Modeling the plasma response to small-scale aerosol particle perturbations in the mesopause region

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1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. D8, 8442, doi: /2002jd002753, 2003 Modeling the plasma response to small-scale aerosol particle perturbations in the mesopause region Ø. Lie-Svendsen, T. A. Blix, and U.-P. Hoppe Norwegian Defence Research Establishment, Kjeller, Norway E. V. Thrane 1 Andøya Rocket Range, Norway Received 12 July 2002; revised 26 September 2002; accepted 23 October 2002; published 13 February [1] We have developed a numerical model that solves the time-dependent, onedimensional, coupled continuity and momentum equations for an arbitrary number of charged and neutral particle species. The model includes production and loss of particles due to ionization, recombination, and attachment of ions and electrons by heavy aerosol particles, and transport due to gravity and multipolar diffusion. The model is used to study the response of the mesopause plasma to small-scale, aerosol particle density perturbations. We find that for aerosol structures on the order of a few meters, electron attachment and ambipolar diffusion are the dominant processes, leading to small-scale electron perturbations that can cause polar mesosphere summer echoes (PMSEs). Moreover, for small aerosol particles, with radii on the order of 10 nm or less, ambipolar diffusion leads to an anticorrelation between electron and ion densities, which is in agreement with most rocket observations. These small-scale structures persist as long as the aerosol layer persists, which will be limited by aerosol particle diffusion. For 10-nm particles, this diffusive lifetime will be on the order of hours. The few instances where rocket observations find instead a correlation between electron and ion densities can be explained either by the aerosol particles becoming large, on the order of 50 nm or more, in which case ion attachment becomes important, or by rapid evaporation of aerosol particles. In the latter case, evaporation must be sufficiently fast to overcome ambipolar diffusion. INDEX TERMS: 0305 Atmospheric Composition and Structure: Aerosols and particles (0345, 4801); 0335 Atmospheric Composition and Structure: Ion chemistry of the atmosphere (2419, 2427); 0341 Atmospheric Composition and Structure: Middle atmosphere constituent transport and chemistry (3334); 0340 Atmospheric Composition and Structure: Middle atmosphere composition and chemistry; KEYWORDS: PMSE, NLC, aerosol, plasma, mesosphere, model Citation: Lie-Svendsen, Ø., T. A. Blix, U.-P. Hoppe, and E. V. Thrane, Modeling the plasma response to small-scale aerosol particle perturbations in the mesopause region, J. Geophys. Res., 108(D8), 8442, doi: /2002jd002753, Introduction [2] At high latitudes in summer, strong VHF radar echoes occur in the height range km. These are called polar mesosphere summer echoes (PMSEs). Their occurrence is related to noctilucent clouds (NLC) or Polar Mesosphere Clouds (PMC) and all three phenomena depend on the cold temperatures occurring near the polar mesopause in summer. While NLC or PMC are an optical phenomenon requiring aerosol particles of a minimum size, the radar phenomenon PMSE requires strong fluctuations in the electron density on scales of half the radar wavelength, the so-called Bragg scale. The characteristics of PMSE, a large number of observations by ground-based radar and 1 Also at Department of Physics, University of Oslo, Norway. Copyright 2003 by the American Geophysical Union /03/2002JD rocket instruments and most of the existing theories have been summarized by, e.g., Cho and Kelley [1993], Hoppe et al. [1994], Cho and Röttger [1997], and references therein. Hoppe et al. [1994] showed that electron density fluctuations as observed by rocket instruments under PMSE conditions can indeed create the observed radar echoes, but the process causing the observed electron density fluctuations and maintaining them against the action of molecular diffusion remained an open question. Czechowsky and Rüster [1997] found that only about 10% of PMSE exhibit the wide spectra and absence of aspect sensitivity that result from turbulence. Lübken et al. [1998] found by in-situ rocket observations that although neutral air turbulence was present in some PMSE, it was totally absent in others. These authors concluded that there must exist processes creating electron density fluctuations and PMSE, which are not directly coupled to neutral air turbulence. All the published explanations for PMSE agree on the fact that charged aerosol particles play a key role, in agreement with PMR 9-1

2 PMR 9-2 LIE-SVENDSEN ET AL.: SMALL-SCALE PLASMA STRUCTURES the observations of Chilson et al. [2000] that PMSE can be weakened by heating the observation volume with radio frequency at high power, and the simulation of such observations by Rapp and Lübken [2000]. [3] Reid [1990] and Klostermeyer [1994] have modeled the effects of such aerosol particles on the distribution of charges between free electrons, ions, and charged aerosol particles and found results similar to observed radar echoes. Rapp and Lübken [2001] and Lübken and Rapp [2001] have refined such simulation studies by introducing disturbances in the electron and ion number density profiles. Along these lines, we wish to understand the behavior of free electrons, ions, and charged aerosol particles, primarily at the Bragg scale of VHF radars. [4] There are at least three likely reasons why aerosol particles are essential to the explanation of PMSE: First, the summer polar mesopause region is cold [Lübken, 1999] and moist [Seele and Hartogh, 1999] enough to allow the growth of ice particles. This is substantiated by the frequent presence of NLC in the same region [e.g., Gadsden and Schröder, 1989] and by common-volume measurements of NLC and PMSE [e.g., Nussbaumer et al., 1996; Von Zahn and Bremer, 1999; Latteck et al., 1999; Stebel et al., 2000]. Second, rocket measurements have directly detected charged aerosol particles with a number density comparable to the local plasma density [Havnes et al., 1996; Blix, 1999]. Thirdly, heavy particles are attractive candidates because their small mobility implies that small-scale structures in their distribution can persist for a long time without being smeared out by diffusion. Free electrons attach themselves easily to such particles, so that small-scale variations in particle density may be accompanied by similar small-scale