g., by Kulmala ally very weak. et al. (1998b, Figs. 3 and 4), Clement and Ford (1999a, (c) Entrainment fluxes can significantly alter the CBL Fig. 2), Birmili and Wiedensohler (2000, Fig. 1), Birmili dynamics. et al. (2000, Figs. 1 and 2), Coe et al. (2000, Fig. 1), Aalto et al. (2001, Figs. 8, 11 and 13), Buzorius et al. (2001, Fig. 6), (d) Countergradient transports can take place in nearly Clement et al. (2001, Fig. 1), Kulmala et al. (2001a, Fig. 1), the entire upper part of the CBL (Sorbjan, 1996; Kulmala et al. (2001b, Fig. 4), Nilsson et al. (2001a, Fig. 4), Sullivan et al., 1998). Such transports are rele- Boy and Kulmala (2002, Fig. 1), Birmili et al. (2003, Figs. 1, vant for turbulent heat, momentum and concen- 2, 4, 5 and 14), Boy et al. (2003b, Fig. 1), Buzorius et al. tration fluxes (Holtslag and Moeng, 1991; Frech (2003, Fig. 6), Stratmann et al. (2003, Figs. 10, 11 and 17), and Mahrt, 1995; Brown, 1996; Brown and Grant, Boy et al. (2004, Figs. 1 and 2), Held et al. (2004, Figs. 1, 2 1997). and 3), Kulmala et al. (2004b, Fig. 1), Kulmala et al. (2004d, Consequently, for unstable conditions non-local closure Fig. 2), O’Dowd et al. (2004, Fig. 3), Siebert et al. (2004, techniques are required even for horizontally homoge- Fig. 3), Steinbrecher and BEWA2000-Team (2004, Fig. 5), neous turbulence over a flat surface and zero mean wind Dal Maso et al. (2005, Figs. 2 and 4), Gaydos et al. (2005, (Ebert et al., 1989; Pleim and Chang, 1992). Never- Figs. 1, 3 and 4) and Kulmala et al. (2005, Fig. 1). theless, even most of state-of-the-art non-local mixing The aim of the present approach is twofold: schemes have difficulties to represent the entrainment processes at the top of even the cloudless boundary layer 1. Reproduction of the typical UCN evolution during a (Ayotte et al., 1996; Siebesma and Holtslag, 1996; Ab- NPF event as represented in Fig. 1; della and McFarlane, 1997, 1999; Mironov et al., 1999).

2. Proposition of a suitable method for the estimation of 3. Turbulent non-local transport was demonstrated to be chemical composition ﬂuxes of the particulate phase, important not only in convective turbulence but also in which is suggested to be a major task for future PBL neutral conditions, hence deserving to be accounted for research (Held et al., 2004). in PBL modelling (Ferrero and Racca, 2004). www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4182 O. Hellmuth: Burst modelling

4. As gradients of chemical concentrations are often more To ensure traceability, the final equations are given in Ap- severe than gradients of heat, moisture and momen- pendix A–E. The appendices are organised as follows: tum, non-local closure is much more stringent for atmo- – Appendix A contains the list of symbols (A1), the list spheric chemistry models than for meteorological ones of constants (A2), the list of parameters (A3), the list of (Pleim and Chang, 1992). In addition, in the CBL reac- annotations (A4), the used abbreviations (A5), the def- tants were found to be normally segregated. Under such inition of the scaling properties (A6) and the definition conditions, chemical transformations depend on turbu- of the turbulence-dissipation length scale (A7). lent mixing, especially when the time scale of the chemical transformations is in the order of the turbulent char- – In Appendix B, the basic model equations are given, acteristic time scale. Then, the mean transformation rate including the description of the meteorological model of multimolecular reactions can be strongly affected by (B1), the description of the chemical model (B2), the co-variances of concentration fluctuations, which de- description of the aerosoldynamical model (B3), the deserves, e.g., an adjustment of the eddy diffusivity or a termination of the condensation coefficient and Fuchs– parameterisation of effective reaction rates, accounting Sutugin correction for the transition regime (B4), the for inefficient mixing due to SGS turbulence in terms of determination of the Brownian coagulation coefficient large-scale grid length (Galmarini et al., 1997; Thuburn (B5), the determination of the humidity–growth factor and Tan, 1997; Verver et al., 1997; Vinuesa and Vila-` (B6), the semi-empirical expression for the determina- Guerau de Arellano, 2005). Thuburn and Tan (1997, see tion of the time-height cross-section of the hydroxyl references therein) demonstrated, that the negligence of radical (B7) and the expression for the large-scale sub- co-variance terms in chemical reaction rates can cause sidence velocity (B8). significant errors in predicted chemical rates. Vinuesa – Appendix C includes the Reynolds averaged model ` and Vila-Guerau de Arellano (2005) demonstrated, that equations including the first-order moment equations heterogeneous mixing due to convective turbulence im- (C1), the second-order moment equations (C2) (i.e., the portantly impacts chemical transformations by slow- velocity correlations (C2.1), the scalar fluxes (C2.2), ing down or increasing the reaction rate, depending on the scalar correlations (C2.3), the interaction of a re- whether reactants are transported in opposite direction active tracer with a passive scalar (C2.4), the inter- or not. action of reactive tracers (C2.5)), the third-order moment equations (C3) (i.e., the turbulent transport of mo- 5. High-order closure becomes more and more common mentum fluxes (C3.1), the turbulent transport of scalar in modelling physical climate processes and their feed- fluxes (C3.2), the turbulent transport of scalar correla- backs (IPCC, 2001, Section 7.2.2.3). A comprehensive tions (C3.3–C3.4)) and the buoyancy fluxes (C4). review and discussion of state-of-the-art parameterisa- – In Appendix D, the initial and boundary conditions are tions of triple correlations and SGS condensation can given, for the first-order moments (D1) (i.e., the ini- be found in Zilitinkevich et al. (1999) and Abdella and tial conditions (D1.1), the lower boundary conditions McFarlane (2001). Recent high-order modelling stud- (D1.2), the similarity functions (D1.3), the skin prop- ies were performed, e.g., by Cheng et al. (2004), Fer- erties (D1.4), the stability functions (D1.5), the upper rero and Racca (2004), Larson (2004), Lewellen and boundary conditions (D1.6)), for the second-order mo- ` Lewellen (2004) and Vinuesa and Vila-Guerau de Arel- ments (D2) (i.e., the initial conditions (D2.1), the lower lano (2005). boundary conditions (D2.2), the upper boundary conditions (D2.3)) and for the third-order moments (D3) (i.e., 5.2 Model description the initial conditions (D3.1), the upper boundary conditions (D3.2)). The closure approach adapted here, inclusive approximations – Appendix E contains a brief description of the numer- (e.g., Rotta’s return-to-isotropy hypothesis for pressure co- ical realisation of the model, including the Adams– variance terms, quasi-normal approximation for the quadru- Bashforth time differencing scheme (E1), the vertical ple correlations, clipping approximation for ad hoc damp- finite-differencing scheme (E2) (i.e., the grid structure ing of excessive growing triple correlations), parameterisa- (E2.1), the standard differencing scheme (E2.2), the tion of turbulence-length scale, numerical model (discretisa- derivatives of the mean variables in the third-order motion, time integration scheme, filtering of spurious oscilla- ment equations (E2.3)). tions), initial and boundary conditions and stability analysis are based on the third-order modelling studies of Andre´ et al. 5.2.1 PBL model (1976a,b, 1978, 1981) and on the second-order ones of Wich- mann and Schaller (1985, 1986) and Verver et al. (1997), in The PBL model includes predictive equations for the hori- each case for the cloudless PBL. zontal wind components, the potential temperature and the

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4183 water vapour mixing ratio (Appendix B1). tions, which are called “spurious oscillations” (Moeng and To calculate the surface fluxes of momentum, heat and hu- Randall, 1984). Wichmann and Schaller (1985) argued, that midity a semi-empirical flux-partitioning scheme proposed spurious oscillation solutions are neither typical for nor re- by Holtslag (1987) is used. It solves the surface energy bud- stricted to third-order closure models and arise from the use get by a simplified Penman–Monteith approach. This scheme of explicit time differencing schemes. Using a second-order was originally developed for daytime estimates of the surface turbulence closure, the authors showed, that spurious oscil- fluxes from routine weather data. The scheme is designed for lations can be suppressed by use of an implicit time differ- grass surfaces, but it contains parameters, which take vari- encing scheme. In the present version, an explicit time dif- able surface properties into account. In the scheme, both the ferencing scheme is retained. To damp spurious oscillations, surface radiation budget (incoming solar radiation, reflected artificial diffusion terms are added to the right-hand sides of solar radiation from the surface, incoming longwave radia- the third-order moment equations as recommended by Mo- tion from the atmosphere, outgoing longwave radiation from eng and Randall (1984). The treatment of spurious oscilla- the surface) (Holtslag, 1987, p. 31–39, Section 3) and the sur- tions are discussed in more detail in Section 6. face energy budget (sensible heat flux, latent heat flux, soil heat flux, surface radiation budget) (Holtslag, 1987, p. 39– 5.2.2 Chemical model 47, Section 4) are employed. The soil heat flux is parameterised in terms of the net radiation. The partitioning of the The chemical model consists of three predictive equations surface energy flux (minus soil heat flux) between the sensi- for NH3, SO2 and H2SO4, in which emission, gas phase ble and latent heat flux is based on the Penman–Monteith oxidation, condensation loss on nucleation mode, Aitken approach (Monteith, 1981). Both fluxes are parameterised mode and accumulation mode particles, molecule loss due in terms of net radiation and soil flux using semi-empirical to homogeneous nucleation and dry deposition are consid- parameters, which are adapted from observations (Holtslag, ered (Appendix B2). To reduce the chemical mechanism, the 1987, p. 40–41). Knowing the heat flux, the Monin–Obukhov OH evolution is diagnostically prescribed (Liu et al., 2001) length scale and the surface momentum flux or shear stress, (see Appendix B7, Eq. (B20) and Appendix D1.1, Eq. (D2)). respectively, is determined (Holtslag, 1987, p. 47–50). This For the pseudo-second order rate coefficient k1 of the OH– −12 way, the Prandtl layer parameterisation is closed, i.e., the re- SO2 reaction to produce H2SO4, a value of k1=1.5×10 3 −1 −1 quired fluxes can be calculated from gridscale variables. Al- cm molecules s according to DeMore et al. (1994) is though this scheme is highly parameterised, it was found to used (see Eq. (B7), Appendix B2). This value was also be suitable for the present purpose. Alternatively, the model used in Liu et al. (2001). In her study, Ervens (2001) −12 3 −1 −1 can be forced using prescribed lower boundary conditions used a higher value of k1=2×10 cm molecules s from observations. taken from Atkinson et al. (1997). In their pseudo- The diabatic heating/cooling rate due to longwave and steady state approximation of the H2SO4 mass balance shortwave radiation, i.e., (∂θ/∂t) =θ/T)×(∂T /∂t) , is equation Birmili et al. (2000) used a lower value of rad rad −13 3 −1 −1 calculated according to Krishnamurti and Bounoua (1996, k1=8.5×10 cm molecules s proposed by DeMore p. 194–207). For the longwave radiation an emissivity tab- et al. (1997). This value is close to that employed in the ulation method is used. In this method, the emissivity is ex- H2SO4 closure study of Boy et al. (2005, Table 2, Reac- pressed as a function of the path length, which in turn de- tion 14, parameter k14, see references therein), who used −13 3 −1 −1 pends on temperature, pressure and relative humidity. The k1=9.82×10 cm molecules s . With respect to the basic emissivity values are given in a look-up-table, from reaction constants reported here, the maximum is by a fac- which the actual value of emissivity is then interpolated for tor of ≈2.35 higher than the minimum. Hence, the choice a given path length. Only absorption and emission by wa- of the higher value of k1 in the present study moves the pre- ter vapour are considered. The calculation of shortwave ra- dicted H2SO4 vapour toward higher concentrations. The con- diation is based on an empirical absorptivity function of sequences for the interpretation of the model results will be water vapour. Aerosols are not considered in the radiation discussed in more detail in Paper IV. model. The radiative transfer calculations are performed at 5.2.3 Aerosoldynamical model each time step. Vertical advection due to large-scale subsidence is consid- As in a third-order turbulence closure the total number of ered by an empirically prescribed subsidence velocity. For equations exponentially increases with the number of predic- fair-weather conditions, anticyclones etc., associated with tive variables (Stull, 1997, p. 198–199, Table 6-1 and 6-2), clear skies and strong nocturnal radiative cooling, Carlson an aerosoldynamical model with a low number of predic- − . −1 and Stull (1986) found vertical velocities of 0 1 ms to tive variables is applied. The aerosol model is based on a − . −1 0 5 ms near the top of the stable boundary layer. monodisperse approach of Kulmala et al. (1995, 1998b)6, As a consequence of the quasi-normal approximation, the third-order moment equations were demonstrated to be of hy- 6The authors used the annotation integral aerosol dynamics perbolic type (wave equation) containing non-physical solu- model. www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4184 O. Hellmuth: Burst modelling

