We have developed a numerical model that solves the time-dependent, one-dimensional, 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.
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 At high latitudes in summer, strong VHF radar echoes occur in the height range 80–90 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 rocket instruments and most of the existing theories have been summarized by, e.g., Cho and Kelley , Hoppe et al. , Cho and Röttger , and references therein. Hoppe et al.  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  found that only about 10% of PMSE exhibit the wide spectra and absence of aspect sensitivity that result from turbulence. Lübken et al.  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 the observations of Chilson et al.  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 .
Reid  and Klostermeyer  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  and Lübken and Rapp  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.
 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.
 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.
 There is also direct evidence for meter-scale variations in the charge carried by aerosol particles: Havnes et al.  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.
 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.
 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.
 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.
Klostermeyer  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.
 The purpose of this paper is to study the role of heavy particles in modifying the electron and ion densities at the 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.
 We shall obtain numerical solutions to the time-dependent 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.
 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 6.
2. The Model
 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 ns as well as the drift velocity us 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).
 The charged particles are minor species in a neutral atmosphere which we assume remain unperturbed by the ions. The density nn(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.
 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 na,1, na,0, na,−1, …na,−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).
 The one-dimensional continuity equation for the particle species s reads
where us is the vertical drift speed, Qs and Ls are rates for production and loss of particles, respectively, and t denotes time.
 We use the reaction rate coefficients for capture of electrons and ions by aerosol particles derived by Natanson  [see also Rapp, 2000]:
Here k and ϵ0 denote Boltzmann's constant and the permittivity of vacuum, respectively. The aerosol charge 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 ψe,0 is the rate for capture of electrons by neutral particles, ψi,q<0 is the rate of capture of ions by a particle with negative charge q = −∣q∣ etc. Furthermore, ra is the aerosol particle radius; is the mean thermal speed of ions or electrons; and γra 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 = −∣q∣ and γ is given by Natanson 
with the numerical solution ranging from γ = 1.62 for q = −1 to γ = 1.22 for the highest charge state of the model, q = −qm = −7.
 The source and loss terms used on the right-hand side of (1) for the light ions, denoted by a subscript i, are then
where Q is the production rate for ions (and electrons), α 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 −qm < q < 1 the source and loss terms are
for positively charged aerosol particles
and for aerosol particles with the maximum charge, q = −qm,
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.
 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 force balance equation in which advection of momentum can be neglected,
Here Ps = nskTs is the partial pressure and Ts the temperature; ms is the mass of the particle; g is the gravitational acceleration; Zs the charge in units of the elementary charge e; E is the electric field; νs is the collision frequency for collisions with neutrals; and w is a prescribed updraft speed of the neutral atmosphere.
 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 ≠ 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 Earth-based frame with w = 0.
 We assume that collisions with neutrals are so efficient in thermalizing the particles, even the electrons, that we can set Ts = T(z). Hence we do not need to solve the equation for the conservation of energy. All production and loss terms (Qs and Ls) and collision frequencies vs 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.
 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:
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 ne = 3 × 109 m−3 and temperature T = 150 K, the electron Debye length is
Since the smallest length scales we shall consider are about 1 m, we can therefore safely treat the plasma as being (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
 Following Cho et al. , we shall assume that ion-neutral collisions can be described as a polarization interaction for particles with a radius smaller than rc = 5 × 10−10 m, with the collision frequency given as [Hill and Bowhill, 1977]
where s ≡ ms/mu with mu being the atomic mass unit and nn must be given in m−3. For particles with radii larger than rc we assume collisions with neutrals can be described as hard-sphere interactions, for which the collision frequency is [e.g., Schunk, 1977]
where mn ≈ 29 mu is the mean molecular mass of air, rs is the radius of the (assumed) spherical particle, rn is the effective radius of the neutral molecule (rn ≈ 0.2 nm), and μs ≡ msmn/(ms + mn) is the reduced mass.
 For the cases we shall consider, (21) will be used for collisions between light (<100 mu), 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 rs ≫ rn and ms ≫ mn so that μs≈ mn.
 As boundary conditions we impose that both the density ns and drift speed us 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.
 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 stripped-down 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].
 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,
where na0, na1, z0, and σa are constants to be chosen.
 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.
 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.  and Hoppe ). (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 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 small-scale structure in a given volume, and hence the radar backscatter from this volume, will be in the form of these quasi-steady structures.
 For the cases we present here we choose to terminate the calculation after 103 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).
 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
 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.
 We choose physical parameters that roughly correspond to an altitude of 85 km, where PMSE are most 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.