variations of the free electron density, which in turn cause the PMSE. We aim to demonstrate that any small-scale structures in aerosol number density will lead to electron density variations that can produce the required backscatter, and to variations in electron and ion density in agreement with in-situ measurements. [5] At large length scales, on the order of hundreds of meters to kilometers, the link between electrons and aerosol particles appears solid. Particularly, the large depletions ( bite-outs ) in electron density that are frequently observed [e.g., Ulwick et al., 1988] together with similarly large amounts of charged, heavy particles, provide conspicuous evidence for this coupling. Although this suggests that aerosol particles may be responsible for PMSE, it is not hard evidence since the large-scale electron depletions cannot cause radar echoes at a few meters wavelengths unless the gradients are very large. However, in conjunction with these large depletions, variations in electron density of the magnitude and at the short length scale that can cause radar echoes are also observed. [6] There is also direct evidence for meter-scale variations in the charge carried by aerosol particles: Havnes et al. [1996] report large and rapid changes in the current carried by such particles over a distance of only a few meters. Their results also indicate an anticorrelation between the electron density and the aerosol charge density, implying that free electrons attach themselves to the aerosol particles. This direct evidence of large aerosol charge variations on short length scales is a strong argument in favor of the hypothesis that small-scale structures in the aerosol particle population can cause PMSE. [7] An unambiguous connection between PMSE and aerosol particles cannot be made, however, without also modeling the plasma response to small-scale perturbations in the particle population. The model provides a controlled environment in which we can monitor how the ambient plasma population responds to the presence of particles. Hopefully, this modeling will show that the particles cause characteristic signatures in the plasma, signatures that we may again look for in the experimental data and which can provide the hard evidence needed to strengthen, or weaken, the link between PMSE and aerosol particles. (It is conceivable that the small-scale electron perturbations are caused, e.g., by plasma instabilities [e.g., Dimant and Sudan, 1995; Blix et al., 1996], unrelated to the presence of aerosol particles.) Comparison between model calculations and in situ rocket measurements can also be used to get some information about what processes create the small-scale particle structures in the first place. For instance, are they essentially static structures or the result of highly dynamic wave activity? As we shall show, if they are essentially static structures ambipolar diffusion will have time to operate and will leave its characteristic mark on the ambient plasma. [8] Several studies of charged particles in the mesosphere have focused on the chemical and physical processes that are important for the large-scale structures (such as the deep electron depletions observed) [e.g., Reid, 1990; Rapp and Lübken, 2001]. At these length scales, charged particle transport plays a minor role as the electrons and ions move a short distance only between the location at which they are produced by ionization and the location at which they recombine. Hence solving the coupled continuity equations for all relevant species will suffice, paying attention to the source and loss rates but neglecting advective or diffusive transport. [9] At the short length scale relevant to PMSE, however, transport effects become important; the short distance the particles may travel between their ionization and recombination may then be comparable to, or larger than, the length scale considered. Several studies have therefore focused on the ambipolar (or multipolar) diffusion occurring on these scales [e.g., Hill, 1978; Cho et al., 1992; Hill et al., 1999; Rapp and Lübken, 2000], and in particular the reduced, effective diffusivity of electrons in the presence of heavy, charged particles. These studies only considered diffusion, thus neglecting the production and loss of particles as well as advective transport (which will be induced by gravity). Neglecting production and loss of particles limits a model to the cases where the diffusion timescale is much shorter than the timescales for production and loss of particles, which in the middle atmosphere may be on the order of a few hundred seconds. [10] Klostermeyer [1994] developed a two-ion ice particle model that obtained the steady state solution to the coupled continuity equations, including particle diffusion, to study radar reflectivity induced by turbulence in the neutral atmosphere, finding that the computed electron density variations were sufficient to produce a reflectivity comparable to what is observed by the mobile sounding system (SOUSY) 53.5-MHz as well as the 0.7 m European Incoherent Scatter (EISCAT) 224 MHz radar. [11] The purpose of this paper is to study the role of heavy particles in modifying the electron and ion densities at the

3 LIE-SVENDSEN ET AL.: SMALL-SCALE PLASMA STRUCTURES PMR 9-3 short length scales that are relevant to PMSE. As mentioned above, comparing such model results with in situ rocket measurements can be used to establish whether the observed profiles agree with what one would expect if they were indeed caused by aerosol particles. The model includes production and loss of electrons and ions (through ionization, recombination, and scavenging of electrons and ions by aerosol particles), as well as ambipolar diffusion and gravity in the force balance equation. Hence it is intended to overcome some of the limitations of previous model studies, in particular at small scales where the force balance becomes important. We shall assume that the number of aerosol particles as well as their size are given, constant quantities, and that the particle density profile is given initially, with an embedded small-scale structure. Hence we do not address how such small-scale structures are formed in the aerosol particles in the first place; rather we shall be concerned with their consequences. Although we do not consider the activation, growth, and evaporation of ice particles, we shall in a simplified manner consider the effect on electrons and ions of rapid evaporation of ice particles. [12] We shall obtain numerical