Pirjola and Kulmala (1998) and Pirjola et al. (1999, 2003). It of H2SO4 and NH3 (Korhonen et al., 1999). This is due to consists of predictive equations for two moments (number the effect, that NH3 is able to stabilise the critical embryo, and mass concentration) in three modes (nucleation mode, i.e., to reduce its size leading to enhanced nucleation rates Aitken mode, accumulation mode) and considers homoge- (Yu, 2003; Weber et al., 1996). neous nucleation, condensation onto particle surfaces, intra- Recently, Berndt et al. (2005) performed laboratory exper- and inter-mode coagulation as well as dry particle deposition iments on NPF in an atmospheric pressure flow tube. Un- (Appendix B3). der irradiation with UV light, H2SO4 was in situ produced from the reaction of OH with SO2. NPF was observed for 6 −3 5.2.4 Nucleation model [H2SO4] ≥7×10 cm . For a temperature of 293 K, a relative humidity ranging from 28–49.5%, a NH3 mixing ratio 7 −3 The calculation of the nucleation rate is based on the classi- below 0.5 pptv and a H2SO4 concentration of ∼10 cm , cal theory of homogeneous nucleation. Thereafter, the rate of the authors observed a nucleation rate of 0.3−0.4 cm−3 s−1 homogeneous nucleation Jnuc=Kkin exp(−Gsp/(kB T )), i.e., (particle size ≥3 nm). This nucleation rate was found to be the number of newly formed critical embryos or nuclei per in line with the lower limit of the nucleation rates observed volume and time unit is a product of both a kinetic and a in the atmosphere. Because of the very low NH3 concen- thermodynamical part. The prefactor Kkin is mainly based on tration of ≤0.5 pptv in the flow tube compared to 100 to nucleation kinetics, and the thermodynamical part is propor- 10 000 pptv occurring in the CBL, the authors called the sub- tional to the Gibbs free energy Gsp, required to form the crit- stantial role of NH3 in the nucleation process into ques- ical cluster (Kulmala et al., 2003). Owing to limited solva- tion. Berndt et al. (2005) compared their experimental nu- tion, small clusters are less stable then the bulk, which leads cleation rates with theoretical ones of Napari et al. (2002a) for moderate supersaturation to the formation of a barrier on and Vehkamaki¨ et al. (2002). The authors concluded: “The the Gibbs free energy surface for cluster growth (Lovejoy H2SO4 concentration needed for substantial binary nucle- et al., 2004). Nucleation according to the classical nucleation ation is ∼1010cm−3, i.e., far too high compared to the ex- 7 theory (CNT) is limited by barrierless nucleation, where the perimental values . Ternary NH3-influenced nucleation does formation energy is zero and only kinetic contribution is in- not explain the observed particle numbers if NH3 mixing ra- cluded. The kinetic part itself is smaller, the lower the vapour tios below 0.5 pptv are assumed [...]. Furthermore, the nu- concentration is (Kulmala et al., 2003). Corresponding to cleation rate seems to behave like H2SO4 concentration to the number of participating species, the applicability of the a power of smaller than 3 to 5, indicating that the nucle- classical binary nucleation theory (CBNT) and the classical ation mechanism depends clearly on H2SO4 concentration, ternary nucleation theory (CTNT) is a subject of a controver- and suggesting a very close to trimer or dimer (kinetically sial discussion. controlled) nucleation mechanism [...]” (Berndt et al., 2005, The CBNT is known for its large uncertainties in ex- p. 700, see references therein). plaining observed nucleation rates. The difference between Weber et al. (1996) derived a semi-empirical expression predicted and observed nucleation rates can exceed several for the collision-controlled nucleation rate with considera- orders of magnitude (Pandis et al., 1995; Kulmala et al., tion of cluster evaporation, non-accommodation for H2SO4 1998a). NPF often occurs at H2SO4 concentrations, which collision and the reaction of a stabilising species such as NH3 are lower than those predicted by the CBNT. The observed with clusters containing one or more H2SO4 molecules and nucleation frequencies are often higher than those predicted H2O. The steady-state rate of particle formation in the ab- by the CBNT (de Reus et al., 1998; Stanier et al., 2003). On sence of cluster scavenging by pre-existing particles reads the other side, binary nucleation is supported by the fact, that 2 2 Jobs ≈ βγ [H2SO4] (1) many observed NPF events are associated with elevated SO2 levels and photochemically induced production of H2SO4 with [H2SO4] being the sulphuric acid concentration in vapour (e.g., Marti et al., 1997) and by the dominating con- [cm−3], β=3×10−10 cm3 s−1 being the collision frequency tribution of sulphate to the total aerosol mass, as shown, e.g., function for the molecular cluster, that contain the stabilised by Muller¨ (1999) for a rural continental test site influenced H2SO4 molecule, and γ being the thermodynamically de- by power plants. Strong arguments for kinetically controlled termined NH3-stabilised fraction of the H2SO4 monomer binary nucleation were provided by Weber et al. (1996) and (γ =0.001 for Hawaii, γ =0.003 for Colorado). Yu (2003). The nucleation rate can be enhanced due to the In the literature, the so-called ion-mediated nucleation has higher stability of embryonic H2O/H2SO4 clusters, which attracted growing attention, because atmospheric gaseous increases the cluster lifetime and hence, the chance of such ions were found to can effectively induce NPF in two steps a cluster to grow into a particle of detectable size (de Reus (Eichkorn et al., 2002): et al., 1998). 1. Ion-induced nucleation: Compared to the CBNT, the CTNT of H2O/H2SO4/NH3 predicts significantly higher nucleation rates and more fre- 7This statement refers to predictions from the theoretical ap- quent nucleation under typical tropospheric concentrations proaches of Napari et al. (2002a) and Vehkamaki¨ et al. (2002).

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4185

(a) Formation of a stable neutral molecular cluster; 2. According to the CNT, NPF performs instantaneously, i.e., the time scale of the growth kinetics of the sub- (b) Attachment of low-volatility trace gas molecules to critical embryos is neglected compared to other relevant small ions leading to large cluster ions; time scales in aerosol evolution (Yu and Turco, 2001b). (c) Ion-ion recombination of positive and negative cluster ions leading to an electrically neutral molec- To overcome these shortcomings, self-consistent kinetic the- ular cluster, which may be sufficiently large to ex- ory is desired. In the kinetic theory, the cluster formation is perience spontaneous growth by condensation of described as a sequence of basic collision steps beginning low-volatility trace gas molecules; with the vapour phase (Yu and Turco, 2001b). Afterwards, molecular-scale coagulation and dissociation act as the driv- 2. Subsequent growth of the freshly nucleated cluster ing processes of aerosol evolution. Condensation and coag- to a small volatile aerosol particle of detectable size ulation are treated analogously, i.e., condensation (evapora- (Dp≥3 nm). tion) is equivalent to coagulation (dissociation) of a molecule or small molecular cluster with (from) a particle (Yu and There are two major effects of the net electric charge of an Turco, 2001b). Kinetic theory can explicitly describe inter- ion: acting systems of vapours, ions, charged and neutral clusters and pre-existing particles at all sizes in a straightforward and – Large stabilisation of a cluster ion compared to an elec- self-consistent manner. Being physically more realistic and trically neutral cluster of the same size and composition; flexible, it is considered to be superior to the classical approach (Yu and Turco, 2001b). However, compared to new – Much faster growth of a cluster ion due to the large ion- theoretical approaches based, e.g., on ab initio molecular dy- molecule collision cross-section resulting from the long namics and Monte-Carlo simulations, the CNT is still the range of the charge-dipole interaction. only one, which can be used in atmospheric modelling, even if molecular approaches are needed to confirm results ob- Favoured conditions for the occurrence of ion-induced nu- tained by the CNT (Noppel et al., 2002). At least, the time cleation are low temperatures and large concentrations of scale of the growth kinetics of critical embryos can be a low-volatility gases. The UT is suspected to be a favoured posteriori considered within the framework of the CNT by place for ion-mediated nucleation under certain conditions adapting the concept of the “apparent nucleation rate” (Ker- (Eichkorn et al., 2002; Lovejoy et al., 2004). A strong ar- minen and Kulmala, 2002). The apparent nucleation rate is gument supporting ion-mediated NPF also results from the the rate, at which newly formed particles appear in the sen- fact, that in both the homogeneous binary and ternary nu- sor detection range. When new embryos, i.e., critical clusters cleation, the growth rate of newly formed nanoparticles from with ∼1 nm diameter, grow in size by condensation and intra- condensation of H2O, H2SO4 and NH3 was found to be often mode coagulation, their number concentration decreases. As far too low to explain the rapid appearance of fresh ultrafine a result, the apparent nucleation rate is lower than the real particles during midday (Weber et al., 1997; Yu and Turco, one derived from nucleation theory. Kerminen and Kulmala 2000). Consequently, Yu and Turco (2000) hypothesised, that (2002) derived an analytical formula to relate the apparent to some of present day discrepancies between theory and obser- the real nucleation rate to cut off the lowest desirable scale vation may be due to the intervention of background ionisa- for the evolution of the aerosol size distribution in explicit tion in NPF. For details the reader is referred, e.g., to We- nucleation schemes of atmospheric models. However, this ber et al. (1997), Horrak˜ et al. (1998), Yu and Turco (2000, formula is not applicable to very intensive nucleation bursts, 2001a,b), Eichkorn et al. (2002), Laakso et al. (2002), Her- to potential nucleation events associated with cloud outflows mann et al. (2003), Laakso et al. (2003), Nadykto and Yu or to nucleation occurring in plumes, which undergo strong (2003), Yu (2003), Lovejoy et al. (2004) and Wilhelm et al. mixing with ambient air. (2004). Based on a critical analysis of the advantages and short- Apart from the question, which species contribute to nu- comings of the classical Gibbs approach to determine the cleation, the CNT suffers from two essential shortcomings: parameters of the critical cluster and the work of critical cluster formation in heterogeneous systems, a gener- 1. Molecular clusters are represented by droplets up to and alised Gibbs approach has been proposed and applied to de- including the critical size, characterised by bulk prop- scribe phase formation processes in multicomponent solu- erties of the condensed phase, such as surface tension, tions by Schmelzer et al. (1999, 2002, 2003, 2004a,b, 2005), density, vapour pressure (Yu and Turco, 2001b). This Schmelzer and Schmelzer Jr. (2002) and Schmelzer and so-called “capillarity approximation” is known to be in- Abyzov (2005). This approach delivers a theoretically well- appropriate for small molecular clusters with ∼1 nm di- founded description not only of thermodynamic equilibrium ameter. It causes large uncertainties in nucleation rates but also of thermodynamic non-equilibrium states of a clus- predicted by the CNT (Lovejoy et al., 2004). ter or ensembles of clusters of arbitrary sizes and composi- www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4186 O. Hellmuth: Burst modelling tions. The new method allows the consideration of a depen- Recently, Kulmala et al. (2006) proposed a cluster acti- dence of the surface tension of the clusters of arbitrary sizes vation theory to explain the linear dependence between the on the state parameters of both coexisting macrophases. The formation rate of 3 nm particles and the H2SO4 concen- generalised Gibbs approach results in relations for the pa- tration. While existing nucleation theories predict a depen- rameters of the critical cluster, which are different, in gen- dence of the nucleation rate on the H2SO4 concentration to eral, from the mechanical, thermodynamical and chemical the power of ≥2, atmospheric observations suggest a power equilibrium conditions derived by Gibbs. Most importantly, between one and two. The activation theory is able to ex- for clusters of nanometric sizes, the generalised Gibbs ap- plain the observed slope. According to the new approach, the proach leads to different results for the determination of the cluster containing one H2SO4 molecule will activate for fur- state parameters of the critical clusters and the work of criti- ther growth due to heterogeneous nucleation, heterogeneous cal cluster formation in comparison with the classical Gibbs chemical reactions including polymerisation or activation of method. Schmelzer et al. (2005) found, that the results of the soluble clusters. In the activation process organic vapours generalised Gibbs approach are in qualitative and partly even are typically needed as condensing agents. This concept is quantitative agreement with the van der Waals square gradi- in general agreement with the theory of TSCs proposed by ent method and more advanced density functional computa- Kulmala et al. (2000). tions of the parameters of the critical clusters and the work A detailed evaluation of the most recent nucleation con- of critical cluster formation. cepts is beyond the scope of the present paper. The inten- Kuni et al. (1999) investigated the kinetics of non-steady tion of the present paper is to discuss mechanistic ways, how condensation on macroscopic solid wettable nuclei. The au- CBL turbulence can affect atmospheric NPF. Therefore, the thors determined kinetic characteristics, such as the number present approach is restricted to the evaluation of classical of supercritical drops formed, the time of start and the time of concepts of homogeneous binary and ternary nucleation. For duration of the stage of effective nucleation of supercritical the calculation of the homogeneous binary H2O/H2SO4 nu- drops as well as the width of the size spectrum of the super- cleation rate and the critical cluster composition the so- critical drops in dependence (i) on initial parameters specify- called “exact” model with consideration of cluster hydra- ing the properties of condensing liquid, the vapour-gas sur- tion effects is implemented (Stauffer, 1976; Jaecker-Voirol roundings and the kind of matter of solid wettable nucleus, et al., 1987; Jaecker-Voirol and Mirabel, 1988, 1989; Kul- (ii) on sizes of condensation nuclei and their initial number mala and Laaksonen, 1990; Laaksonen and Kulmala, 1991; in the vapour-gas medium as well as (iii) on the rate of ex- Kulmala et al., 1998a; Seinfeld and Pandis, 1998, Chap- ternally determined establishment of the vapour supersatura- ter 10). For the ternary H2O/H2SO4/NH3 nucleation rate tion. and the critical cluster composition a state-of-the-art param- Shchekin et al. (1999) investigated the thermodynamic eterisation of Napari et al. (2002a,b) is used. The limits of conditions of the process of heterogeneous nucleation the validity of the ternary nucleation parameterisation are 4 9 −3 through the formation of thin, uniform and non-uniform, T =240−300 K, RH=5−90%, [H2SO4]=10 −10 cm , −5 6 −3 −1 wetting films on solid insoluble macroscopic nuclei, which [NH3]=0.1−100 ppt and Jter=10 −10 cm s . The pa- allows, e.g., the determination of the work of wetting using rameterisation can not be used to obtain the binary a relatively small number of, in general, quantifiable param- H2O/H2SO4 or H2O/NH3 limit (Napari et al., 2002b). When eters. The obtained expressions can be easily incorporated the vapour concentrations fall below their lower parameteri- into an analytical theory, which describes the possibility of sation limits, they were kept at their corresponding minimum barrierless heterogeneous nucleation, developed for thermo- concentrations. dynamics and kinetics of heterogeneous nucleation occurring via formation of films of uniform thickness. Recently, 5.2.5 Condensation flux model Shchekin and Shabaev (2005) presented a thermodynamic theory of deliquescence of small soluble solid particles in an (a) Treatment of H2SO4 vapour undersaturated vapour, which allows, e.g., the determination The condensation flux represents the vapour molecular de- of the work of droplet formation under different assumptions position rate onto spherical droplets of a certain radius. It regarding the size dependence and chemical state of the sol- 8 results from the solution of the mass transport equation in uble solid core . the continuum regime, i.e., when the particle is sufficiently 8The authors defined deliquescence as follows: “The stage of large compared to the mean free path of the diffusing vapour nucleation by soluble particles in the atmosphere of solvent vapour, molecules. The solution obtained by Maxwell (1877) de- when arising droplets consist of a liquid film around incompletely scribes the total flow of vapour molecules toward an aerosol dissolved particles, is called deliquescence stage. In a supersatu- particle by diffusion on a molecular scale (Seinfeld and Pan- rated vapour, this stage is the initial one in the whole condensation process, and the particles inside the droplets will completely rium between droplets of different sizes, with residues of the parti- dissolve in the growing droplets with time. In an undersaturated cles (solid cores within the droplets) varying in size down to com- vapour, this stage finishes by establishing the aggregative equilib- plete dissolution”(Shchekin and Shabaev, 2005, p. 268).