 We choose a temperature T = 150 K and a neutral atmosphere density nn ≈ 2.3 × 1020 m−3 [Lübken, 1999].
 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 , we therefore adopt a light ion mass mi = 50 mu and an electron-ion recombination rate coefficient α = 10−12 m3 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 ne = ni ≡ n0 = 6 × 109 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 = αn02 = 3.6 × 107 m−3 s−1, which is not unreasonable for the summertime polar mesopause [Rapp and Lübken, 2001].
 We start the model run with electron and positive ion densities equal to their chemical equilibrium values in the absence of aerosol particles, ne(t = 0, z) = ni(t = 0, z) = n0 = 6 × 109 m−3. For the aerosol particles we choose na0 = 105 m−3, na1 = 6 × 108 m−3, and σa = 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.
 The aerosol particles have a mass ma = 2.3 × 106mu and a radius ra = 10 nm, corresponding to a density ρa ≈ 900 kg m−3, about equal to the density of ice. We set the maximum negative charge of the particles to qm = 2, which turns out to be sufficient for 10-nm particles.
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 na,1 < 4 × 105 m−3 at all times, and it is therefore insignificant in this case.
 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
With the parameters chosen above, (2) gives an electron attachment rate ψe,0 = 1.2 × 10−10 m3 s−1, and hence ta ≈ 1.4 s.
 After the initial rapid electron depletion, which is equal in magnitude to the increase in na,−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 ni 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.
 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
with na,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 = ta is approximately
where we have also used that na1 ≪ na0 in the reference case. This maximum occurs at a distance from the center of the perturbation z0. We then define the diffusion timescale td,i as the time it takes to fill the volume between z = z0 and with as many positive ions as there are negatively charged aerosol particles at t = ta. From the ion continuity equation (1) this time is roughly (neglecting factors on the order of unity)
With the chosen neutral air density and ion mass, νi ≈ 7 × 104 s−1. With σa = 3 m, (27) gives a diffusion timescale td,i ≈ 25 s.
 Since, from (27), td,i ∝ σa2, 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, td,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 approximation be modeled assuming that all species are in chemical equilibrium [e.g., Reid, 1990].
 In this example, the attachment and diffusion timescales, ta and td,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.
 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 tr, defined as the time needed, through recombination alone, to achieve an increase in ni comparable to the dip in ne is roughly
or tr ≈ 170 s. Hence this process is significantly slower than the diffusion process in this particular example. Since td,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.
 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
with ta,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.
 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), ua ≈ g/νa. For 10-nm particles the hard-sphere approximation gives a collision frequency νa ≈ 400 s−1 using (22), and thus ua ≈ 2.5 cm s−1. We define the gravitational timescale, tg, 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
For σa = 3 m, tg ≈ 120 s. Since this timescale is about five times longer than the ion diffusion timescale td,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)tg ∝ ra−1 so that the two timescales should become comparable for 50-nm particles.
 The attachment rate (5) for electrons onto singly charged negative particles is much lower than ψe,0, ψe,−1 ≈ 2.4 × 10−13 m3 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
or ta2 ≈ 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 ψi,−2 ≈ 1.8 × 10−12 m3 s−1, the timescale for loss of doubly negative particles is
or ta2,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 na,−2(eq.) ≈ na,−1ψe,−1/ψi,−2 ≈ 0.13 na,−1, where we have used that ne ≈ ni 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 na,−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 ). 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.
 Finally, similarly to the ion diffusion timescale td,i defined above, the timescale for diffusion of aerosol particles can be estimated as
For the reference case we then have td,a ≈ 7 × 103 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)maνa ∝ ra2, 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.
 To summarize, we have identified eight different timescales, listed in Table 1, corresponding to different physical processes. The timescales range from ta ≈ 1 s, the attachment of free electrons onto neutral aerosol particles, to td,a ≈ 104 s, the diffusion timescale of the particles. These two extreme scales can be neglected for our purpose: 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 na,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.
Table 1. Timescales and Their Values in the Reference Case
Reference case value
Ion-double neg. attachment
 In Figure 1 we also show the solution when an updraft speed w = 3 cm s−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 ≠ 0 that discussion must be carried out in a reference frame moving at speed w.