solutions to the timedependent transport equations. As we shall see, in many cases simple analytical expressions provide good approximations to the modeled density structures. The main advantage of using a rather complex numerical model is that it does the bookkeeping for us. The analytic results are only valid in certain limits, for instance when particles are in near diffusive equilibrium, and care must then be taken to ensure that we do not apply them outside these limits. The numerical model on the other hand is designed to handle a very wide range of conditions. Moreover, in many intermediate cases it is difficult, or impossible, to solve the equations analytically. Finally, we want to study the time evolution of the solution, which cannot easily be done analytically. [13] The paper is organized as follows: In section 2 we present the time-dependent continuity and momentum equations that we solve numerically, as well as the source and loss rates, due to ionization, recombination, and attachment, that are included in the model. Section 3 presents a reference case, in which we show one particular solution and use this solution to discuss the timescales of the various processes taking place. In this section, analytical solutions in certain limiting cases are also presented. In section 4 we perform sensitivity studies, to see how the plasma structures vary as the properties of the imposed aerosol particle structure is varied. In section 5 we discuss how the results compare with rocket measurements, as well as the effect of rapid evaporation of particles. Finally, the main results are summarized in section The Model [14] In solving the coupled fluid transport equations for charged particles, we find it convenient to separate the continuity and force balance equations, so that we obtain solutions for the density n s as well as the drift velocity u s of a particle species s. We shall only consider vertical transport, hence assuming that horizontal gradients are much smaller than vertical gradients. Effectively, this means that we do not consider horizontal neutral winds and assume that the magnetic field is vertical (otherwise falling particles would experience the Lorentz force which would induce a horizontal velocity component). [15] The charged particles are minor species in a neutral atmosphere which we assume remain unperturbed by the ions. The density n n (z) and temperature T(z) of the neutral atmosphere as a function of altitude z are therefore provided as a given background for the charged particles. [16] For the present study we restrict ourselves to a plasma consisting of positive (light) ions and one class of aerosol particles with a given size and mass. In other words, we shall not consider a size distribution of particles, nor shall we consider a change in size of the particles with time. By not allowing for negative ions, we restrict ourselves to the sunlit atmosphere, which is dominated by positive ions. We include charge states of the aerosol particles ranging from singly positive, through neutral and up to seven negative charges for the cases with the largest particles, with the respective densities denoted by n a,1, n a,0, n a, 1,...n a, 7. Since charge states are treated as separate species in the model, we then obtain solutions for up to 10 species, in addition to electrons. With the adopted ionization and reaction rates, the dominant aerosol particle charge state for the smaller particles will be singly negative, although a significant fraction of charge then also exists in the form of doubly negative particles. Neutral particles will be important mostly when the aerosol particle density is higher than the ambient electron density (so that there are not enough electrons available to charge all particles). [17] The one-dimensional continuity equation for the particle species s ð n su s Þ ¼ Q s L s where u s is the vertical drift speed, Q s and L s are rates for production and loss of particles, respectively, and t denotes time. [18] We use the reaction rate coefficients for capture of electrons and ions by aerosol particles derived by Natanson [1960] [see also Rapp, 2000]: y e;0 ¼ pr 2 a c e y i;0 ¼ pr 2 a c i 1 þ 1 þ s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! e ktr a s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! e ktr a y e;q>0 ¼ pra 2 c e 1 þ qe2 4p 0 ktr a y e;q<0 ¼ pra 2 jje q 2 g2 c e exp 4p 0 ktr a g 1 1 2gðg 2 1Þjj q ð1þ ð2þ ð3þ ð4þ ð5þ y i;q<0 ¼ pra 2 c i 1 þ jq je2 : ð6þ 4p 0 ktr a Here k and 0 denote Boltzmann s constant and the permittivity of vacuum, respectively. The aerosol charge

4 PMR 9-4 LIE-SVENDSEN ET AL.: SMALL-SCALE PLASMA STRUCTURES in units of the elementary charge e is denoted by q, and the two subscripts denote the electron or ion being attached to the aerosol particle and the charge q of the aerosol, respectively. Hence y e,0 is the rate for capture of electrons by neutral particles, y i,q<0 is the rate of capture of ions by a particle with negative charge q = jqj etc. Furthermore, r a q ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8kT= pm i;e is the aerosol particle radius; c i;e is the mean thermal speed of ions or electrons; and gr a is the critical distance at which the induced attractive (dipole) force is stronger than the Coulomb repulsive force of the aerosol particle. The relation between a negative aerosol charge q = jqj and g is given by Natanson [1960] q ¼ 2g2 1 gg ð 2 1Þ 2 ; ð7þ with the numerical solution ranging from g =1.62forq = 1 tog = 1.22 for the highest charge state of the model, q = q m = 7. [19] The source and loss terms used on the right-hand side of (1) for the light ions, denoted by a subscript i, are then L i ¼ n i Q i ¼ Q ð8þ! an e þ X0 y i;q n a;q q¼ q m ; ð9þ where Q is the production rate for ions (and electrons), a is the electron-ion recombination coefficient, and the sum in (9) represents the loss due to ion attachment by the various aerosol particle charge states. For aerosol particles with charge q m < q < 1 the source and loss terms are Q a;q ¼ y e;qþ1 n e n a;qþ1 þ y i;q 1 n i n a;q 1 ð10þ L a;q ¼ n a;q y e;q n e þ y i;q n i ; ð11þ force balance equation in which advection of momentum can be þ n sm s g n s Z s ee ¼ n s m s n s ðu s wþ: ð16þ Here P s = n s kt s is the partial pressure and T s the temperature; m s is the mass of the particle; g is the gravitational acceleration; Z s the charge in units of the elementary charge e; E is the electric field; n s is the collision frequency for collisions with neutrals; and w is a prescribed updraft