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4187 dis, 1998, p. 596–597, Maxwellian flux). When the mean free in defining the growth rate. The authors argued: “There ap- path length of diffusing vapour becomes comparable to the peared recently to be a theoretical and experimental con- particle diameter, i.e., when the Knudsen number is Kn≈1, sensus that sticking probabilities for water and sulphuric the phenomena are said to lie in the transition regime (Se- acid on liquid droplets are small (Itoh, 1990; Van Dingenen infeld and Pandis, 1998, p. 601). Then, the Maxwellian flux and Raes, 1991). Values quoted for sulphuric acid ranged needs to be adjusted by a transitional correction factor. This between 0.02 and 0.09, and, in their model of the rela- correction factor extends the growth rate from the continuum tionship between DMS flux and CCN concentration in re- regime into the transition regime. To match the continuum mote marine regions, Pandis et al. (1994) used a base case and free molecule fluxes, several approaches have been pro- value of 0.02 with a test value of 0.05. However, experi- posed, known, e.g., as Fuchs theory, Fuchs and Sutugin ap- mental and theoretical evidence from a wide field of surface proach, Dahneke approach, Loyalka approach, Sitarski and science, including molecular beam experiments and molec- Nowakowski approach. For a description and additional ref- ular dynamics simulations, suggests that such small val- erences the reader is referred to Seinfeld and Pandis (1998, ues are extremely unlikely at normal atmospheric temper- p. 601–605). Following previous attempts, e.g., that of Liu atures (Clement et al., 1996). Direct experiments on grow- et al. (2001), here the so-called Fuchs–Sutugin factor in ing droplets of n-propanol/water mixtures give results con- terms of Knudsen number has been applied (Fuchs and Sutu- sistent with Sp=1, the theoretically most likely value for gin, 1971). all except very light atoms or molecules on dense mate- In the present approach, the condensation flux param- rials at normal temperatures. Recent measurements of the eterisation is applied to nucleation mode, Aitken mode growth rates of ultrafine particles at remote marine and and accumulation mode particles. The Maxwellian flux is continental sites (Weber et al., 1996, 1997) are consistent proportional to the driving force, i.e., the difference between with Sp=1 for sulphuric acid. Since the evidence for Sp the partial pressure of the condensable vapour far from the that is much less than unity is generally reliant on complex particle (ambient conditions) and the vapour pressure at the modelling subject to alternative interpretations, we much droplet surface, the latter being a product of the equilibrium prefer values close to unity” (Clement and Ford, 1999a, partial pressure at ambient temperature and the acid activity p. 478). Clement and Ford (1999a, Fig. 1) investigated in the condensed phase. If the driving force is positive, the the influence of the sticking probability of H2SO4 onto flow of vapour molecules is directed toward the particle the droplet growth rate in the range of droplet size radii surface and if negative vice versa. In the present approach, Rp=0.001−10 µm. The Fuchs–Sutugin transitional correc- maximum Maxwellian flux is considered, i.e., the ratio of tion factor (and therewith the growth rate of droplets with the vapour pressure at the droplet surface to the ambient radius Rp) for Rp=10 nm is by approximately two or- partial pressure is assumed to be much lower than 1. This ders of magnitude lower at Sp=0.02 than at Sp=1. At the means, that the diffusive flux is always directed to the smaller sizes, the Fuchs–Sutugin transitional correction fac- particle surface. This assumption is accepted to be valid for tor is proportional to Sp. For droplets with Rp=1 µm the low-volatile vapours, such as sulphuric acid (Clement and respective correction factors differ by one order of mag- Ford, 1999a,b; Liu et al., 2001; Boy et al., 2003b; Krejci nitude. As the most of the atmospheric aerosol is in the et al., 2003; Held et al., 2004; Gaydos et al., 2005) and for size range 0.01−1 µm, growth rates have an uncertainty of organic molecules, such as dicarboxylic acids and for pinon over a factor of 10 for nearly all the aerosol if Sp is unde- aldehyde as the main product of α-pinene oxidation (Boy termined (Clement and Ford, 1999a). The authors empha- et al., 2003b; Held et al., 2004). sised, that the change in the slope of the Sp=1 and Sp=0.1 curves in the accumulation mode region Rp=0.1−1 µm, de- (b) H2SO4 mass accommodation coefficient (or sticking picted in Clement and Ford (1999a, Fig. 1), has an influ- probability) ence on the acid condensation on the aerosol in this mode relative to that on smaller sized aerosols. From their mod- The Fuchs–Sutugin transitional correction factor depends on, elling study on sulphate particle growth in the MBL, Ker- among others, the so-called mass accommodation coefficient minen and Wexler (1995) concluded the necessity to use αgas (or sticking probability Sp) (see Eq. (B16) in Appendix high values of Sp for smaller particles. The choice of the B4). In their box modelling study Kulmala et al. (1998b) sticking probability is highly important for the determination investigated the sensitivity of the model results against the of the molecular removal onto aerosol, which often speci- sticking probability. Varying this parameter between 0.001 fies the molecular concentration in the atmosphere. Clement and 1.0, the authors concluded, that the change of the sticking and Ford (1999a, p. 485) concluded: “For sulphuric acid probability on accumulation mode particles does not change molecules there is a large difference in removal rates used at the growth properties of nucleation mode particles signifi- present arising from differing values for the molecular stick- cantly. ing probability. For theoretical reasons we favour a value of In opposite to this, Clement and Ford (1999a) considered the unity, which is the value supported by experiments on other uncertainty in the sticking probability as the main problem molecules such as water.” For barrierless nucleation Clement www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4188 O. Hellmuth: Burst modelling and Ford (1999b) found a strong influence of the sticking (c) Treatment of NH3 probability on the characteristic time scale for a homoge- With respect to atmospheric NH , the assumption of max- neous nucleation process to affect the vapour concentration 3 imum Maxwellian flux is questionable. As demonstrated by (time scale for burst). This time scale will be reduced by a Nenes et al. (2000), the vapour pressure at the droplet surface factor of 0.074, when S is reduced from 1 to 0.02. Cor- p strongly depends, e.g., on droplet composition. In their NPF respondingly, an increase of the characteristic time scale for modelling study, Gaydos et al. (2005) argued, that H SO removal on pre-existing aerosols is expected as a result of this 2 4 condensation alone produces growth, that is similar to obser- change. Hence, low values of the sticking probability favour vations, hence NH condensation by molecular diffusion was the occurrence of NPF events. 3 not considered explicitly. Therefore, the NH gas phase con- Evaluating the pseudo-steady state approximation of the 3 centration was diagnostically determined from the assump- H SO mass balance to calculate the H SO concentration, 2 4 2 4 tion, that total ammonia, total nitrate and total sulphate, taken Birmili et al. (2000) assumed a sticking probability of unity from measurements, are always in thermodynamic equilib- to estimate the condensation sink. rium with gas phase and particulate phase concentrations In their mesoscale aerosol modelling study Liu et al. (2001) (Ansari and Pandis, 1999, model GFEMN). Here, a simi- used a sticking probability of 0.3 for H SO according to 2 4 lar approach is applied. The gas phase NH concentration is Worsnop et al. (1989). Liu et al. (2001, p. 9700) commented 3 diagnostically determined for given total ammonium and to- this value as follows: “Our value for the accommodation co- tal sulphate concentration (gas phase plus particle phase) us- efficient is an intermediate value in the range which extends ing the inorganic aerosol thermodynamic equilibrium model from ∼0.1 to 1.0 (Jefferson et al., 1997). More extreme val- ISORROPIA (meaning “equilibrium” in Greek) of Nenes ues could significantly modify the magnitude of nucleation et al. (2000). The total ammonium and total sulphate con- rates although spatial patterns are unlikely to be altered.” centrations are prognostically determined. In their analytical tool to derive formation and growth properties of nucleation mode aerosols (diameter growth 5.2.6 Humidity growth rate, aerosol condensation and coagulation sinks, concentration of condensable vapours and their source rate, 1 nm par- The water uptake of dry aerosol is considered by applying ticle nucleation rates) from measured aerosol particle spec- the empirical humidity–growth factor of Birmili and Wieden- tral evolution, Kulmala et al. (2001a, Fig. 3a) used a stick- sohler (2004, personal communication, see Appendix B6, ing probability of unity. Performing a sensitivity study with Eq. (B19)). varying sticking probabilities the authors demonstrated, that for Sp=0.01 the total sink is significantly smaller than that 5.2.7 Particle deposition model for Sp=1. Not only the magnitude of the condensation was shown to be significantly affected by the choice of the stick- The size-segregated particle dry deposition velocity is pa- ing probability, but also the influence of supermicron parti- rameterised according to Zhang et al. (2001). cles was demonstrated to contribute more significantly to the total removal of condensable vapours and a bimodal sink re- 5.2.8 Alternative approaches sults. The sticking probability also significantly affects the calculated vapour concentration. Dividing the sticking prob- Alternatively to the third-order closure approach presented ability by a factor of 100 (Sp=0.01) requires 100 times more here, a “mixed closure model” can be realised, e.g., run- vapour to reproduce the observed aerosol growth. ning the third-order closure for PBL dynamics and the first- In their tropospheric multiphase chemistry mechanism order closure for chemistry and aerosol dynamics, respec- CAPRAM2.4 Ervens (2001) and Ervens et al. (2003) tively. Then, a higher number of predictive physicochemical (see also http://projects.tropos.de:8088/capram/capram24. variables can be considered, such as organic compounds and pdf, Table 3b: Mass accommodation coefficients and gas size bins of a sectional aerosoldynamical model. Doing so, phase diffusion coefficients) used a H2SO4 sticking proba- special care has to be taken to appropriately parameterise bility of Sp=0.12 according to Schwartz (1986). For com- the turbulent fluxes. First-order closure is based on diag- parison the authors also referred to Sp=0.07 proposed by nostic flux gradient relations, i.e., the fluxes are represented Davidovits et al. (1995). by the downgradient approach, eventually semi-empirically The low H2SO4 sticking probability used in the present corrected for countergradient flows. For the eddy diffusiv- study is expected to lead to a quite low condensation loss ity of scalars one can either use an ad hoc approach (e.g., of nucleating vapours onto newly formed and pre-existing O’Brien, 1970), diagnostic relations based on diagnostic flux particles, hence resulting in high amounts of H2SO4 vapour, equations (Holtslag and Moeng, 1991; Holtslag et al., 1995) that will favour NPF. This fact will be discussed in more or semi-empirical expressions based on the turbulence spec- detail in Paper IV, where the model results are compared trum, such as proposed and evaluated, e.g., by Degrazia et al. with observed NPF events. (1997a,b, 1998, 2001). Generally, the closure type depends mainly on the question and scale of interest as well as on