 The timescales and corresponding physical processes that remain as most important for the plasma density structure are ambipolar diffusion, with a timescale td,i ≈ 25 s, gravity with a scale tg ≈ 100 s, and ion-electron recombination with a scale tr ≈ 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 ne and ni. Let nac be the total density of negative charges carried by the aerosol particles, which in the reference case becomes
since na,1 ≈ 0. We regard nac(z) as a known quantity, and seek to express ne and ni as functions of nac. In the reference case without gravity and particle diffusion, and neglecting doubly charged particles (which is questionable, from the results presented above), nac(z) ≈ na,0(t = 0, z) with na,0(t = 0, z) taken from (23). However, the expressions derived below apply to an arbitrary nac(z). Diffusive equilibrium for electrons and ions imply that ue = ui = 0. Assuming furthermore that the temperature is constant (or weakly varying), equation (16) for electrons and ions simplify to
These may be trivially combined and integrated to yield
where ne(zs) and ni(zs) are electron and ion densities at some arbitrary starting point zs 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, ni = ne + nac, (37) may be solved for ne as a function of nac(z):
When the amplitude of the perturbation is small, more precisely when ∣nac(z) − nac(zs)∣ nac(zs)/ne(zs) ≪ ne(zs), (38) can be Taylor-expanded in this term, giving
For the parameters used in the reference case (38) may be further simplified. Choosing zs at the lower (or upper) boundary where ne = n0 and nac = 0, and using that everywhere nac ≪ n0, the diffusive equilibrium solution for the reference case is
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 nac, is obtained when the “ambient” aerosol charge density, given by nac(zs), is low. When the ambient aerosol charge density is much higher than the ambient electron density, nac(zs) ≫ ne(zs), the perturbation in electron density will be much smaller than the aerosol particle perturbation, while from (40) the ion perturbation in this case will be of the same magnitude as the aerosol particle perturbation.
 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.
 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 na,0 = na,1 = 0 (all particles have become negatively charged), and regarding na,−1 and na,−2 as given (e.g., from the numerical solution), the solution is, in the limit when the aerosol number density is much smaller than n0,
We note that if ion attachment onto aerosol particles could be neglected, that is, if ψi,−1 = ψi,−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, ψi,−1/α ≈ 1 and ψi,−2/α ≈ 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.
 In Figure 2 we compare the full numerical solution at the last time step with the simple analytical expressions (41) and (42), with nac 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.
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.
 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).
 From (16), the frictional force acting on the charged aerosol particles must be balanced by gravity, the pressure gradient force, and the electric force. For the singly charged particles we find that at the end of the run, the pressure gradient force is everywhere less than 10% of the gravitational force, while the electric force is much smaller still, being everywhere smaller than 10−3 times the gravitational force. Hence the charge of the aerosol particles neither affects their fall speed nor their diffusion speed. Because gravity is so dominant, the different particle charge states therefore fall together, explaining why na,−2/na,−1 is close to its chemical equilibrium value as discussed above.
4. Varying the Aerosol Layer Properties
 In the following subsections we shall vary the parameters specifying the initial aerosol particle structure, such as the size and number of particles and the width of the initial disturbance. (In section 5 we also consider the effect of rapidly evaporating aerosol particles.) In sections 4.1–4.4, except for the one parameter being varied, all other parameters remain unchanged from the reference case.
4.1. Width of the Disturbance
 In the reference case ne and ni were close to their diffusive equilibrium values, as expected since the ion diffusive timescale, td,i ≈ 25 s, was much shorter than the other timescales of the system (except the initial electron attachment scale, ta). Here we vary σa in (23) from 1 to 100 m, which allows us to probe the range from diffusion-dominated density structures to structures determined by chemical processes.
Figure 3 shows the resulting electron and ion densities. If diffusive equilibrium were to hold, equations (41) and (42) predict that the electron and ion density perturbations should have the same magnitude but opposite sign, which is a good approximation for σa ≲ 3 m. As the width of the disturbance increases, the electron depletion becomes deeper and the ion enhancement smaller, with the σa = 100 m densities being essentially equal to their chemical equilibrium values, given by (43) and (44). From (27) we note that the ion (and electron) diffusion timescale increases as σa2; hence we expect a rapid transition from a diffusion-dominated density structure to a chemical equilibrium structure as σa is increased, with σa = 10 m being an intermediate case in which the diffusive and recombination timescales, td,i and tr, are comparable.
 Note also that for the smallest structure, σa = 1 m, aerosol diffusion is important. From (33), td,a has decreased from 7 × 103 s in the reference case to td,a ≈ 800 s for σa = 1 m, which is comparable to our chosen integration time. Hence the structure seen in Figure 3 for σa = 1 m has widened considerably compared to the imposed structure at t = 0. This effect accounts for the small amplitude of the electron and ion density perturbations in this case. It also illustrates that, although very small structures are desirable in order to produce radar echoes, the lifetime of the structure decreases rapidly as the width of the structure is reduced; from (33), td,a ∝ σa2.