speed of the neutral atmosphere. [21] With the exception of one example, shown in Figure 1, we shall always set w = 0. Neglecting neutral atmosphere updraft in the model does not imply that updraft is unimportant for the aerosol particle layers formed near the mesopause. Modeling of breaking gravity waves [Garcia and Solomon, 1985] indicates an updraft speed on the order of a few cm/s in the summer mesosphere, which is of the same order of magnitude as the fall speed of 10-nm particles. Hence updraft can keep aerosol particles suspended and possibly increase their lifetime significantly. However, the results presented in this paper will be essentially identical whether updraft is included or not. If w 6¼ 0 we can simply move into a reference frame moving at the speed w relative to Earth, and in this frame the equations (and their solution) will be exactly the same as in an Earthbased frame with w =0. [22] We assume that collisions with neutrals are so efficient in thermalizing the particles, even the electrons, that we can set T s = T(z). Hence we do not need to solve the equation for the conservation of energy. All production and loss terms (Q s and L s ) and collision frequencies v s have a temperature dependence at most on the order of T ±1/2. The results therefore will not be very sensitive to changes in temperature. [23] We require that the plasma remains quasi-neutral at all times, and that there are no currents. These requirements determine the electron density and flow speed: for positively charged aerosol particles n e ¼ X s Z s n s ð17þ Q a;1 ¼ y i;0 n a;0 n i L a;1 ¼ y e;1 n a;1 n e ð12þ ð13þ u e ¼ 1 X Z s n s u s ; n e s ð18þ and for aerosol particles with the maximum charge, q = q m, Q a;q ¼ y e;qþ1 n e n a;qþ1 L a;q ¼ y i;q n a;q n i : ð14þ ð15þ Since we shall obtain the electron density from the requirement of charge neutrality (see below) we do not need the explicit expressions for the corresponding electron source and loss terms. [20] The high collision frequency with the neutral atmosphere implies that the particles quickly accelerate to their terminal speed, so that inertial effects will not be important. The momentum equation for species s then simplifies to a where the sum extends over all particle species (excluding electrons) included in the model. These equations replace the continuity and momentum equations for electrons and also determine the electric field necessary to maintain zero current at all times. The requirement of quasi charge neutrality is only appropriate at length scales longer than the Debye length. With a typical electron density n e = m 3 and temperature T = 150 K, the electron Debye length is sffiffiffiffiffiffiffiffiffi kt 0 l e ¼ n e e 2 1:5 cm: ð19þ Since the smallest length scales we shall consider are about 1 m, we can therefore safely treat the plasma as being

5 LIE-SVENDSEN ET AL.: SMALL-SCALE PLASMA STRUCTURES PMR 9-5 (quasi) neutral. (Even radars that operate at GHz frequencies, corresponding to 15 cm Bragg wavelength, should be safely outside this limit.) The polarization electric field provides an essential coupling between the charged particles; since they are minor species, collisions among them can be neglected and their only means of communication are through this electric field (and the source and loss terms in the continuity equation (1)). The electric field can be extracted from the electron momentum equation. To a very good approximation we can neglect the frictional force on electrons due to collisions with neutrals. From the electron momentum equation the electric field is then E kt e 1 n : ð20þ [24] Following Cho et al. [1992], we shall assume that ion-neutral collisions can be described as a polarization interaction for particles with a radius smaller than r c = m, with the collision frequency given as [Hill and Bowhill, 1977] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n s ¼ 2: n n 0:78 1:74 ~m s þ þ 0:21 ~m s þ 28 28~m s ~m s þ 32 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:57 ~m s þ 32 þ 0:01 32~m s 40 ~m s þ 40 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1:64 ~m s þ 40 40~m s ; ð21þ where ~m s m s /m u with m u being the atomic mass unit and n n must be given in m 3. For particles with radii larger than r c we assume collisions with neutrals can be described as hard-sphere interactions, for which the collision frequency is [e.g., Schunk, 1977] sffiffiffiffiffiffiffiffi n s ¼ p 8 3 ffiffiffi n nm n 2kT pðr s þ r n Þ 2 ; ð22þ p m s þ m n m s where m n 29 m u is the mean molecular mass of air, r s is the radius of the (assumed) spherical particle, r n is the effective radius of the neutral molecule (r n 0.2 nm), and m s m s m n /(m s + m n ) is the reduced mass. [25] For the cases we shall consider, (21) will be used for collisions between light (<100 m u ), positive ions, and the neutral air, while (22) will be used for collisions between heavy aerosol particles and air molecules. In the latter case we shall always have that r s r n and m s m n so that m s m n. [26] As boundary conditions we impose that both the density n s and drift speed u s of all species are constant through the lower and upper boundaries. Since it is force balance, rather than advection of momentum, that determines the speed, the precise choice of momentum boundary condition is not very important. The computational domain is chosen sufficiently thick so that the small-scale structure that we want to follow at all times is many grid points from the boundary. Assuming constant density and drift speed through the boundary then effectively simulates an open boundary. Since particles are settling, the upper boundary is most important as new particles are advected by gravity into the slab here. Imposing no gradients in density and drift speed here simulates moving this boundary to infinity. At the lower boundary, this open boundary condition merely allows the falling particles to move out of the slab. We shall always stop the time integration before the structure we want to follow has reached the lower boundary. [27] The time-dependent numerical solution to equations (1), (16), (17), and (18) for the four particles species and electrons that we consider, is obtained using a strippeddown version of a large solar wind model [see Hansteen and Leer, 1995; Lie-Svendsen et al., 2001]. The main advantage for the present application offered by this code is that it obtains the time-dependent solution for multipolar diffusion of an arbitrary number of charged (and neutral) particle species, the