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4189 the flow characteristics, e.g., on atmospheric stability (see and Lacarrere` (1989) and Verver et al. (1997). However, there Subsection 5.1). However, the present approach opens a suit- was no a priori guarantee, that the underlying PDEs are phys- able way to investigate turbulence-related NPF. But when fo- ically stable. In cases they would not, the search for a stable cussing on “organic” NPF scenarios, e.g., occurring in bo- differencing scheme must fail. To ensure the physical and nu- real forests, where a large number of organic compounds merical stability and to evaluate the accuracy of the numer- and, perhaps, size bins must be considered, then either a pure ical scheme, a number of tests have been carried out, which first-order closure or even a mixed third/first-order closure is are described below. An important result of the present ap- recommended to use. proach is the objective evidence, that the considered modelling concept is conditionally stable12 with respect to the treatment of the physicochemical processes. 6 Numerical realisation and basic tests 6.1 Amplitude error The partial differential equations (PDEs) of the present model represent a mixed class of two types of initial- value problems. The advection terms in the governing The satisfaction of the Courant–Friedrichs–Lewy stability equations belong to the class of flux-conservative initial- criterion (CFL) (Press et al., 1996, p. 829) was ensured by value problems (Press et al., 1996, p. 824–838, Chap- empirical adjustment of the integration time step accord- ter 19.1, Eq. (19.1.6)), and the diffusion terms in the ing to the prescribed vertical grid resolution. To investigate parabolic parts belong to the class of diffusive initial-value the sensitivity of the model results against the time inte- problems (Press et al., 1996, p. 838–844, Chapter 19.2, gration scheme, the first-order forward Euler time differencing scheme was compared with the third-order Adams– Eq. (19.2.1)). The PDEs can be numerically solved simplest 13 by means of the Forward Time Centred Space (FTCS) rep- Bashforth time differencing scheme , which is known to resentation (Press et al., 1996, p. 825–827, Eqs. (19.1.1), have attractive stability characteristics (Durran, 1999, p. 65– (19.1.11)). Thereafter, the time derivative is numerically 72, Table 2.1). The differences were visually inspected and represented by the forward Euler differencing9, which is found to be small and hence, being not relevant for the only first-order accurate in 1t (Press et al., 1996, p. 826, question of interest. This agrees with the findings of Dur- [ ] Eq. (19.1.9)). For the space derivative a centred space differ- ran (1999, p. 65), who argued: “ ... A major reason for the encing10 of second-order accuracy in 1z is applied (Press lack of interest in higher-order time differencing is that in et al., 1996, p. 827, Eq. (19.1.10)). The FTCS scheme is many applications the errors in the numerical representation an explicit scheme11. Regarding the requirements of a nu- of the spatial derivatives dominate the time discretisation er- merical scheme, Press et al. (1996, p. 830) argued, that ror, and as a consequence it might appear unlikely that the “the [numerical] inaccuracy is of a tolerable character when accuracy of the solution could be improved through the use the scheme is stable.” Later on the authors added an argu- of higher-order time difference.” The studies presented in Pa- ment, “that annoys the mathematicians: The goal of numer- per II and III were carried out using the third-order Adams– ical simulation is not always ’accuracy’ in a strictly math- Bashforth scheme. ematical sense, but sometimes ’fidelity’ to the underlying physics in a sense that is looser and more pragmatic. In such 6.2 Transport error contexts, some kinds of error are much more tolerable than other” (Press et al., 1996, p. 832). The numerical realisation The large-scale subsidence is considered by additional of the model closely follows pragmatic arguments as well vertical-advection terms in the governing PDEs. The dis- as the recommendations given in the basic papers of Andre´ cretisation of these terms can cause transport errors (Press et al. (1976a,b, 1978, 1981), Bougeault (1981a,b), Moeng et al., 1996, p. 831). It was found, that the application of and Randall (1984), Andre´ and Lacarrere` (1985), Bougeault the centred-differencing scheme does work well for meteo- (1985), Wichmann and Schaller (1985), Bougeault and rological advection, but not for advection of physicochemi- Andre´ (1986), Wichmann and Schaller (1986), Bougeault cal properties, where at very low mean concentrations first- order moments can become negative. To avoid this effect, an 9 The forward Euler time scheme has the advantage, that the upwind differencing is generally used for the advection terms quantities at time step n + 1 can be calculated in terms of only (Press et al., 1996, p. 832, Eq. (19.1.27)). This scheme is only quantities known at time step n. 10 first-order accurate in the calculation of the spatial deriva- The centred space differencing also uses quantities known at tives, but it was found to be appropriate for the discretisation time step n only. of advection terms. 11Explicit scheme means, that the quantities at the time step n+1 at each level can be calculated explicitly from the quantities that are already known. The FTCS approach is also classified as a single- 12To ensure physical stability, the third-order moment PDEs for level scheme, since only values at time level n have to be stored to the physicochemical variables must be modified. See Paper III. find values at time level n+1. 13There exists also a second-order version of this scheme. www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4190 O. Hellmuth: Burst modelling

6.3 Phase error and non-linear instability damps physicochemical oscillations effectively but not sufficiently. 6.3.1 Inspection of clipping approximation 6.3.3 Inspection of space differencing The clipping approximation was found to be both a necessary and sufficient condition to ensure the stability of the meteo- Performing a third-order modelling study of a marine stra- rological model. This is fully in accordance with Andre´ et al. tocumulus layer, Bougeault (1985) pointed out, that a modi- (1978, p. 1881, see their conclusions), who concluded, that fication of the vertical finite-differencing scheme is required the realisability inequalities are absolutely necessary “since to handle the strong gradients in the temperature and mois- if one ceases to enforce them at any time the model blows ture profile at the inversion level. During the simulation of up very rapidly.” In opposite to this, the clipping approxima- stratocumulus a very sharp inversion can develop, featuring a tion was found to be a necessary but insufficient condition to strong change over one grid length with negligible gradients stabilise the physicochemical part of the model (see below). immediately above and below. The vertical derivatives of the first-order moments appear both in second- and third-order 6.3.2 Inspection of spurious oscillations moment equations. Bougeault (1985) argued, that owing to the staggered structure of the grid15, the derivatives evalu- Moeng and Randall (1984) demonstrated, that the third-order ated by the centred finite-difference scheme at the half levels equations proposed by Andre´ et al. (1978) are of hyperbolic can differ by a factor 2. This creates problems in the third- type. The solution of these equations exhibits oscillations, order moment equations, leading to the appearance of nega- which arise from the mean gradient and buoyancy terms of tive values of the variances at the half level just below the in- the triple-moment equations. In the present approach, the version. The ultimate way to solve this problem is a variable author followed the attempt of Moeng and Randall (1984, grid spacing with higher resolution near the inversion. To by- Eq. (3.5)) to damp these oscillations by introducing artifi- pass a variable grid, the author proposed a more empirical cial diffusion terms into the triple-moment equations. With solution. The centred differencing of the derivatives of the a reasonable choice of the diffusion coefficient, these os- first-moment variables in the third-order moment equations cillations can be additionally damped in the cloudless case is replaced by a geometric approximation, while all other considered here. This is fully in accordance with Moeng and derivatives are still evaluated by centred finite-differencing Randall (1984). It was found, that the physical stability of (Bougeault, 1985, p. 2827–2828, Fig. 1, Eqs. (2)–(3)). In the the numerical model is a prerequisite to damp spurious os- present study it was found, that the revised finite-differencing cillations. When the numerical model is not stable, physical scheme for the mean variables in the third-order moment stability can not be achieved by considering artificial diffu- equations – originally designed to be applied near a stra- sion terms. From a visual inspection of the triple moments tocumulus top – is not necessary for the simulation of the it was found, that the amplitude of the spurious oscillations meteorological fields in the cloudless case. This is plausi- can be effectively damped by artificial diffusion terms, but ble, as near the cloudless PBL top such large gradients of not be prevented. It should be noted, that spurious oscilla- temperature and humidity, as observed near stratocumulus tions – even being non-physical – are a part of the mathemat- cloud tops, are unlikely. Anyway, for physicochemical prop- ical solution of the underlying hyperbolic PDEs. These parts erties the situation is completely different. In both the cloud- of the solution can be traced back to an insufficient phys- less and the cloudy case much higher vertical gradients of ical parameterisation of quadruple correlations. The spuri- physicochemical properties can be expected, compared to ous oscillations, occurring in the meteorological part of the those appearing in the meteorological profiles. Consequently, third-order moment PDEs, could be successfully damped by the revised finite-differencing scheme was applied. This way, the recommendations of Moeng and Randall (1984). In con- the physical stability of the physicochemical model could trary to this, the damping of these spurious oscillations is ex- be effectively enhanced. To be consistently, the approach ceptionally difficult in the physicochemical part. Apart from proposed by Bougeault (1985) is used both in the physico- modelling passive tracers, for which spurious can be suc- chemical and meteorological model. As for artificial diffu- cessfully damped, these oscillations were found to amplify sion, the physical stability of the numerical model is a pre- themselves, when physicochemical interactions are consid- requisite for the stabilising effect of the revised differencing ered (reactive tracers). This causes instability of the physico- 14 scheme. When the model is physically unstable, instability chemical part . The introduction of artificial diffusion terms can be damped, but full stability can not be achieved by the proposed alteration of the differencing scheme. 14The same numerical model and subroutines are used in both the meteorological and the physicochemical model. Self-amplifying oscillations can be hardly called “spurious”. However, it seems, that 15In a staggered grid, first- and third-order moments are calcu- the amplification directly originate from spurious oscillations, or at lated at the main levels, and second-order ones are calculated at the least are strongly related to them. half levels.

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4191

6.3.4 Effect of artificial retardation terms alteration of prescribed fine-tuning parameters etc.). A detailed re-evaluation of this issue is intended to be performed To prevent amplifying oscillations of physicochemical third- in the context of the replacement of the clipping approxi- order moments, several attempts have been made, which mation by a state-of-the-art parameterisation of third-order consider additional empirical retardation terms in the corre- moments, recently proposed in the literature. sponding PDEs. Such terms secure an evolutionary tendency of the predicted fields toward some prescribed values, such 6.5 Sensitivity against changes in the turbulence parame- as realised in the classical nudging concept. Runs were per- terisation formed, e.g., with different retardation time scales. It was found, that such terms can prolongate the “characteristic run- 6.5.1 Extension of PDE numbers time” of an executable, that exhibits amplifying oscillations. The model equations, proposed by Andre´ et al. (1978), are However, it was not possible to prevent the amplification of originally based on a number of simplifications, e.g., the gov- such oscillations at all. erning equations for the kinematic humidity fluxes, inclusive the corresponding equations of the related triple correlations, 6.3.5 Remaining stability problems are neglected. In the present model version, these simplifications were omitted to generalise the approach. Although To prevent the amplification of oscillations, which desta- the consideration of the additional equations is numerically bilise the model, physical arguments must be considered for more expensive compared to the base case, and although it the modification of the third-order moment equations of the does presently not lead to an evaluable gain of information physicochemical model. This is explained in detail in Pa- with respect to the question of interest, the model was found per III. Finite-difference schemes for hyperbolic equations to behave “well-tempered” against that important extension can exhibit purely numerical dispersion (phase errors) and of the number of equations, i.e., stability remains ensured non-linear instability (Press et al., 1996, p. 831). The ques- and the results are qualitatively similar. tions, to what extent numerical phase errors and non-linear instability can be separated from spurious oscillations, or 6.5.2 Fine-tuning turbulence parameters how they interact, could not be answered for the time being. This issue deserves a special evaluation in a separate The fine-tuning parameters, recommended to use by Andre´ work. et al. (1978), should not be changed without reasonable care, especially without a physically motivated revision of the un- 6.4 Mass conservation derlying parameterisation. The model was found to be no- ticeably sensitive against changes of these parameters. How- The mass conservation was investigated in connection with ever, corresponding tests were not systematically performed, the prevention of amplifying oscillations in the physico- because they are ill-motivated at the present stage of work. chemical part. Three tests using a passive tracer have been performed. Firstly, a scenario with zero-background tracer 6.5.3 Length scale parameterisation concentration and a source with a time-independent unit strength located at the ground was considered. Sinks were The mixing length effectively affects the dissipation rate. As excluded. In the course of the day, the source continuously already stated by Moeng and Randall (1984), there exists no replenished the mixing layer with the tracer. The result was proper way to determine the length scale. Existing formula- a tracer profile exponentially decreasing with height. The tions are more or less arbitrary, or their validity is restricted to tracer concentration near the ground increased when the mix- special conditions, such as certain ranges of atmospheric sta- ing layer height collapsed in the late afternoon. Secondly, bility, respectively. Tests using several commonly accepted a number of runs were performed with different height- parameterisations of the mixing length scale and the mixing dependent initial profiles without a ground source. In each layer height (used as scaling properties) were carried out. In case, the tracer concentration became stationary and height- the present case, the mixing length parameterisation was ex- independent after a couple of model hours, i.e., the tracer tensively evaluated in the context of the treatment of spuri- was fully diluted throughout the boundary layer. Thirdly, ous oscillations. Bougeault and Andre´ (1986) argued, that the runs were performed with a height-independent initial pro- mixing length formulation of Mellor and Yamada (1974) is file. This profile was found to remain nearly constant during known to grossly overestimate the mixing length near the in- the runs, i.e., the fluctuations around the initial profile were version and therefore leads to a gross underestimation of the found to be negligibly small with respect to the question of dissipation rate. From a linear stability analysis the authors interest. Such kinds of tests were performed several times, concluded, that oscillations can only develop in a region of e.g., when numerically relevant revisions were conducted unstable stratification, and that in this case, they can be ef- (replacement of the time integration scheme and the discreti- ficiently controlled by decreasing the mixing length. These sation scheme, modification of the clipping approximation, findings are very important, especially for the cloud-topped www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4192 O. Hellmuth: Burst modelling