 The presence of steep electron density gradients, leading to appreciable changes in electron density over a distance comparable to the Bragg scale, is necessary in order to produce PMSE. However, Hoppe et al.  demonstrate that PMSE are likely caused by a large number of such structures, each of them similar to the smaller-scale structures we simulate here.
4.2. Amplitude of the Disturbance
 In this case we keep the (initial) width of the disturbance fixed at the reference value, σa = 3 m, but increase the amplitude, na1, from the reference value (6 × 108 m−3) up to a maximum na1 = 2 × 1010 m−3. Thus we go from the “linear” case in which the aerosol particle perturbation, and hence the resulting electron depletion, is much smaller than the ambient plasma density, to the “extreme” limit in which the perturbation is much larger than the ambient plasma density.
Figure 4 shows the electron and ion densities. As expected, the electron and ion perturbations increase in response to an increasing aerosol particle perturbation. As long as the perturbations are small, the perturbations in ne and ni should be directly proportional to na1, in accordance with equations (41) and (42) (the diffusive equilibrium result), which also predict that the perturbations should have the same magnitude, but the opposite sign. We note that even for the na1 = 6 × 109 m−3 case, in which the particle perturbation equals the ambient (unperturbed) plasma density, this is not a bad approximation.
 The na1 = 6 × 109 m−3 case also illustrates that even with a particle perturbation sufficiently large to produce a nearly complete electron depletion, if electron attachment were the only process taking place, this is not what we get: All the particles have become negatively charged (most of them with a single charge), but the drop in electron density is only about half of the particle charge density nac. As explained in section 3, electrons and positive ions flow into the perturbed region by ambipolar diffusion, and hence reduce the magnitude of the electron perturbation.
 This is even more apparent in the na1 = 2 × 1010 m−3 case, where we get a fairly deep, but also wide, electron depletion. Figure 5 shows the densities of all six species that we solve for, for this case. Comparing the upper, left panel with the left panel of Figure 2, we note that the density profile now departs somewhat more from the diffusive equilibrium value, being deeper and wider than predicted by (38). However, assuming diffusive equilibrium for the ions and electrons is still a reasonable approximation, even for such large amplitude perturbations, since the timescale needed to achieve diffusive equilibrium is still of the same order of magnitude as the reference estimate (27).
 The lower panels of Figure 5 show that not only the absolute densities of the various aerosol particle charge states are very different (as anticipated from the discussion in section 3), but also that the shape of the perturbation is rather different: The negatively charged particles have a markedly wider distribution, in particular the doubly charged particles which also display a double-humped distribution. These differences are for the most part not a result of diffusion (since we have not run the model long enough for particle diffusion to become important), but are rather a result of the reactions taking place. The double-humped distribution of doubly charged particles is a consequence of the severe electron depletion: The rate of production of na,−2 is proportional to nena,−1, so the production rate is actually reduced near the center of the aerosol particle perturbation. Note also that the electron distribution is significantly wider than the distribution of the dominant charge state, na,−1. This implies that even if we tried to create large electron density gradients by imposing very large aerosol particle density gradients, the electron gradient would be smaller than the imposed particle perturbation because of (ambipolar) inflow of electrons from the surroundings.
4.3. Background Aerosol Particle Density na0
 Instead of varying the amplitude of the disturbance, we now vary the “background” aerosol particle density na0 (which was essentially zero in the reference case). Figure 6 shows the electron and ion densities at the end of the run. In the na0 = 2 × 109 m−3 case, we get results that are very similar to the reference case; the electron density outside the perturbed region has decreased significantly, to approach the new chemical equilibrium densities in the presence of aerosol particles. However, the change in density caused by the perturbation is almost the same as in the reference case, both for electrons and ions. For the na0 = 6 × 109 m−3 case, when the background aerosol particle density is comparable to the equilibrium plasma density without aerosol particles, the ion distribution is still close to the reference case, while now not only has the overall electron density decreased, but also the difference between the electron density outside and inside the aerosol particle perturbation has decreased by roughly a factor 6 relative to the reference case. This is in qualitative agreement with the discussion in section 3, where the diffusive equilibrium approximate results (39) and (40) predict that the perturbation in ne should be reduced while the perturbation in ni should remain constant as the background aerosol density increases.