number only limited by the available computer memory and time. Second, the time integration is implicit, thus avoiding the restrictions imposed by the Courant stability criterion [see, e.g., Press et al., 1992]. [28] As stated in the introduction, our goal is to study how small-scale structure in the aerosol particle population, whose origin is beyond the scope of this paper, induces density variations in the ambient plasma and hence (possibly) PMSE. We shall therefore start the model with a plasma with a constant density, but with a population of neutral aerosol particles which has an initial, small-scale density perturbation, for simplicity modeled as a Gaussian, " n a;0 ðt ¼ 0; zþ ¼ n 0 a þ n1 a exp z z # 2 0 ; ð23þ s a where n a 0, n a 1, z 0, and s a are constants to be chosen. [29] Assuming that all aerosol particles are neutral at the start of the model run is probably unrealistic for the conditions of the mesosphere: The growth of ice particles most likely takes an hour or more before they reach the size that we have assumed [Turco et al., 1982]. Attachment of electrons, as we shall see, happens on a timescale of seconds. The particles will therefore likely be in near equilibrium with the ambient electrons at all times during their growth. However, we find that starting the model with all particles neutral is useful in order to elucidate how the introduction of aerosol particles into the plasma modifies the plasma. The distribution of charges at the end of the calculation will be independent of this particular initial state. [30] Because ambipolar (or more correctly, multipolar) diffusion and gravity are included, the system will never reach a truly steady state. The integration therefore has to be stopped at some time, either when a quasi steady state has been reached or when the structure has completely vanished by diffusion. The purpose of this paper is not to estimate the lifetime of these small-scale aerosol particle structures; the lifetime can easily be estimated from the diffusivity of the particles, and the low diffusivity of heavy particles when collisions can be approximated by the hard sphere model is well known (see, e.g., the discussions by Cho et al. [1992] and Hoppe [1993]). (We shall find that the electric field is too weak to significantly affect the force balance of the particles and hence to modify their ability to diffuse, so that the charge state of the particles will not matter for their diffusion.) For heavy aerosol particles the lifetime of the structure may eventually be limited by sedimentation since the particles (in the absence

6 PMR 9-6 LIE-SVENDSEN ET AL.: SMALL-SCALE PLASMA STRUCTURES Figure 1. Densities of electrons, positive ions, and negative and doubly negative aerosol particles for the reference case at t = 0, 10, 100, and 1000 s. The thick solid curves show the corresponding solution after 1000 s for w =3cms 1. of neutral air updraft) will eventually reach warmer, subsaturated air in which they sublimate. This process is not our focus here, and would require detailed knowledge of both the temperature and humidity structure of the atmosphere. Rather, we want to focus on the structure imposed on the plasma, primarily the electrons, by the particles, on an intermediate timescale that is shorter than the lifetime of the aerosol particle structure, but still long enough so that the plasma has had time to reach a quasi steady state. As we shall see, there is a very short timescale, on the order of seconds, in which these structures are formed and reach such a steady state, while the lifetime of the structures themselves (limited by particle diffusion or sedimentation) may be on the order of hours. Unless some very vigorous process creates a significant number of these small-scale structures within a few seconds, it should be the intermediate timescale that we are interested in that is most important for the radar echoes, since most of the smallscale structure in a given volume, and hence the radar backscatter from this volume, will be in the form of these quasi-steady structures. [31] For the cases we present here we choose to terminate the calculation after 10 3 s, which will be more than sufficient to reach a quasi steady state, while aerosol particle diffusion will not have had time to smear out the structure significantly (except for the smallest, 1 nm, particles that we shall consider). [32] For the altitude range, we only need a range large enough to contain the small-scale structure that we want to follow over the integration time. For that purpose, an altitude range of 100 m will suffice in most cases, although for the largest (100 nm) particles that we consider a 300 m altitude range is necessary in order to contain the structure at all times. 3. A Reference Case [33] It is useful to define a reference case, both to discuss the physical processes taking place and their timescales, and as a benchmark with which other cases may be compared. [34] We choose physical parameters that roughly correspond to an altitude of 85 km, where PMSE are most

7 LIE-SVENDSEN ET AL.: SMALL-SCALE PLASMA STRUCTURES PMR 9-7 frequently observed, while still keeping the system as simple as possible. The initial aerosol particle perturbation is chosen such that it leads to electron and ion density structures that are in reasonable agreement with rocket observations and which will be sufficient to cause PMSE. [35] We choose a temperature T = 150 K and a neutral atmosphere density n n m 3 [Lübken, 1999]. [36] The chosen altitude is roughly in the transition region where the positive charge of the plasma changes from being carried by heavy water-cluster ions to light (predominantly NO + ) ions. Following Reid [1990], we therefore adopt a light ion mass m i =50m u and an electron-ion recombination rate coefficient a =10 12 m 3 s 1, which correspond to a mixture of light ions and heavy cluster ions. In the absence of aerosol particles, we require that the model yields a plasma density n e = n i n 0 = m 3, which is a typical summertime ambient plasma density measured by rockets at these altitudes [Blix, 1999]. For the model to produce this chemical equilibrium density the electron-ion production rate must be chosen as Q = an 0 2 = m 3 s 1, which is not unreasonable for the summertime polar mesopause [Rapp and Lübken, 2001]. [37] We start the model run with electron and positive ion densities equal to their chemical equilibrium values in the absence of aerosol particles, n e (t =0,z) =n i (t =0,z) =n 0 = m 3. For the aerosol particles