PBL, but perhaps even though for physicochemical processes respective 10 an 10 cm−3, used in the present study, cor- near the top of the cloudless PBL. Sensitivity tests show, respond to the values, used in the rural (R1, R2), the ur- that the mixing length parameterisation used in Andre´ et al. ban (U1, U2) and the marine case scenarios (M1, M2, (1978) is appropriate to simulate the meteorological evolu- M3) of the modelling study performed by Pirjola and tion of the cloudless PBL. But owing to the more sophis- Kulmala (1998, Fig. 1). In the binary nucleation study ticated adjustment of their length scale parameterisation to performed by Pirjola et al. (1999, Table 1), e.g., an ini- buoyancy-induced instabilities, the approach of Bougeault tial number concentration of 100 cm−3 was assumed for and Andre´ (1986) was also evaluated. With respect to the me- Aitken mode particles and 10 cm−3 for accumulation teorological model, the results using different schemes were mode particles (see their cases 1 and 2). found to be qualitatively very similar, but to differ quantitatively. Without a more detailed PBL verification study, no 2. As discussed in Section 6, spurious oscillations are un- scheme could be ad hoc favoured over the other one. Consid- wanted solutions of the non-linear PDE system. They ering the physicochemical model it appears, that the revised were found to be not critical in the meteorological approach proposed by Bougeault and Andre´ (1986) enhances part. In opposite to this, such oscillations can become the model stability. However, as previously discussed, when critical in the physicochemical model. Especially near the numerical model is not physically stable, than stability the CBL top, at very low concentrations of condens- can also not be achieved by alteration of the mixing length ing vapours and pre-existing aerosols the physicochem- parameterisation. The observed sensitivity of the model re- ical variables could cross zero owing to spurious oscil- sults against the length scale parameterisation shows, that its lations. Hence, a scenario with low background concen- evaluation in a model like the present one remains an impor- tration is reasonable for the evaluation of the model be- tant task, even after years of great efforts on this subject. haviour. The considered scenarios will be quantitatively characterised 7 Reference case scenarios in Paper III. They are not claimed to represent the variety of observations. A systematic evaluation of other possible NPF To select meaningful reference scenarios for the present fea- scenarios and the direct evaluation of a dedicated measure- sibility study, both phenomenological and numerical argu- ment campaign is beyond the scope of the present paper and ments are considered. In Paper III, two scenarios will be in- deserves further studies. vestigated with an emission source at the ground but very low background concentrations of aerosol, henceforth called 8 Conclusions “clean air mass” scenarios. Such situations are possible to occur in anthropogenically influenced CBLs depleted from Based on previous findings on high-order modelling, an at- air pollutants in connection with frontal air mass change and tempt is made to describe gas-aerosol-turbulence interactions post-frontal advection of fresh polar and subpolar air, respec- in the CBL. The approach can be characterised as follows: tively. 1. A third-order closure is self-consistently applied to 1. Firstly, a ground emission source is required to pro- a system of meteorological, chemical and quasi- vide precursor gases for the production of nucleating linearised aerosoldynamical equations under horizon- vapours. A low background concentration of aerosol tally homogeneous conditions. particles is required to ensure the absence of condensation sinks, which condensable vapours prevent from 2. The model is designed to predict time-height profiles of nucleating or TSCs prevent from growing to detectable meteorological, chemical and aerosoldynamical mean- size (see Kulmala et al., 2000). Such a scenario was state variables, variances, co-variances and triple corre- hypothesised and observed as a prerequisite for NPF lations in the CBL. (Nilsson et al., 2001a; Birmili et al., 2003; Buzorius et al., 2003). For example, in a comprehensive NPF- 3. The approach might be instrumental in interpreting ob- CBL evolution study Nilsson et al. (2001a, p. 459, Sub- served particle fluxes from in situ measurements and/or section 4.1, item (1)) hypothesised: “On the days when remote sensing. In addition, it provides information a dilution of the preexisting aerosol number and con- about gas-aerosol-turbulence interactions, that can not densation sink was observed before nucleation, this may be directly observed in the CBL. itself be enough to trigger nucleation by decreasing the 4. Though highly parameterised, the model configuration sink of precursor gases at the same time that the precur- considers the basic processes, which are supposed to be sor production may be increasing due to increasing pho- involved in the evolution of NPF bursts in the CBL. tochemical activity. Such a scenario would form favor- able conditions for nucleation”. The low initial Aitken 5. The model provides input information required for a mode and accumulation mode number concentrations of more sophisticated parameterisation of the effect of

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SGS turbulence on the homogeneous nucleation rate, Appendix A such as variances and co-variances of temperatures, humidity and the condensable vapours H2SO4 and NH3 List of symbols, annotations, scaling properties, con- (Easter and Peters, 1994; Hellmuth and Helmert, 2002; stants, parameters and abbreviations Shaw, 2004; Lauros et al., 2006). A1. Symbols 6. Provided, that the model is validated, it may serve as a tool, e.g., to perform conceptual studies and to ver- Latin letters ify parameterisations of SGS processes in large-scale chemistry and aerosol models (effective reaction rates, A − Replacement variable turbulence-enhanced condensation etc.). a − Dual−use character (replacement variable and constant) Known shortcomings of the nucleation approach and con- a , a − Empirical parameters for the densation flux parameterisation are discussed. To consider 1 2 incoming solar radiation turbulent density fluctuations, the governing equations for B − Replacement variable the first-, second- and third-order moments must be re- b − Dual−use character derived on the base of a scale analysis (Bernhardt, 1964, (replacement variable and constant) 1972; Bernhardt and Piazena, 1988; Foken, 1989; Venka- b , b − Empirical parameters for the tram, 1993; van Dop, 1998). In the subsequent Papers II 1 2 incoming solar radiation and III, the model capability to predict the CBL evolu- C − Replacement variable tion in terms of first-, second- and third-order moments c − Dual−use character and to simulate the evolution of UCNs during a NPF event (replacement variable and constant) will be demonstrated within the framework of a conceptual C ,C ,C −C − Adjustment parameters for study. Furthermore, the model results are discussed with re- 1 2 4 11 the turbulence closure scheme spect to previous observational findings. Ccoag − Brownian coagulation coefficient [m3s−1] 3 −1 Ccond,gas − Condensation coefficient [m s ] − CT0 Parameter for the buoyancy fluxes c1, . . . , c3 − Empirical parameters for the surface energy budget cG − Empirical parameter for the soil heat flux cR − Parameter for the radiative destruction rate ch, cm − Scaling constants for the universalfunctions cp − Specific heat capacity at constant pressure(dry air) − DH2SO4 Diffusion coefficient of H2SO4 Di,Dj − Stokes−Einstein expression for the diffusion coefficient Dp − Particle diameter ˜ Ddown − Downward finite differencing ˜ Dup − Upward finite differencing d − Constant Escf − Water vapour mass flux e − Turbulent kinetic energy F − Replacement function F − Modified Fuchs−Sutugin factor fc − Coriolis parameter Gsoil − Soil heat flux Gsp − Gibbs free energy of cluster formation GF − Humidity−growth factor g − Acceleration of gravity gi, gj − Parameter functions www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4194 O. Hellmuth: Burst modelling

− − HMi Characteristic scale height Mi,sfc Mode i particle mass concentration of particle mass concentration Mi at the surface (i = 1,..., 3) − − HNi Characteristic scale height m Index of particle number concentration Ni mi − Mode i mean dry particle mass − [ ] = HH2SO4 Characteristic scale height of H2SO4 kg (i 1,..., 3) − − [ ] HNH3 Characteristic scale height mH2SO4 Mass of one H2SO4 molecule kg − [ ] of total NH3 mNH3 Mass of one NH3 molecule kg HOH − Characteristic scale height OH N − Total number of physicochemical − HSO2 Characteristic scale height SO2 variables Hsfc − Sensible heat flux at the surface Ncld − Total cloud cover Hw − Scale height of large−scale subsidence Ni − Mode i particle number concentration h − Integration time step [m−3] (i = 1,..., 3) i − Index Ni,sfc − Mode i particle number concentration −3 −1 Jnuc − Nucleation rate [m s ] at the surface (i = 1,..., 3) −3 −1 Jobs − Observed nucleation rate [m s ] n − Index −3 −1 Jter − Ternary nucleation rate [m s ] nOH − Exponent in the OH representation j − Index nw − Exponent in the representation of Kh,Km − Eddy diffusivity coefficients the large−scale subsidence n? , n? − Number of NH and H SO molecules for heat and momentum NH3 H2SO4 3 2 4 K↓ − Incoming solar radiation at ground level per newly formed embryo ↓ − [OH]{ / } − Minimum/maximum of the Kclear Incoming solar radiation at ground level min max in clear skies OH concentration − Kkin − Kinetic prefactor in the P Generation rate of, the nucleation rate expression eddy kinetic energy e Kn − Knudsen number Pdiagonal − Diagonal part of the pressure triple term − − 0 0 k Index of vertical main level Pia Generation rate of the, scalar flux uia k1, k14 − Pseudo−second order rate coefficient for Pij − Generation rate of reaction of OH with SO 0 0 2 the Reynolds stress uiuj k − B Boltzmann constant Prapid,Prelax − Rapid and relaxation part α β α β K , K = (kmn), (kmn) of the pressure triple term ”Coupling matrices” of the rate constants Km(ζ = 0) Prt = for the interaction between tracer χm Kh(ζ = 0) and χnin reaction equations α, β Turbulent Prandtl number ((m, n) = 1,..., N; (α, β) = 1,..., N) p − Air pressure L − Monin−Obukhov length scale Qα − Source term in the governing equation L0 − Length scale (dual−use variable) of reactant α (α = 1,..., N) LBlackadar − Blackadar’s length scale for Qα,emission − Emission strength of reactant α neutral and unstable stratification (α = 1,..., N) ? LD − Turbulence−length scale for Q − Net radiation at the surface stable stratification q − Water vapour mixing ratio Ln − Length scale qs − Saturation specific humidity Lturb − Turbulence−length scale q? − Kinematic humidity scale − Lv Latent heat of water vapourisation RAχα − Rate of change of the − 0 0 Lv,0 Latent heat of water vapourisation second−order moment A χα due to at T = 273.15 K the interaction of a reactive tracer χα LvEsfc − Latent heat flux with a nonreactive scalar A L↓ − Incoming longwave radiation (A ≡ replacement variable) from the atmosphere (α = 1,..., N) L↑ − Outgoing longwave radiation Raχα − Rate of change of the from the surface 0 0 second−order moment a χα due to − li Parameter function the interaction of a reactive tracer χα Mgas − Molar mass of gas molecules with a nonreactive scalar a Mi − Mode i particle mass concentration (α = 1,..., N) [ −3] = kg m (i 1,..., 3) Rp − Particle radius

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Ru − Universal gas constant zk=1 − Geometrical height of the first main level − − Ruiχα Rate of change of the zp Prandtl layer height (first main level zk=1) − 0 0 zs − Height level just above the surface second order moment uiχα due to the interaction term in the governing (screening height) equation of the tracer flux Greek letters (i = 1,..., 3; α = 1,..., N) α − Dual−use variable (index and − Rα Interaction term in the governing replacement variable) − equation of the first order moment χ α α = 1/Pr = 0 t (α 1,..., N) Reciprocal of turbulent Prandtl number Rχ χ − Rate of change of the α β αgas − Mass accommodation coefficient − 0 0 second order moment χαχβ due to or sticking probability of the interaction of the reactive tracers χα a condensing gas and χβ ((α, β) = 1,..., N) αPM − Empirical parameter for RH − Relative humidity Penman−Monteith approach r0,dry − Reference radius of the dry particle αsfc − Surface albedo in the humidity growth factor αT − Coefficient of thermal expansion ri,dry, ri,wet − Mode i mean radius of the dry and wet β − Dual−use variable (replacement variable particle [m] (i = 1,..., 3) and collision frequency function for Sp − Sticking probability, accommodation molecular clusters coefficient βbuo = αT × g s − Slope of curve of the saturation Buoyancy parameter specific humidity curve βPM − Empirical parameter for T − Temperature Penman−Monteith approach T0 − Reference temperature for the coefficient 0d − Dry−adiabatic lapse rate of thermal expansion γ − Dual−use variable (replacement variable Tscr − Temperature at screening height (1−2 m) and NH3−stabilised fraction of the t − Time (Local Standard Time) H2SO4 monomer t0 − Initial time (Local Standard Time) γPM − Psychrometric constant U − Horizontal wind velocity δ − Replacement variable ui, uj − Components of three−dimensional δij − Kronecker delta ((i, j) = 1,..., 3) wind vector ((i, j) = 1,..., 3) (Stull, 1997, p. 57) u − x−component of horizontal wind ijk − Alternating unit tensor ((i, j, k) = 1,..., 3) ug − x−component of geostrophic wind (Stull, 1997, p. 57) ui, uj, uk − Components of the wind vector ε − Dissipation rate ((i, j, k) = 1,..., 3) εR − Radiative destruction rate u? − Friction velocity εab − Dissipation rate in the governing equation − = 0 0 Vχα Deposition velocity (α 1,..., N) of the scalar correlations a b − − v y component of horizontal wind εabc − Dissipation rate in the governing equation vg − y−component of geostrophic wind of the triple moments a0b0c0 v − gas Mean speed of gas molecules εuab − Dissipation rate in the governing equation vi, vj − Mean speed of particles with the 0 0 0 = of the triple moments uia b (i 1,..., 3) respective masses mi, mj εuua − Dissipation rate in the governing equation ((i, j) = 1,..., 3) of the triple moments u0u0a0 ((i, j) = 1,..., 3) w − Vertical velocity i j εuuu − Dissipation rate in the governing equation w? − Convective velocity scale 0 0 0 = w − Large−scale subsidence velocity of the triple moments uiujw ((i, j) 1,..., 3) wH − Large−scale subsidence velocity at Hw ζ = z/L − Normalised height coordinate x − Normalised scaling height stability parameter xj, xj − Geometrical co−ordinates (three dimensions) η1, . . . , η3 − Fit parameter for humidity growth factor ((i, j) = 1,..., 3) θ − Potential temperature z − Geometrical height θv − Virtual potential temperature z0 − Surface roughness length θ? − Kinematic temperature scale zi − Mixing layer height (∂θ/∂t)rad − Diabatic heating/cooling rate due to zk − Geometrical height of main level k radiative flux divergency κ − VonKarm´ an´ constant www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4196 O. Hellmuth: Burst modelling