 In the na0 = 2 × 1010 m−3 case, the high background aerosol particle density leads to complete electron bite-out, so that not only is the overall electron density extremely low, but the dip introduced by the disturbance is even less. Such a high aerosol particle density will not cause PMSE of course, mainly because there are (essentially) no free electrons to scatter the electromagnetic wave.
4.4. Aerosol Particle Size
 Next we vary the size and mass (assuming constant aerosol particle mass density ρa ≈ 900 kg m−3) of the aerosol particles, from ra = 1 nm to 100 nm.
 In order to accommodate the higher charge states of large particles, the maximum negative charge state included in the model is increased to qm = 5 for the case with ra = 30 nm, and to qm = 7 for ra = 50 and 100 nm. For ra = 30 nm most particles are in the q = −2 and q = −3 charge states; for ra = 50 nm in the q = −3 and q = −4 states; and for ra = 100 nm q = −4, −5, and −6 are the dominant charge states.
Figure 7 shows the electron and positive ion densities after 103 s integration time. For the heavier particles, the altitude grid had to be extended downward since these particles have a much higher sedimentation speed.
 For the smallest, ra = 1 nm, aerosol particles, the structure has almost vanished after 103 s, and only a small-amplitude, wide perturbation remains. From (2) and (24), the aerosol particle diffusion timescale td,a ∝ ra2. Hence going from ra = 10 nm to 1 nm, td,a is reduced from approximately 104 s in the reference case to less than 100 s for 1-nm particles, which explains why the structure has vanished. From (22) and (33) the attachment timescale ta ∝ ra−3/2 for small particles. The 1-nm particles therefore have an attachment timescale ta ≈ 45 s, in other words comparable to the particle diffusion timescale, but longer than the ion diffusion timescale td,i which in this case is the shortest timescale of the system. The charging of the aerosol particles is therefore slow for the smallest particles, and the production of doubly negative particles can be entirely neglected.
 For the 3-nm particles the particle diffusion timescale is comparable to the integration time, so the structure has not had time to vanish by the time the integration is stopped. However, the structure has significantly widened, and the amplitude of the perturbation is reduced.
 As the aerosol particle size is increased further, particle diffusion ceases to be important (at least for the 103 s we consider), and their motion is dominated by gravity. From (16) and (22), neglecting the pressure gradient and electric forces, the fall speed ua ∝ ra−1, so 100-nm particles fall 10 times as far as the 10-nm particles of the reference case.
 From Figure 7 we note that the electron depletion becomes deeper and more asymmetric as the aerosol particle size increases. At the same time the ion density becomes less positive, forming a “W”-shaped distribution, and eventually, for ra ≳ 30-nm particles, forming a density depletion. In other words, while for small particles we have an anticorrelation between electrons and ions, mainly as a result of ambipolar diffusion as discussed above, for heavy particles the anticorrelation becomes less pronounced, and for really heavy particles we end up with a correlation.
 What is the cause of this change from anticorrelation to correlation? To understand this, let us look at the timescale for the most “extreme” case, with ra = 100 nm. The rate of attachment onto neutral and singly charged particles, ψe,0 and ψe,−1, are in this case comparable, with timescales ta ∼ ta2 ≈ 0.05 s. Even the timescale for electron attachment onto the dominant charge state, q = −5, ta5 ≡ 1/(ψe,−4ne) ≈ 0.8 s. Hence all particles become multiply charged almost instantly. The gravitational timescale, that is, the time needed to fall a distance equal to the width σa of the perturbation, is, from (30), tg ≈ 13 s. This time is significantly shorter than the ion (and electron) diffusion timescale td,i ≈ 25 s. Hence the particles are falling too fast for the electrons and ions to be able to establish diffusive equilibrium. It should therefore be no surprise that the anticorrelation resulting from ambipolar diffusion is no longer seen. The cause of the ion depletion is simply attachment of ions onto the aerosol particles. The rate coefficient for ion attachment onto q = −5 particles, ψi,−5 ≈ 5.2 × 10−11 m3 s−1 for 100-nm particles, which is more than 20 times the ion attachment rates in the reference case. The corresponding ion attachment timescale is then ta5,i ≡ 1/(ni ψi,−5) ≈ 3 s. Thus the ion attachment timescale is even shorter than the gravitational timescale tg, which means that a significant number of ions will have time to attach themselves onto the aerosol particles in the time the particles need to pass through a given volume. At the same time there is a decrease in the ion-electron recombination rate, because of the decrease in electron density around the perturbation, which should favor an increase in ion density. However, as discussed in section 3, the corresponding timescale is tr ≈ 170 s, and hence insignificant by comparison. Since the ion attachment timescale is significantly shorter than the diffusion timescale td,i, the recombination timescale tr, and the gravitational timescale tg, this explains why in this case we get a correlation between ne and ni.