we choose n a 0 =10 5 m 3, n a 1 = m 3, and s a = 3 m. In other words, outside the small perturbed region the particle density is essentially negligible compared to the ambient plasma density, and the particles therefore have no effect on electrons and positive ions outside this small region, while the maximum perturbation is 10% of the ambient plasma density. Initially, the perturbation is located slightly above the middle of the 100-m-thick slab. [38] The aerosol particles have a mass m a = m u and a radius r a = 10 nm, corresponding to a density r a 900 kg m 3, about equal to the density of ice. We set the maximum negative charge of the particles to q m = 2, which turns out to be sufficient for 10-nm particles. [39] Figure 1 shows the densities of electrons, ions, and two of the particle species at three times during the integration. The density of positively charged aerosol particles is not shown because n a,1 < m 3 at all times, and it is therefore insignificant in this case. [40] The first thing to note is that electrons quickly attach themselves to the neutral aerosol particles; all particles have essentially become singly charged within the first 10 s. As a rough estimate all particles become charged in a time t a 1 y e;0 n e : ð24þ With the parameters chosen above, (2) gives an electron attachment rate y e,0 = m 3 s 1, and hence t a 1.4 s. [41] After the initial rapid electron depletion, which is equal in magnitude to the increase in n a, 1, the dip in electron density actually becomes shallower and wider instead, as seen in the t = 100 s graph. This is mostly the effect of ambipolar diffusion: As seen in the upper right panel of Figure 1, there is a corresponding increase in n i at the center of the perturbed region. The initial dip in electron density creates a pressure gradient force in the electron gas that attempts to move electrons into the depletion. The initial movement of electrons toward the hole then instantly creates a (very small) charge imbalance. The resulting polarization electric field then impedes the flow of electrons, at the same time creating an electric force on the positive ions that pulls them into the electron hole. The net result is then ambipolar diffusion: Electrons and ions flow together (with zero net current) into the perturbed region, partially filling in the electron depletion while at the same time creating an enhancement in the density of positive ions. (On this timescale aerosol particles can essentially be regarded as immobile, although their movement is included in the model.) This is evident in the upper right panel, where we note that the increase in ion density at the center of perturbation is accompanied by a decrease in ion density adjacent to the perturbation, showing that ions are being pulled into this region from the surroundings. [42] The ambipolar diffusion timescale may be estimated as follows. We assume that the initial charging of the aerosol particles is complete before diffusion has had time to act. The electron density after attachment is then approximately n e ðt ¼ t a ; zþ n 0 n a;0 ðt ¼ 0; zþ ð25þ with n a,0 (t =0,z) given by (23). Using (20) for the electric field, and neglecting gravity and the pressure gradient force for the ions, the maximum ion diffusion speed at t = t a is approximately u i;max ðt ¼ t a ; zþ kt n 1 a ; m i n i s a n 0 ð26þ where we have also used that n 1 a n 0 a inpthe reference case. This maximum occurs at a distance s a = ffiffiffi 2 from the center of the perturbation z 0. We then define the diffusion timescale t d,i as the time pit takes to fill the volume between z = z 0 and z ¼ z 0 þ s a = ffiffiffi 2 with as many positive ions as there are negatively charged aerosol particles at t = t a. From the ion continuity equation (1) this time is roughly (neglecting factors on the order of unity) t d;i m in i kt s2 a : ð27þ With the chosen neutral air density and ion mass, n i s 1. With s a = 3 m, (27) gives a diffusion timescale t d,i 25 s. [43] Since, from (27), t d,i / s a 2, the diffusion timescale is very sensitive to the length scale of the aerosol particle perturbation. For the small-scale perturbations that are relevant to PMSE, t d,i is short and diffusion is a dominant process. Conversely, for large-scale perturbations such as the deep electron bite-outs that have frequently been observed with length scales of several hundred meters, ambipolar diffusion plays no role and the density perturbations must be understood in terms of chemical reactions only. In the latter case, densities may to a good approx-

8 PMR 9-8 LIE-SVENDSEN ET AL.: SMALL-SCALE PLASMA STRUCTURES imation be modeled assuming that all species are in chemical equilibrium [e.g., Reid, 1990]. [44] In this example, the attachment and diffusion timescales, t a and t d,i, are not sufficiently far apart that the amplitude of the electron depletion ever becomes equal to the amplitude of the aerosol particle perturbation. In the numerical simulation, attachment causes the electron density to decrease to about 3/4 of the value predicted by (25), whence diffusion sets in and reduces the depletion, as explained above. [45] The ion density will also increase because the attachment of electrons by aerosol particles leads to a decrease in the ion loss by electron-ion recombination. The recombination time t r, defined as the time needed, through recombination alone, to achieve an increase in n i comparable to the dip in n e is roughly t r 1 n 0 a ; ð28þ or t r 170 s. Hence this process is significantly slower than the diffusion process in this particular example. Since t d,i is so sensitive to the length scale of the perturbation, we shall see that for aerosol particle perturbations on longer length scales this process will be more important than diffusion. [46] Negative aerosol particles can be neutralized by attachment of positive ions. Similarly to the above estimate for electron-ion recombination, negative particles are neutralized at a timescale t a;i 1 n 0 y i; 1 ; ð29þ with t a,i 170 s. However, because this timescale is a factor 100 longer than the electron attachment timescale, this process can be neglected: The particles that become neutralized will instantly be charged by attachment of a free electron. [47] The whole density perturbation is settling due to gravity. For the heavy particles, both the electric force and the pressure gradient force may to a good approximation be neglected when estimating the fall speed, which is then, from (16), u a g/n a. For 10-nm particles the hard-sphere