λair − Mean free path of air C9 = −0.67 λgas − Mean free path of gas C10 = 6.0 µair − Viscosity of air C11 = 0.2 −13 −2 −6 ρ0 − Standard value of density c1 = 5.31 × 10 W m K −2 ρair − Air density c2 = 60 W m ρp − Particle density c3 = 0.12 σ − Stefan−Boltzmann constant cg = 0.1 8 − Replacement variable ch = α0 κ 8k − Replacement variable at main level k cm = κ −1 −1 8sun − Solar elevation cpa = 1006 J kg K 8n − Replacement variable at time step n = × −5 2 −1 DH2SO4 1.2 10 m s − −4 −1 φgeo Geographical latitude fc = 2ω sin φgeo ≈ 1.117 × 10 s φh, φm − Similarity (universal) functions for g = 9.80665 m2 s−1 heat and momentum −18 3 −1 −1 k1 = 1.5 × 10 m molecules s χ , χ − Reactive tracer and/or aerosol parameter −23 −1 m n kB = 1.381 × 10 JK ((m, n) = 1,..., N) 3 −1 Lv,0 = 2515 × 10 J kg χα, χβ , χγ − Reactive tracer and/or aerosol parameter = × −3 −1 MSO2 64.06 10 kg mol ((α, β, γ ) = 1,..., N) = × −3 −1 MNH3 17.0318 10 kg mol χ − − − α? Characteristic scaling property of M = 98.08 × 10 3 kg mol 1 reactive tracer and/or aerosol H2SO4 N = 6.022 × 1023 molecules parameter χ (α = 1,..., N) A α r = 59.49 nm 4 − Parameter function for humidity 0,dry R = 287.955 J kg−1 K−1 growth factor d R = 462.520 J kg−1 K−1 9 − Replacement function v R = 8.314 J mol−1 K−1 9 , 9 − Stability functions for u H M T = 285 K heat and momentum 0 ω − Angular velocity of the Earth [H2SO4] − Sulphuric acid vapour concentration Greek letters [NH3] − Ammonia concentration [OH] − Hydroxyl radical concentration = αH2SO4 0.12 [SO ] − Sulphur dioxide concentration −3 −1 2 αT = 1/T0 = 3.51 × 10 K F − Correction formula for the transition gas αPM = 1 regime of diffusion of gas molecules in αsfc = 0.23 a background gas to a particle −2 βPM = 20 W m ϒ1 − Similarity function for kinematic η1 = 0.097 heat, humidity and tracer ﬂuxes η2 = 0.204 ϒ2 − Similarity function for variances η3 = 5.5826 and co−variances of temperature, [ ] −4 gram water −1 humidity and tracer concentration γPM = cp/Lv ≈ 4 × 10 K [gram]air 0d = g/cp ≈ 1 K/100 m κ = 0.41 A2. Constants −8 λair = 6.98 × 10 m µ = 1.83 × 10−5 kg m−1 s−1 Latin letters air ω = 7.27 × 10−5 s−1 3 −3 ρp = 1.5 × 10 kg m −2 − − − a1 = 1041 W m σ = 5.67 × 10 8 W m 2 K 4 −2 a2 = −69 W m b1 = 0.75 b2 = 3.4 C2 = 2.5 C4 = 4.5 C5 = 0 C6 = 4.85 C7 = 0.4 C8 = 8.0

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A3. Parameters A5. Abbreviations BIOFOR − Biogenic Aerosol Formation 3 in the Boreal Forest HOH = 6.9 × 10 m = × 3 BLMARC − Boundary layer mixing, aerosols, HSO2 1.2 10 m 3 radiation and clouds HNH = 1.2 × 10 m 3 BVOC − Biogenic volatile organic compound H = 1.2 × 103 m H2SO4 CAPRAM − Chemical Aqueous Phase H = 1 × 103 m,(i = 1,..., 3) Ni Radical Mechanism H = 1 × 103 m,(i = 1,..., 3) Mi CBL − Convective boundary layer H = 1.2 × 103 m w CBNT − Classical binary nucleation theory [H SO ] = 1 × 1011 molecules m−3 2 4 min CCN − Cloud condensation nucleus [ ] = × 12 −3 H2SO4 max 1 10 molecules m CFL − Courant, Friedrichs, Lewy = nOH 6 CLAW − Charlson, Lovelock, Andrea, Warren = nw 2.4 CNT − Classical nucleation theory [ ] = × −1 −3 NH3 tot,sfc 1 10 µg m CSL − Convective surface layer 6 −3 [N1]sfc = 1 × 10 m CTNT − Classical ternary nucleation theory 6 −3 [N2]sfc = 10 × 10 m DMS − Dimethyl sulphide 6 −3 [N3]sfc = 10 × 10 m EU − European Union 11 −3 [OH]min = 2 × 10 molecules m FT − Free troposphere 12 −3 [OH]max = 10 × 10 molecules m FTCS − Forward Time Centred Space −3 [SO2]sfc = 5 µg m GAW − Global Atmosphere Watch −1 ug = 5 m s GFEMN − Gibbs Free Energy Minimization −1 vg = 0 m s MBL − Marine boundary layer = × −2 −1 NPF − New particle formation VSO2 0.8 10 m s = × −2 −1 nss − non−sea−salt VNH3 1 10 m s = × −2 −1 PAQS − Pittsburgh Air Quality Study VH2SO4 1 10 m s −2 −1 − wH = −1 × 10 m s PBL Planetary boundary layer − zs = 1 m PDE Partial differential equation − zp = 10 m QUEST Quantiﬁcation of Aerosol Nucleation in the European Boundary Layer SGS − Subgridscale TSC − Thermodynamically stable cluster UCN − Ultraﬁne condensation nucleus A4. Annotations UT − Upper troposphere

() − Average over grid cell volume and integration time step ()e − Integration variable ()0 − Turbulent deviation from the average ()? − Surface energy budget property ()? − Surface layer scaling property ()max,()min − Maximum, minimum value ()dry − Dry particle property ()rad − Radiation−induced ()reac − Chemical reaction−induced ()sfc − Surface variable ()top − Model top property ()tot − Total (gas phase + particle phase) concentration of species ()wet − Wet particle property

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A6. Scaling properties Appendix B

2 2 1/4 0 0 0 0 Model equations u? = w u + w v zs zs 0 0 u(zk=1) 2 B1. Meteorological model w u = − u? zs U(zk=1) v(zk=1) du w0v0 = − u2 = fc(v − vg) (B1) ? dt zs U(zk=1) p 2 2 U(zk=1) = u(zk=1) + v(zk=1)  1/3 dv 0 0 0 0 = −fc(u − ug) (B2)  βbuo w θ zi , w θ > 0 dt w = zs zs ?  0 , w0θ 0 < 0 zs dθ ∂θ 3 (A1) = (B3) u? L = − dt ∂t rad 0 0 κβbuo w θ zs 0 0 dq θ? = − w θ u? = 0 (B4) zs dt 0 0 q? = − w q u? zs 0 0 B2. Chemical model χα? = − w χα u? zs 0 0 dχα w θ = −u?θ? = Rα + Qα + Qα,emission zs dt 0 0 N N (B5) w q = −u?q? X X α zs Rα = kmn χmχn , α = 1,..., N m=1 n=m A7. Turbulence-dissipation length scale dχ d[NH ] 1 = 3 tot (References: Andre´ et al., 1978) dt dt (B6) = Q1,emission e3/2 ε = C1(Lturb) Lturb dχ d[SO ] (A2) 2 = 2 L = + turb dt dt C1(Lturb) 0.019 0.051 (B7) LBlackadar = − k1 [OH][SO2] +Q2,emission | {z } Q2

Lturb = Min(LBlackadar,LD) dχ3 d[H2SO4] κz = dt dt LBlackadar = z + κ = − [ ] 1 Ccond,H2SO4 (r1,wet) H2SO4 N1 L0 | {z } | {z } |{z} ∞ 3 χ3 χ4 Z √ k34 ez dz − C (r ) [H SO ] N (A3) cond,H2SO4 2,wet 2 4 2 0 | {z } | {z } |{z} L0 = 0.1 ∞ 3 χ3 χ5 (B8) Z √ k35 e dz − [ ] Ccond,H2SO4 (r3,wet) H2SO4 N3 0 | {z } | {z } |{z} 3 χ3 χ6 v k u e 36 = u ? LD 0.75u + k1 [OH][SO2] − Jnuc n t ∂θ H2SO4 βbuo | {z } ∂z Q3

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dχ dM B3. Aerosoldynamical model 8 = 2 dt dt = [ ] Ccond,H2SO4 (r2,wet) mH2SO4 H2SO4 N2 (References: Pirjola et al., 1999) | {z } | {z } |{z} 8 χ3 χ5 k35 + (B13) dχ dN Ccoag(r1,wet, r2,wet) m1 N1 N2 4 1 |{z} |{z} = | {z } χ χ dt dt k8 4 5 1 45 = − Ccoag(r1,wet, r1,wet) N1 N1 − Ccoag(r2,wet, r3,wet) m2 N2 N3 2 |{z} |{z} |{z} |{z} | {z } χ4 χ4 | {z } χ χ 4 k8 5 6 k44 56

− Ccoag(r1,wet, r2,wet) N1 N2 |{z} |{z} | {z } χ χ (B9) dχ dM k4 4 5 9 = 3 45 dt dt − C (r , , r , ) N N coag 1 wet 3 wet 1 3 = Ccond,H SO (r3,wet) mH SO [H2SO4] N3 | {z } |{z} |{z} 2 4 2 4 χ4 χ6 | {z } | {z } |{z} k4 9 χ3 χ6 46 k36 + Jnuc + C (r , r ) m N N (B14) | {z } coag 1,wet 3,wet 1 1 3 | {z } |{z} |{z} Q4 9 χ4 χ6 k46

+ Ccoag(r2,wet, r3,wet) m2 N2 N3 dχ5 dN2 | {z } |{z} |{z} = 9 χ5 χ6 dt dt k56 1 = − Ccoag(r2,wet, r2,wet) N2 N2 2 |{z} |{z} | {z } χ5 χ5 (B10) B4. Condensation coefﬁcient, Fuchs–Sutugin correction for 5 k55 transition regime

− Ccoag(r2,wet, r3,wet) N2 N3 | {z } |{z} |{z} (References: Seinfeld and Pandis, 1998, pp. 452–459, 596– 5 χ5 χ6 k56 605; Fuchs and Sutugin, 1971; Clement and Ford, 1999b; Liu et al., 2001) dχ dN 6 = 3 dt dt Ccond,gas(ri,wet) = 4 π Fgas(ri,wet)Dgas ri,wet , (B15) 1 (B11) = − Ccoag(r3,wet, r3,wet) N3 N3 (”gas” = H2SO4; i = 1,..., 3) 2 |{z} |{z} | {z } χ6 χ6 6 k66 Fgas(ri,wet) = f (Kngas) dχ dM 7 = 1 1 dt dt 1 + 1.333 Kngas f (Kngas) − 1 αgas = [ ] Ccond,H2SO4 (r1,wet) mH2SO4 H2SO4 N1 | {z } | {z } |{z} f (Kngas) = 7 χ3 χ4 k34 1 + Kngas 2 − Ccoag(r1,wet, r2,wet) m1 N1 N2 1 + 1.7 Kngas + 1.333 (Kngas) (B16) | {z } |{z} |{z} λ 7 χ4 χ5 (B12) = gas k45 Kngas(ri,wet) ri,wet − Ccoag(r1,wet, r3,wet) m1 N1 N3 3 D |{z} |{z} = gas | {z } χ χ λgas 7 4 6 vgas k46 s + ? + ? 8R T Jnuc(nH SO mH2SO4 nNH mNH3 ) u 2 4 3 vgas = | {z } πMgas Q7 www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4200 O. Hellmuth: Burst modelling

B5. Brownian coagulation coefﬁcient B8. Large-scale subsidence

(References: Seinfeld and Pandis, 1998, p. 445–456, 474,  z nw 661, 662, Fig. 12.5)  wH 1 − 1 − , z/Hw ≤ 1 w = Hw (B21)  wH , z/Hw > 1 Ccoag(ri,wet, rj,wet) = 4π Di + Dj ri,wet + rj,wet r + r × i,wet j,wet Appendix C + + 2 + 2 1/2 ri,wet rj,wet (gi gj ) (B17) 4(D + D ) −1 Reynolds averaged equations + i j , (v2 + v2)1/2 (r + r ) i j i,wet j,wet (References: Andre´ et al., 1976a,b, 1978, 1981) (i, j) = 1,..., 3 C1. First-order moment equations kB T Di = 6πµairri,wet ∂u ∂u ∂w0u0 ! + w = − + f (v − v ) (C1) 5 + 4Kn + 6Kn 2 + 18Kn 3 ∂t ∂z ∂z c g × i i i 2 5 − Kni + (8 + π)Kni ∂v ∂v ∂w0v0 λair + = − − − Kni = w fc(u ug) (C2) ri,wet ∂t ∂z ∂z 1 3 (B18) g = (2 r , + l ) ! i 6 r l i wet i ∂θ ∂θ ∂w0θ 0 ∂θ i,wet i + w = − + (C3) − 2 + 2 3/2 − ∂t ∂z ∂z ∂t (4 ri,wet li ) 2 ri,wet rad s 8kB T vi = ∂q ∂q ∂w0q0 πmi + w = − (C4) ∂t ∂z ∂z 8Di li = πvi ∂χ ∂χ ∂w0χ 0 α + w α = − α + R + Q , B6. Humidity growth factor ∂t ∂z ∂z α α N N (References: Birmili and Wiedensohler, pers. comm., 2004) X X α 0 0 (C5) Rα = kmn χ mχ n + χmχn , = Mi m=1 n m mi = , i = 1,..., 3 Ni α = 1,..., N 1/3 3mi ri,dry = 4πρp C2. Second-order moment equations ri,wet = ri,dry × GF(ri,dry, RH) (B19) C2.1 The six velocity correlations ( u0u0, u0v0, u0w0, v0v0, −4(ri,dry) RH 0 0 0 0 GF(ri,dry, RH) = (1 − RH) v w , w w ) η1 − η2 4(r ) = η + 0 0 0 0 i,dry 2 η3 ri,dry ∂uiuj ∂uiuj 1 + + w = r0,dry ∂t ∂z 0 0 0 ∂u u w ∂u ∂u B7. Hydroxyl radical − i j − u0w0 j + u0w0 i ∂z i ∂z j ∂z (References: Liu et al., 2001, see references therein) + 0 0 + 0 0 (C6) βbuo δ3juiθv δ3iujθv z [ ] = [ ] + [ ] − 0 0 0 0 OH OH min OH max exp + fc ik3u u + jk3u u HOH j k i k (B20) nOH ! π 1 ∂p0 ∂p0 2 × sin t − 0 + 0 − × ui uj δijε 24 3600 ρ0 ∂xj ∂xi 3