 When the heavy, ∼100 nm, negatively charged particles fall into a previously undisturbed plasma, the electric field which accompanies the disturbance will rapidly push electrons upwards to maintain charge neutrality, causing the drop in ne. Ambient positive ions then attach themselves onto aerosol particles, causing the decrease in ion density. When the aerosol particle structure has passed this volume, electrons stream back into the volume to ensure neutrality. But because there is now a reduction in the ion density, there must also be a reduction in ne. Eventually, electron and ion densities will increase due to ionization back to their equilibrium values ne = ni = n0. However, the timescale for this process is again given by the recombination timescale, tr ≈ 170 s. Hence this final readjustment is much slower than the gravitational and ion attachment processes that formed the density depletions in the first place. This explains the asymmetry seen, e.g., in ne for the 100-nm case; the trailing electron (and ion) depletion is a consequence of the slow recombination process. Note also that the slow electron-ion recombination rate makes the electron and ion density perturbations very wide, and much wider than the aerosol particle structure that causes the perturbations (for the heavy aerosol particles diffusion is negligible so that they maintain the narrow (∼3 m) distribution during the course of the model run).
 For the heavier particles the electron depletion is actually larger than the imposed particle perturbation. For instance, for the ra = 100 nm case we get a drop in ne of about 3.7 × 109 m−3, while the initial particle perturbation na1 = 6 × 108 m−3. In other words, the drop in electron density is much larger than the corresponding increase in aerosol charge, nac. This is mostly a consequence of the multiple charging of aerosol particles. In addition, ion attachment contributes by neutralizing aerosol particles, thus making them available for additional electron attachment, while without ion attachment electron attachment quickly stops when the particles have become sufficiently negatively charged to repel electrons.
 Finally, we shall consider the effect of changing the altitude at which the aerosol particle density perturbation occurs. Instead of just varying one of the input parameters to the model, we shall now vary several, in order to simulate the change in ion composition with altitude. For the reference case we chose parameters appropriate to an altitude of approx. 85 km. We shall here repeat the calculation, choosing 82 and 88 km altitude, which are near the approximate lower and upper limits of where PMSE have been observed. The main effect of varying the altitude will be the change in ion composition: At 82 km we shall assume that the positive ions are (H3O)+(H2O)3 cluster ions, with mi = 73 mu and an ion-electron recombination rate α = 7 × 10−12 m3 s−1, while at 88 km we assume the positive ion component to be NO+ with a mass mi = 30 mu and a recombination rate α = 6 × 10−13 m3 s−1, where the rates have been taken from the compilation in Table 1 of Rapp and Lübken . For the neutral density we choose nn ≈ 4.2 × 1020 m−3 at 82 km and nn ≈ 1.1 × 1020 m−3 at 88 km [Lübken, 1999], while for the ambient plasma density (that is, with no aerosol particles present) we choose n0 = 3 × 109 m−3 at 82 km and n0 = 1010 m−3 at 88 km [Blix, 1999]. As in the reference case these requirements fix the ionization rate Q = αn02, which is then approximately 6 × 107 m−3 s−1 at both 82 and 88 km. Apart from these changes, all parameters are identical to the reference case. Particularly, the aerosol density perturbation is identical to the reference case.
Figure 8 shows the resulting densities after the model has been run for the usual 103 s in these cases. Since not only the absolute distance is different in the different cases, but the aerosol particles will also fall at different speeds, we have simply juxtaposed the densities so that their maxima are located at the same place. Moreover, because the ambient density n0 is so different, we have subtracted n0 from ne and ni in order to facilitate the comparison.
Figure 8 shows that, crudely, the results are not very sensitive to the altitude or ion composition: We get the same anticorrelation between electrons and ions, and similar charging of the aerosol particles. Perhaps the most noticeable difference is the reduction in the ion density perturbation at 88 km, and the rather large depletion in ion (and hence also electron) density outside the aerosol layer at this altitude. This is a consequence of ion attachment onto aerosols: The lighter NO+ ions have a higher attachment rate and the background plasma density n0 is higher than in the reference case, together causing a reduction in the ion attachment timescale ta,i from 170 s in the reference case to ∼80 s at 88 km. This value is comparable to the gravitational timescale tg (which is also only half of its reference case value due to the lower neutral density). Hence the ions have time to attach themselves onto the aerosols, causing the reduction in ni. At 82 km the ion attachment rate is much lower, while the electron-ion recombination rate α is much higher. At this altitude ion (ambipolar) diffusion and electron-ion recombination therefore dominate, and ne and ni are close to both their diffusive and chemical equilibrium values. (As we remarked in section 3, when ψi,−1/α ≪ 1 and the aerosol density is much lower than the ambient plasma density, the diffusive equilibrium and chemical equilibrium densities are the same.)