approximation gives a collision frequency n a 400 s 1 using (22), and thus u a 2.5 cm s 1. We define the gravitational timescale, t g, then as the time it takes for an aerosol particle to fall a distance on the order of the width of the perturbation. Hence we have t g s a n a g : ð30þ For s a =3m,t g 120 s. Since this timescale is about five times longer than the ion diffusion timescale t d,i, the ions have time to readjust themselves as the heavy particles are falling. For heavy aerosols we shall see that this is no longer the case. From (22) t g / r a 1 so that the two timescales should become comparable for 50-nm particles. [48] The attachment rate (5) for electrons onto singly charged negative particles is much lower than y e,0, y e, m 3 s 1. Hence the time needed to produce a significant number of doubly charged aerosol particles is much longer than the time needed to acquire a single electron; a rough estimate would be t a2 1 y e; 1 n e ; ð31þ or t a2 700 s. In other words, our chosen integration time is barely sufficient for the doubly charged negative particles. However, the loss of doubly charged particles, caused by ion attachment, has a significantly shorter timescale. With an ion attachment rate y i, m 3 s 1,the timescale for loss of doubly negative particles is t a2;i 1 n i y i; 2 ; ð32þ or t a2,i 100 s. In chemical equilibrium the rate of electron attachment onto negative particles must balance loss due to attachment of positive ions. This equilibrium density is approximately n a, 2 (eq.) n a, 1 y e, 1 /y i, n a, 1, where we have used that n e n i since there are so few aerosol particles. Although particles are falling due to gravity and diffusing, so that they will never attain true chemical equilibrium, it is nevertheless a good approximation for the doubly negative particles in this case. The reason is that all aerosol particles are essentially falling at the same speed irrespective of charge (which means that both the electric force and the pressure gradient force on them can be neglected), so that singly and doubly charged particles always stay together and therefore have time to come to equilibrium. The actual values for n a, 2 at the end of the run therefore deviate only by about 10% from the simple estimate above (which also agrees with the calculation by Rapp and Lübken [2001]). Although most aerosol particles are still singly charged, this case shows that even for particles as small as 10 nm about 1/4 of the aerosol particle charge sits on doubly charged particles, so that these cannot be entirely neglected. [49] Finally, similarly to the ion diffusion timescale t d,i defined above, the timescale for diffusion of aerosol particles can be estimated as t d;a m an a kt s2 a : ð33þ For the reference case we then have t d,a s. In other words, this timescale is much longer than the integration time, so that the aerosol particle perturbation is essentially falling as a frozen structure. With a fall speed on the order of 3 cm/s, 10-nm particles may then fall about 200 m before particle diffusion becomes important. From (22) m a n a / r a 2, so that particle diffusion is very sensitive to the size of the particles. We shall see that for 1-nm particles, the whole perturbation quickly vanishes due to diffusion. [50] To summarize, we have identified eight different timescales, listed in Table 1, corresponding to different physical processes. The timescales range from t a 1s, the attachment of free electrons onto neutral aerosol particles, to t d,a 10 4 s, the diffusion timescale of the particles. These two extreme scales can be neglected for our purpose:

9 LIE-SVENDSEN ET AL.: SMALL-SCALE PLASMA STRUCTURES PMR 9-9 Table 1. Timescales and Their Values in the Reference Case Process Timescale Reference case value Electron-neutral attachment t a 1 y e;0 ne 1.4 s Ion/electron diffusion t d;i mini kt s2 a 25 s Ion-double neg. attachment t a2;i 1 niy i; s na Gravity t g s a g 120 s Electron-ion recombination t r 1 n0a 170 s Ion-neg. attachment t a;i 1 n0y i; s Electron-neg. attachment t a2 1 y e; 1 ne 700 s Particle diffusion t d;a mana kt s2 a 7000 s The attachment is so rapid that it merely ensures that all particles are charged at all times, so we may to a good approximation set n a,0 (t, z) = 0. Aerosol particle diffusion is important as it limits the lifetime of the small-scale structures, but, as emphasized, it is not the focus of this paper as our concern is what happens to the plasma when small-scale particle structures are present. [51] In Figure 1 we also show the solution when an updraft speed w =3cms 1 [Garcia and Solomon, 1985] is included. This updraft speed us just slightly larger than the 2.5 cm s 1 fall speed of the particles, and hence will cause the whole structure to move a small distance upwards. Apart from this trivial difference in location of the structure, the figure illustrates that the density perturbations are the same whether w = 0 or not (the small differences reflect small errors in the numerical advection scheme), in accordance with the discussion in the previous section. The timescales discussed above will not change either; when w 6¼ 0 that discussion must be carried out in a reference frame moving at speed w. [52] The timescales and corresponding physical processes that remain as most important for the plasma density structure are ambipolar diffusion, with a timescale t d,i 25 s, gravity with a scale t g 100 s, and ion-electron recombination with a scale t r 200 s. Since in this particular example the diffusion timescale is significantly shorter than the other two, it would be reasonable to assume that electrons and positive ions remain in near diffusive equilibrium at all times except for the first minute or so when this equilibrium is established. If we then neglect both gravity and ion-electron recombination, and only consider ambipolar diffusion, we can obtain analytical solutions for n e and n i. Let n ac be the total density of negative charges carried by the aerosol particles, which in the reference case becomes n ac n a; 1 þ 2n a; 2 ð34þ since n a,1 0. We regard n ac (z) as a known quantity, and seek to express n e and n i as functions of n ac. In the reference case without gravity and particle diffusion, and neglecting doubly charged particles (which is questionable, from the results presented above), n ac (z) n a,0 (t =0,z) with n a,0 (t = 0, z) taken from (23). However, the expressions derived below apply to an arbitrary n ac (z). Diffusive equilibrium for electrons and ions imply that u e = u i = 