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4201

1 ∂p0 ∂p0 ε 0 0 − 0 + 0 = εab = C2 a b (C13) ui uj e ρ0 ∂xj ∂xi ε 2 2 The radiative destruction rate ε appears only in the tempera- − C u0u0 − δ e − C P − δ P R 4 e i j 3 ij 5 ij 3 ij ture variance equation. It is parameterised according to Stull (1997, p. 132, see references therein): = 0 0 + 0 0 (C7) Pij βbuo δ3juiθv δ3iujθv 0 0 εR = cRθ θ 0 ∂uj 0 ∂ui − u w0 + u w0 (C14) i ∂z j ∂z m ε 1 cR ≈ 0.036 , cR = s e3/2 s 0 0 0 0 ∂u 0 0 ∂v P = βbuow θv − u w − v w ∂z ∂z The interaction term between passive and reactive scalar = = 1 (a (θ, q), b χα) follows from Eq. (C20), i.e., e = u0 u0 k k 2 ∂a0χ 0 3/2 (C8) α e = Raχ ε = C (L ) ∂t α 1 turb reac Lturb (C15) N N X X α 0 0 = = = k χ a0χ 0 + χ a0χ 0 + a0χ 0 χ 0 C2.2 Scalar ﬂuxes ( uia , (ui) (u, v, w), a (θ, q, χα), mn m n n m m n α=1,..., N ) m=1 n=m The interaction between two reactive scalars ∂u0a0 ∂u0a0 i + w i = ((a, b)=(χα, χβ )), results from Eq. (C25), i.e., ∂t ∂z 0 0 0 0 0 ∂u w a ∂a ∂u ∂χαχβ − i − 0 0 + 0 0 i = R . (C16) uiw w a χαχβ ∂z ∂z ∂z ∂t (C9) reac 1 ∂p0 + 0 0 + 0 0 − 0 δ3iβbuoθva ik3fcuka a C2.4 Interaction of a reactive tracer χα with a nonreactive ρ0 ∂xi = scalar A (ui, θ, q) ∂u0a0 + i N N ∂t ∂χα X X α reac = k χmχn | × A ∂t mn reac m=1 n=m 1 ∂p0 ε ∂A − a0 = −C u0a0 − C P + = 0 | × χ ρ ∂x 6 e i 7 ia α 0 i (C10) ∂t reac (C17) 0 0 0 0 ∂ui Pia = δ3iβbuoθva − w a ∂z N N ∂Aχα X X α = k Aχmχn | () The reaction term in the tracer ﬂux equation (a=χ ) follows ∂t mn α reac m=1 n=m from Eq. (C20), i.e.,

0 0 ∂ 0 0 ∂u χ Aχ α + A χα = i α = R ∂t reac ∂t uiχα reac N N (C11) X X α + 0 0 N N kmnA χ mχ n χmχn (C18) = X X α 0 0 + 0 0 + 0 0 0 m=1 n=m kmn χ muiχn χ nuiχm uiχmχn m=1 n=m N N X X α 0 0 0 0 0 0 0 + kmn χ mA χn + χ nA χm + A χmχn 0 0 C2.3 Scalar correlations ( a b , (a, b)=(θ, q, χα, χβ ), m=1 n=m (α, β)=1,..., N ) N N (C19) 0 0 0 0 ∂χ α X X α ∂a b ∂a b = k χ χ + χ 0 χ 0 | × A + w = ∂t mn m n m n ∂t ∂z reac m=1 n=m ! ∂A ∂w0a0b0 ∂b ∂a + = 0 | × χ − − w0a0 + w0b0 ∂t α ∂z ∂z ∂z (C12)

0 0 N N ∂a b ∂Aχ X X α − ε − ε + α = k A χ χ + χ 0 χ 0 | () ab R ∂t ∂t mn m n m n reac reac m=1 n=m www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4202 O. Hellmuth: Burst modelling

Subtracting Eq. (C19) from Eq. (C18): Contribution of the mean values to the total change rate:

∂A0χ 0 N N (C24) R = α ∂χ α X X α 0 0 Aχα = k χ χ + χ χ | × χβ ∂t ∂t mn m n m n reac reac m=1 n=m N N N N (C20) ∂χ β X X β X X α 0 0 0 0 + = k χ χ + χ 0 χ 0 | × χ = kmn χ mA χn + χ nA χm mn m n m n α ∂t reac = m=1 n=m m=1 n m N N 0 0 0 ∂χ αχ β X X α +A χ χ = k χ χ χ + χ 0 χ 0 m n ∂t mn β m n m n reac m=1 n=m C2.5 Interaction of reactive tracers χα and χβ N N X X β 0 0 + kmnχ α χ mχ n + χmχn (C21) m=1 n=m ∂χ N N α = X X α | × kmnχmχn χβ Subtraction of Eq. (C24) from Eq. (C23) to extract the con- ∂t reac = n=m m 1 tribution of the tracer co-variance to the total change rate: N N ∂χβ X X β + = k χmχn | × χα 0 0 ∂t mn ∂χαχβ reac m=1 n=m R = χαχβ ∂t reac N N N N ∂χαχβ X X α X X α 0 0 = k χβ χmχn = k χ χ χ ∂t mn mn m β n reac m=1 n=m m=1 n=m (C25) N N + 0 0 + 0 0 0 X X β χ nχβ χm χβ χmχn + kmnχαχmχn | () m=1 n=m N N X X β 0 0 + kmn χ mχαχn m=1 n=m N N ∂χαχβ X X α + 0 0 + 0 0 0 = k χβ χmχn χ nχαχm χαχmχn ∂t mn reac m=1 n=m N N C3. Third-order moment equations + X X β kmnχαχmχn 0 0 0 m=1 n=m C3.1 Turbulent transport of momentum ﬂuxes ( u u w , u0v0w0, u0w0w0, v0v0w0, v0w0w0, w0w0w0 ) Expanding the last equation using: 0 0 0 0 0 0 (C26) 0 0 0 ∂u u w ∂u u w ABC = (A + A )(B + B )(C + C ) i j + w i j = ∂t ∂z = A B C + A B0C0 + B A0C0 ∂u ∂u − u0w0w0 j + u0w0w0 i +C A0B0 + A0B0C0 i ∂z j ∂z 0 0 0 0 0 0 χαχmχn = χ αχ mχ n + χ αχmχn + χ mχαχn (C22) ∂u u 0 0 i j 0 0 0 0 0 − w w +χ nχαχm + χαχmχn ∂z 0 0 ! χ χ χ = χ χ χ + χ χ 0 χ 0 + χ χ 0 χ 0 ∂u w ∂u0w0 β m n β m n β m n m β n − 0 0 j + 0 0 i uiw ujw + 0 0 + 0 0 0 ∂z ∂z χ nχβ χm χβ χmχn 0 0 0 0 0 0 0 0 0 + βbuo u u θ + δ3ju w θ + δ3iu w θ ∂ i j v i v j v + 0 0 χ αχ β χαχβ ! ∂t reac 1 ∂p0 ∂p0 ∂p0 − u0u0 + u0w0 + u0w0 N N ρ i j ∂z i ∂x j ∂x X X α 0 0 0 j i = kmn χ β χ mχ n + χ β χmχn m=1 n=m − εuuu +χ χ 0 χ 0 + χ χ 0 χ 0 + χ 0 χ 0 χ 0 (C23) m β n n β m β m n 1 ∂p0 ∂p0 ∂p0 − 0 0 + 0 0 + 0 0 N N uiuj uiw ujw + X X β + 0 0 ρ0 ∂z ∂xj ∂xi kmn χ αχ mχ n χ αχmχn (C27) m=1 n=m = Prelax + Prapid 0 0 0 0 0 0 0 | {z } | {z } +χ mχαχn + χ nχαχm + χαχmχn Andre et al., 1978 Andre et al., 1981

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4203 ε P = −C u0u0w0 C3.3 Turbulent transport of scalar correlations (“fluxes of relax 8 e i j correlations”) P = −C β u0u0θ 0 + δ u0w0θ 0 + δ u0w0θ 0 0 0 0 = = rapid 11 buo i j v 3j i v 3i j v ( uia b , (a, b) (θ, q, χα, χβ ), (α, β) 1,..., N ) ε = 0 uuu ∂u0a0b0 ∂u0a0b0 i + w i =   ∂t ∂z r 2  u02 u02 w02 + u0w0  !  i j j  0 0 0 ∂ui 0 0 0 ∂b 0 0 0 ∂a  r  − w a b + u w a + u w b  2  ∂z i ∂z i ∂z u0u0w0 ≤ Min u02 u02 w02 + u0w0 (C28) i j j i i ! (C32)  r  0 0 ∂u0b0 ∂u0a0  2  0 0 ∂a b 0 0 i 0 0 i  02 02 02 + 0 0  − u w + w a + w b  w ui uj uiuj  i ∂z ∂z ∂z ! 1 ∂p0 C3.2 Turbulent transport of scalar fluxes (“fluxes of fluxes”) 0 0 0 0 0 + δ3iβbuoθva b − a b − εuab 0 0 0 = = ρ0 ∂xi ( uiuja , a (θ, q, χα), α 1,..., N )

0 ! (C29) 1 0 0 ∂p ∂u0u0a0 ∂u0u0a0 − a b = Prelax + Prapid (C33) i j + w i j = ρ0 ∂xi | {z } | {z } ∂t ∂z Andre et al., 1978 Andre et al., 1981 ∂a ∂u ∂u − 0 0 0 + 0 0 0 j + 0 0 0 i ε uiujw uiw a ujw a P = −C u0a0b0 ∂z ∂z ∂z relax 8 e i 0 0 0 0 ! ∂u u ∂u a ∂u0a0 P = −δ C β a0b0θ 0 − 0 0 i j + 0 0 j + 0 0 i rapid 3i 11 buo v w a uiw ujw ∂z ∂z ∂z εuab = 0 0 0 0 0 0 0 + βbuo δ3ju θva + δ3iu θva  r  i j 0 2  u 2 a02 b02 + a0b0  0 0 !  i  1 0 0 ∂p 0 0 ∂p   − u a + u a  r 2  i j u0a0b0 ≤ Min 02 02 02 + 0 0 (C34) ρ0 ∂xj ∂xi i a ui b uib  r  − ε  2  uua  02 02 02 + 0 0   b ui a uia  1 ∂p0 ∂p0 − 0 0 + 0 0 uia uja C3.4 Turbulent transport of scalar correlations ρ0 ∂xj ∂xi 0 0 0 (C30) ( a b c , (a, b, c)=(θ, q, χα, χβ , χγ ), = Prelax + Pdiagonal + Prapid (α, β, γ )=1,..., N ) | {z } | {z } Andre et al., 1978 Andre et al., 1981 ∂a0b0c0 ∂a0b0c0 + w = ε 1 ∂t ∂z P = −C u0u0a0 − δ u0 u0 a0 ! relax 8 e i j 3 ij k k ∂c ∂b ∂a − w0a0b0 + w0a0c0 + w0b0c0 ε ∂z ∂z ∂z P = δ C u0 u0 a0 (C35) diagonal ij 9 e k k 0 0 0 0 0 0 ! 0 0 0 0 0 ∂b c 0 0 ∂a c 0 0 ∂a b Prapid = −C11βbuo δ3ju θ a − w a + w b + w c i v ∂z ∂z ∂z 0 0 0 2 0 0 0 +δ3iu θ a − δijw θ a j v 3 v − εabc 0 0 0 ε ukuka ε εuua = δijC10 ε = C a0b0c0 (C36) e 3 abc 10 e  r   r  2 2  u02 u02 a02 + u0a0   a02 b02 c02 + b0c0   i j j     r   r   2   2  u0u0a0 ≤ Min 02 02 02 + 0 0 (C31) a0b0c0 ≤ Min 02 02 02 + 0 0 (C37) i j uj ui a uia b a c a c  r   r   2   2   02 02 02 + 0 0   02 02 02 + 0 0   a ui uj uiuj   c a b a b  www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4204 O. Hellmuth: Burst modelling