 With the exception of very large aerosol particles, with radii on the order of 30–100 nm or larger, all our model results point to an anticorrelation between electron and positive ion densities. For the small-scale structures that are most relevant to PMSE, this anticorrelation is a result of ambipolar diffusion. The same anticorrelation in the small-scale structures is also found, almost invariably, in rocket measurements. An example is shown in Figure 9. (The ion probe used on this rocket has been described by Blix et al. .) Here the relative fluctuations, and , are shown, where and are the mean electron and ion densities averaged over a wider distance. The figure shows that the electron and ion densities can be well anticorrelated over many consecutive perturbations, each of which having a spatial extent of only a few meters. This agreement between model and observations lends support to the hypothesis that heavy, charged particles are responsible for the electron and ion density perturbations seen in the rocket data, as well as PMSE radar echoes. Moreover, since the ambipolar diffusion timescale, td,i, is on the order of a few tens of seconds, the aerosol particle perturbations must have been present at least as long, in order for the anticorrelation to have time to be established. This would indicate that the aerosol particle structures that caused the perturbations shown in Figure 9 must have been essentially static structures.
 In all the model calculations we chose to start with an aerosol particle enhancement, i.e., with a positive value for na1. If we had chosen a negative value for na1 instead, we would have obtained essentially the same results, except for the obvious sign reversal in the perturbations. In particular, the electron-ion anticorrelation would persist. In that case the particle depletion would imply that initially more electrons would attach themselves to particles outside the perturbed region than inside. Hence we would initially have an electron density enhancement. The electron pressure gradient would then try to push electrons out of this perturbed region. Again, this would result in a polarization electric field which would force electrons and positive ions to move together, with the result that an equal number of ions and electrons would be forced out of the perturbed region until approximate diffusive equilibrium had been established. Hence we would end up with a reduced electron enhancement together with an ion depletion in that region, so that ne and ni would still be anticorrelated.
 Would the anticorrelation persist even if the particles were positively charged? Again the answer is yes. To obtain a significant number of positively charged aerosol particles (apart from the small number produced by ion attachment) in the model, we would need to include ionization of neutral particles, probably caused by UV-radiation, which has not been considered here. (Presumably, this would correspond to ice particles consisting of “dirty” ice with a lower work function than pure ice.) A particle enhancement in a small region would then initially produce an enhanced electron density in the same region (due to the ionization of the initially neutral particles). The electron pressure gradient would then again try to push electrons out of this region, and the resulting electric field would push the same number of ions out, leading to anticorrelation between ne and ni.
 Only for very large aerosol particles, with ra ≳ 30 nm, have we been able to produce a correlation between electron and ion densities at small length scales. (In the absence of any particles, there must of course be a correlation, simply from charge neutrality.) In that case it was due to the rapid ion attachment onto such large particles. However, one would expect that the number density of such large particles is low, so that in most cases the aerosol particle charges will be carried by smaller particles, in which case ne and ni are still anticorrelated. Moreover, because of the high fall speed of such large particles (about 25 cm/s for 100-nm particles) their lifetime could be small since they may quickly fall to an altitude where they evaporate. On the other hand, a strong neutral updraft can significantly increase their lifetime (for 50-nm particles an updraft w ≈ 12 cm s−1 will keep them suspended).
 This difficulty creating a correlation between ne and ni is consistent with the available observational data, since the rocket measurements almost without exception find that ne and ni are anticorrelated on small length scales. However, in a couple of instances such a correlation has been found. Preliminary data from one such rocket flight, made near 85 km altitude, is shown in Figure 10. In this case the perturbations in ne and ni are well correlated at length scales down to a few meters. (Because these data have not been fully analyzed yet, the amplitudes of the perturbations in ne and ni remain to be determined and should therefore not be inferred from the figure.) These few instances have always been found near the bottom of the aerosol particle layer. Is it conceivable that aerosol particle density perturbations can be responsible for these structures, too? We have already seen that heavy particles can create a correlation, and this is indeed one possible explanation. The fact that correlations are only observed near the bottom of the particle layer supports this conjecture: Because of gravitational settling, one would expect that the largest particles are to be found near the bottom of the layer.