0. Assuming furthermore that the temperature is constant (or weakly varying), equation (16) for electrons and ions simplify to kt dn e dz þ n eee ¼ 0 kt dn i dz n iee ¼ 0: These may be trivially combined and integrated to yield n e ðþn z i ðþ¼n z e ðz s Þn i ðz s Þ ¼ constant; ð35þ ð36þ ð37þ where n e (z s ) and n i (z s ) are electron and ion densities at some arbitrary starting point z s for the integration. In other words, if diffusive equilibrium holds the product of the electron and ion densities should be constant across the perturbation. Using the neutrality condition, n i = n e + n ac, (37) may be solved for n e as a function of n ac (z): n e ðþ¼ z 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ac ðþ z 2 þ 4n e ðz s Þ½n e ðz s Þþn ac ðz s ÞŠ: n ac ðþ z : ð38þ 2 When the amplitude of the perturbation is small, more precisely when jn ac (z) n ac (z s )j n ac (z s )/n e (z s ) n e (z s ), (38) can be Taylor-expanded in this term, giving 1 n e ðþ¼n z e ðz s Þ ðn 2 þ nacðzsþ ac ðþ n z ac ðz s ÞÞ ð39þ n e ðz s Þ n i ðþ¼ z n e ðz s Þþ n ac ðz s Þþ 1 þ nac ð z s Þ n e ðz s Þ 2 þ n ðn acðz s Þ ac ðþ z n ac ðz s ÞÞ: ð40þ n e ðz s Þ For the parameters used in the reference case (38) may be further simplified. Choosing z s at the lower (or upper) boundary where n e = n 0 and n ac = 0, and using that everywhere n ac n 0, the diffusive equilibrium solution for the reference case is n e ðþn z n acðþ z n i ðþn z 0 þ 1 2 n acðþ: z ð41þ ð42þ This simple result in other words says that when the aerosol density is low the decrease in electron density triggered by the attachment of charge onto aerosol particles is only half of the aerosol particle charge density, while the same number of ions have moved into the perturbed region to maintain charge neutrality. Comparing (39) and (41), we note that the largest electron density perturbation that can be obtained, for a given aerosol charge perturbation n ac, is obtained when the ambient aerosol charge density, given by n ac (z s ), is low. When the ambient aerosol charge density is much higher than the ambient electron density, n ac (z s ) n e (z s ), the perturbation in electron density will be much smaller than the aerosol particle perturbation, while from (40) the ion perturbation

10 PMR 9-10 LIE-SVENDSEN ET AL.: SMALL-SCALE PLASMA STRUCTURES Figure 2. Electron and ion densities at t = 1000 s in the reference case using the full numerical solution (solid curves), the diffusive equilibrium approximations (41) and (42) (dotted curves), and the chemical equilibrium approximations (43) and (44) (dashed curves). in this case will be of the same magnitude as the aerosol particle perturbation. [53] Note that the diffusive equilibrium result is independent of temperature. Although the pressure gradient force, which depends on temperature, balances the electric force, the tendency of ions to move into the perturbed region is unaltered as long as electrons and ions have the same temperature. When the temperature increases, thus increasing the pressure force that resists the accumulation of ions, the electric field increases, too, from (20), and these two effects cancel exactly. [54] Although assuming chemical equilibrium is not a valid approximation for the reference case, it is nevertheless instructive to see what the structure would have looked like if the particles had had time to come to chemical equilibrium. (As mentioned, most previous studies have only obtained the chemical equilibrium solution.) Setting n a,0 = n a,1 = 0 (all particles have become negatively charged), and regarding n a, 1 and n a, 2 as given (e.g., from the numerical solution), the solution is, in the limit when the aerosol number density is much smaller than n 0, n eq e n eq i ðþn z 0 n a; 1 2 ðþn z 0 þ n a; þ y i; 1 a 1 y i; 1 a n a; 2 þ n a; 2 1 þ y i; 2 2a ð43þ 1 y i; 2 : ð44þ 2a We note that if ion attachment onto aerosol particles could be neglected, that is, if y i, 1 = y i, 2 = 0, then the chemical equilibrium result is identical to the diffusive equilibrium result (41) (42). However, the attachment rates are only negligible for very small particles; for the 10-nm particles that we have assumed, the ion attachment rate is comparable to the electron-ion recombination rate, y i, 1 /a 1 and y i, 2 /a 2. The decrease in electron density predicted by (43) will therefore be significantly larger than the diffusive equilibrium prediction (41). In other words, because the ion attachment rate is comparable to the ion-electron recombination rate, the ion chemical equilibrium density will not change that much in the presence of aerosol particles: The increase in density effected by the decrease in the ion-electron recombination loss rate is mostly compensated for by increased ion loss due to attachment onto (negative) particles. [55] In Figure 2 we compare the full numerical solution at the last time step with the simple analytical expressions (41) and (42), with n ac obtained from the numerical solution, and with the corresponding chemical equilibrium result (43) and (44). As anticipated from the timescales discussed above, the figure confirms that electrons and ions are close to being in diffusive equilibrium. Adjacent to the aerosol particle structure the analytic result slightly overestimates the electron density while slightly underestimating it at the center of the perturbation. For the ions, the diffusive equilibrium result slightly overestimates the amplitude of the perturbation, and does not capture the decrease in ion density adjacent to the structure. Although the initial diffusion happens within 25 s, diffusion slows down considerably after the initial movement of electrons and ions into the perturbed region. Chemical reactions and the fact that the whole structure is moving downward due to gravity then ensures that full diffusive equilibrium will never be reached. [56] Figure 2 shows that assuming chemical equilibrium overestimates the electron depletion (and hence would overestimate the radar echo from the structure) and underestimates the ion enhancement. Although both diffusive and chemical processes predict an anticorrelation between electron and ion density perturbations, the relative magnitude is very different in the two processes. [57] To conclude, as a good approximation the electron and ion density perturbations in the reference case are close to their diffusive equilibrium values, given by (41) and (42). [58] From (16), the frictional force acting on the charged aerosol particles must be balanced by gravity, the pressure