C4. Buoyancy ﬂuxes D1.2 Lower boundary conditions (“constant-ﬂux layer” hypothesis) = + = + θv θ (1 0.61q) θ CT0 q ≈ CT0 0.61 T0 0 0 0 0 ∂u (C38) u w ≈ u w ≈ −Km 0 0 = 0 0 + 0 0 zs zp ∂z a θv a θ CT0 a q zp 0 0 0 0 0 0 0 0 0 ∂v a b θ = a b θ + CT a b q 0 0 0 0 v 0 v w ≈ v w ≈ −Km z z ∂z s p zp

0 0 0 0 ∂θ w θ ≈ w θ ≈ −Kh (D3) zs zp ∂z Appendix D zp

0 0 0 0 ∂q w q ≈ w q ≈ −Kh z z ∂z Initial and boundary conditions s p zp

0 0 0 0 ∂χα w χα ≈ w χα ≈ −Kh , α = 1,..., N D1. First-order moments z z ∂z s p zp

D1.1 Initial conditions

cmu?z (References: Liu et al., 2001, for chemical variables) Km = φ (ζ ) m (D4) chu?z Initial proﬁles of meteorological variables, i.e., Kh = φh(ζ ) u(z, t0), v(z, t0), θ(z, t0), q(z, t0) (D1) ∂u u(zk=1) u? can be either derived from observations or synthetically pre- = φm(ζ ) ∂z z U(zk=1) cmzp scribed. The geostrophic wind components ug, vg and the p large-scale subsidence velocity wH are prescribed. ∂v v(zk=1) u? = φm(ζ ) The ﬁrst-order moments of physicochemical variables are ∂z c z zp U(zk=1) m p initialised as follows: ∂θ θ = ? φh(ζ ) (D5) ∂z chzp z zp [OH](z, t0) = [OH]min + [OH]max exp − H ∂q q? OH = φ (ζ ) n h π OH ∂z z chzp × sin t p × 0 24 3600 ∂χ α χα? = φh(ζ ) , α = 1,..., N z ∂z c z zp h p [SO2](z, t0) = [SO2]sfc exp − HSO2 z [NH3]tot(z, t0) = [NH3]tot,sfc exp − HNH 3 D1.3 Similarity functions [H2SO4](z, t0) = [H2SO4]min + [H2SO4]max (D2) z × exp − (References: Dyer and Hicks, 1970) HH2SO4 π × sin t 24 × 3600 0 z (1 − 16ζ )−1/4 , ζ < 0 N (z, t ) = N exp − , i = 1,..., 3 φm = i 0 i,sfc H (1 + 5ζ ) , ζ ≥ 0 Ni (D6) z (1 − 16ζ )−1/2 , ζ < 0 M (z, t ) = M exp − , i = 1,..., 3 φ = i 0 i,sfc h + ≥ HMi (1 5ζ ) , ζ 0

Atmos. Chem. Phys., 6, 4175–4214, 2006 www.atmos-chem-phys.net/6/4175/2006/ O. Hellmuth: Burst modelling 4205

D1.4 Skin properties D2. Second-order moments

(References: Holtslag, 1987, p. 55–56, 70) D2.1 Initial conditions

0 0 w θ At the starting time, second-order moments are set to zero. zs θ? = − u? −1 D2.2 Lower boundary conditions zp zp zs = κ1θ ln − 9H + 9H zs L L (References: Holtslag, 1987, p. 23–46) w0q0 zs (D7) Surface energy budget: q? = − u? ? = + + −1 Q Hsfc LvEsfc Gsoil zp zp zs = κ1q ln − 9H + 9H Q? = (1 − α )K↓ + L↓ − L↑ zs L L sfc 1 1θ = θ(zp) − θ(zs) Q? = (1 − α )K↓ 1 + c sfc = − 3 1q q(zp) q(zs) 6 4 +c1(Tscr) − σ (Tscr) + c2Ncld θ(zs) = θ(zp) ↓ θ? zp zp zs ↓ b2 − − + K = K 1 − b1(Ncld) ln 9H 9H (D8) clear κ zs L L ↓ q K = a sin 8 + a q(z ) = q(z ) − ? θ(z ) − θ(z ) clear 1 sun 2 s p p s ? (D12) θ? Gsoil = cGQ

1θ 1T g 1z (1 − αPM) + (γPM/s) ? = + Hsfc = (Q − Gsoil) − βPM θ T cp T (D9) 1 + (γPM/s) αPM ? 1T ≈ 1θ − 0d (zp − zs) LvEsfc = (Q − Gsoil) + βPM 1 + (γPM/s) D1.5 Stability functions cp γPM = Lv (References: Paulson, 1970, unstable case; Carson and ∂qs Richards, 1978, stable case; Holtslag, 1987, p. 56, 71, 101) s = ∂T  2 ! 1 + x 1 + x (D10) L = . − . (T − . ) × 6 −1  2 ln + ln v 2 5 0 00236 273 15 10 J kg   2 2  ◦  π T[ C] -5 0 5 10 15 20 25 30 35 = −2 arctan(x) + , L < 0 9M 2 γ /s 2.01 1.44 1.06 0.79 0.60 0.45 0.35 0.27 0.21  PM    c bc Friction velocity:  − aζ + b ζ − exp(−dζ ) + , L > 0  d d  ! (a) For unstable conditions (References: Holtslag, 1987, 1 + x2  2 ln , L < 0 p. 99–100, 106, 130) 9H = 2  3 9M , L > 0 u? L = − g x = (1 − 16ζ )1/4 , a=0.7, b=0.75, c=5, d=0.35 0 0 κ (w θ )zs T (zk=1) D1.6 Upper boundary conditions (D13) u? = κU(zk=1) −1 (D11) zp zp z0 × ln − 9M + 9M z L L ∂u ∂v ∂θ ∂q 0 = 0 , = 0 , = 0 , = 0 ∂z ∂z ∂z ∂z top top top top Given the surface layer heat flux, the computation starts with u?=κU(zk=1)/ ln(zp/z0) for 9M =0 (L=∞). This way, an ∂χ α = 0 , α = 1,..., N estimation of L is obtained. With this estimate, u? is recal- ∂z top culated using improved 9M and so on. The iteration stops, www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4206 O. Hellmuth: Burst modelling when u? differs less than 5% from its anterior value. Kinematic and convective humidity fluxes (References: Andre´ et al., 1978): (b) For stable conditions (References: Holtslag, 1987, p. 130) 0 0 u(zk=1) 0 0 u q = w q × ϒ1(ζ ) (D14) zs U(zk=1) zs v(z )  1/2 0 0 k=1 0 0 (Ln − L0) + [Ln(Ln − 2L0)] ,Ln ≥ 2L0 v q = w q × ϒ1(ζ ) (D19)  zs U(zk=1) zs L = 1/2 Ln Esfc  L0 ,Ln < 2L0 w0q0 = 2 zs ρair 5z L = p 0 zp Variances, co-variances of temperature and humidity (Refer- ln( ) ences: Andre´ et al., 1978): z0 2 κU(z = ) T (z = )   L = k 1 k 1 n 2 2 zp 0 0  u?  2gθ ln θ θ   = ϒ2(ζ ) ? z 2 z0 s  w0θ 0  1/3 zs g 0 0   u? = −κ (w θ )zs L T (zk=1) 2 0 0  u?  (D20) q q   = ϒ2(ζ ) The computation starts with θ?≈0.09 K to obtain an estima- z 2 s  w0q0  tion of L. With this estimate, θ? is recalculated using im- zs proved 9H and so on. The iteration stops, when θ? differs   2 less than 5% from its anterior value. 0 0 u? θ q   = ϒ2(ζ ) z  0 0 0 0  Surface Reynolds stresses (References: Andre´ et al., 1978): s w θ w q zs zs u(z = ) 0 0 k 1 2 −2/3 w u = − u? 4(1 − 8.3ζ ) , ζ < 0 zs U(z ) ϒ (ζ ) = (D21) k=1 2 4 , ζ > 0 v(z ) 0 0 k=1 2 (D15) w v = − u? zs U(zk=1) Kinematic and convective tracer fluxes (dry deposition) (Ref- erences: Andre´ et al., 1978; Verver et al., 1997): u0v0 = 0 zs u(z = ) u0χ 0 = k 1 w0χ 0 × ϒ (ζ )  2 + 2 0 0 α α 1  4u? 0.3w? , w θ > 0 zs U(zk=1) zs u0u0 = zs , z 2 0 0 0 0 v(zk=1) 0 0 s  4u? , w θ < 0 v χ = w χ × ϒ (ζ ) (D22) z α α 1 s zs U(zk=1) zs  0 0 1.75u2 + 0.3w2 , w θ > 0 0 0  ? ? (D16) w χ = −V χ (z = ) 0 0 zs α χα α k 1 v v = , zs z 2 0 0 s  1.75u? , w θ < 0 zs Variances, co-variances of temperature, water vapour mix- 2/3 2 0 0 1.75 + 2(−ζ ) u? , ζ < 0 ing ratio and tracer concentration (References: Andre´ et al., w w = 2 zs 1.75u? , ζ > 0 1978; Verver et al., 1997): Kinematic and convective heat fluxes (References: Andre´   et al., 1978): u2 0 0  ?  = θ χα ϒ2(ζ ) zs  0 0 0 0  0 0 u(zk=1) 0 0 w θ w χα u θ = w θ × ϒ1(ζ ) zs zs zs U(zk=1) zs   v(z = ) u2 0 0 = k 1 0 0 × 0 0  ?  = v θ w θ ϒ1(ζ ) (D17) q χα ϒ2(ζ ) (D23) zs U(zk=1) zs zs  0 0 0 0  w q w χα H zs zs w0θ 0 = sfc   z ρ c s air p u2 χ 0 χ 0  ?  = ϒ (ζ ) −1/4 −1/2 α β   2 −3.7(1 − 15ζ ) (1 − 9ζ ) , ζ < 0 zs w0χ 0 w0χ 0 ϒ1(ζ ) = (D18) α β −3 , ζ > 0 zs zs

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D2.3 Upper boundary conditions E2.1 Grid structure

0 0 A staggered grid is used with ﬁrst- and third-order correla- ∂uiuj (t) = 0 tions calculated at main levels and second-order ones calcu- ∂z top lated at half levels.

∂u0a0 i (t) = 0 (D24) E2.2 Standard differencing scheme ∂z top ∂8 1 0 0 = Deup + Dedown ∂a b ∂z 2 (t) = 0 k ∂z − top 8k+1 8k Deup = (E2) zk+1 − zk D3. Third-order moments 8k − 8k−1 Dedown = zk − zk−1 D3.1 Initial conditions E2.3 Derivatives of mean variables in third-order moment At the starting time, third-order moments are set to zero. equations A geometric approximation is used to determine the deriva- D3.2 Upper boundary conditions tives of the mean variables in the third-order moment equa- 0 0 tions to avoid the appearance of negative values of variances ∂u u w0 i j = just below the inversion: (t) 0 ∂z top ∂8 Deup + Dedown ≈ 2 D D 0 0 0 eup edown 2 (E3) ∂u u a ∂z k D + D i j = eup edown (t) 0 ∂z top (D25) Acknowledgements. The work was realised at the IfT Modelling

∂u0a0b0 Department, headed by E. Renner, within the framework of IfT i (t) = 0 main research direction 1 “Evolution, transport and spatio-temporal ∂z top distribution of the tropospheric aerosol”. Special thanks go to the following people for the use of their Fortran source codes: ∂a0b0c0 = A. Nenes (inorganic aerosol thermodynamical equilibrium model (t) 0 ∂z ISORROPIA), M. Wilck (exact binary nucleation model), I. Na- top pari (parameterisation of the ternary nucleation rate) and L. Zhang (parameterisation of the particle dry deposition velocity). For the motivation and numerous discussions on the subject of the present Appendix E Paper I am very indebted to D. Mironov, E. Renner, K. Bernhardt, E. Schaller, M. Kulmala, A. A. Lushnikov, M. Boy, R. Wolke, Numerics F. Stratmann, H. Siebert, T. Berndt and J. W. P. Schmelzer. Sincer- est thanks are given to M. Kulmala and M. Boy for the opportunity E1. Adams–Bashforth time differencing scheme to present and discuss previous results at the Department of Physics at the University of Helsinki as well as to J. W. P. Schmelzer for the chance to discuss selected aspects of cluster formation under at- (References: Durran, 1999, p. 68, Table 2.1) mospheric conditions, generalisation of classical Gibbs theory and phase transformations in multicomponent systems during the Xth d9 Research Workshop on Nucleation Theory and Applications at the = F (9) dt Bogoliubov Laboratory of Theoretical Physics of the Joint Insti- 8n = 9(n 1t) tute for Nuclear Research (JINR), Dubna. In this context, I want to express special thanks also to A. K. Shchekin, A. S. Abyzov, h (E1) 8n+1 = 8n + 23 F (8n) H. Heinrich and I. S. Gutzow. Many thanks go to the editor and 12 to the three reviewers for their helpful comments, additional references and suggestions to improve the manuscript. Special thanks go −16 F (8n−1) + 5 F (8n−2) also to N. Otto and N. Deisel for their strong support during the technical processing as well as to M. Reichelt for the proofreading of the manuscript. Last but not least, it should be mentioned, that E2. Vertical ﬁnite-differencing scheme the pioneering works of J. C. Andre,´ P. Bougeault, G. H. L. Verver and their co-workers served as the conditio sine qua non for the (References: Andre´ et al., 1976a,b; Bougeault, 1985) approach presented here. www.atmos-chem-phys.net/6/4175/2006/ Atmos. Chem. Phys., 6, 4175–4214, 2006 4208 O. Hellmuth: Burst modelling

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