 However, near the bottom of the layer one may also expect evaporation of ice particles, which we have so far neglected, to become important. Assuming that evaporation of negatively charged particles will eventually, when the particles evaporate completely, lead to the release of free electrons into the plasma, this could create a positive correlation: Before evaporation, the positive ion density will be correlated with the particle density (because ambipolar diffusion has pulled ions into the perturbed region). The larger aerosol number density in this region than in the surroundings will then imply that more electrons are released inside this region than outside. If the release is rapid enough, not only would this increase in electron density be enough to fill in the electron depletion that existed before evaporation, but it could also create an enhancement. The process must be rapid, however; if it is too slow, ambipolar diffusion will simply cause a new diffusive equilibrium to be established with a lower particle density and hence lower electron and ion density perturbations, but with the anticorrelation intact. For the 3 m structure that we considered in the reference case, we estimated the ambipolar diffusion timescale to be td,i ≈ 25 s. The timescale of evaporation must be shorter than td,i for this process to produce a correlation.
 In order to illustrate the effect of rapid evaporation of charged particles we have modeled the process in a highly simplified manner, by simply assuming that all the aerosol particles have a specified decay rate, la. The corresponding loss rate for the negatively charged particles will then be lana,−1, which is added to the other loss terms in (11) (similarly for the other charge states). As the (negatively) charged particles disappear, their charge is released as free electrons. We mimic evaporation by starting the model from the reference solution at t = 103 s, shown in Figure 1, suddenly switching on the evaporation term. The model is then run for t = 3/la, that is, for three times the aerosol particle evaporation lifetime, which is sufficient to cause almost complete evaporation. Figure 11 shows the resulting electron and ion densities for various choices for la. For slow evaporation, la = 0.01 s−1, we are not able to produce a correlation: The anticorrelation then persists at all times, while the whole electron-ion density perturbation slowly vanishes due to diffusion. Increasing la to 0.1 s−1, we do get a correlation between ne and ni, although the amplitudes of the density perturbations are much smaller than in the reference case we started from (shown in the upper left panel). In this case, diffusion has still had time to smear out the structure somewhat. For even more rapid evaporation, la ≥ 1 s−1, the correlation is very pronounced. Hence this numerical simulation just confirms our simple estimate above: The aerosol particle lifetime due to evaporation must be shorter than the ambipolar diffusion timescale in order to create a correlation between ne and ni.
 This simple modeling of evaporation has only demonstrated what evaporation rates are required in order to produce a correlation. However, we have not shown that such rates are realistic, or even attainable. To do so, one would need to know not only the rate at which an individual particle evaporates (which can be obtained from, e.g., the formulas of Hesstvedt , provided that the temperature and humidity is known), but also the size distribution of aerosol particles. A complete study of the role of evaporation is therefore outside the scope of this paper.
 Although evaporation can produce a correlation, the correlation will not last long: Once the particles have evaporated, “ordinary” ambipolar diffusion will quickly, on a timescale td,i ∼ 1/2 min, cause the perturbation to vanish. In order to see this correlation, a rocket would have to pass through the volume after evaporation is almost complete but before diffusion has removed the structure.
 To conclude, we find that both heavy aerosol particles and rapid evaporation of particles can explain a possible positive correlation between electron and ion densities. However, both processes require rather “extreme” conditions: The heavy particles must be so large that ion attachment becomes more rapid than ion diffusion, or the evaporation must be more rapid than ion diffusion.
 With the exception of the heavy aerosol particle cases, electrons and ions will be in near diffusive equilibrium around small-scale aerosol particle structures, with the electron density to a good approximation given by (38). Once (approximate) diffusive equilibrium has been established, this electron density structure will remain “glued” to the particle density structure, and will persist as long as the particle structure persists. In other words, the lifetime of the particle structure, which will be limited by diffusion or gravitational settling and subsequent evaporation, is also the lifetime of the accompanying electron density structure. In that sense, we may say that the diffusivity of electrons is (drastically) reduced in the presence of aerosol particles. However, in the initial ambipolar diffusion of electrons and ions in the model, shortly after electron attachment by particles is complete, the diffusion of electrons is not retarded by the particles. Rather, as emphasized by Hill , electron diffusion in the presence of negatively charged particles is enhanced, while it is retarded in the presence of positively charged particles. For negatively charged particles, the positive ion density must by higher than the electron density (charge neutrality). The zero current requirement then implies that ui = uene/ni < ue, while ui = ue without aerosol particles present. Since the ion speed is proportional to the electric field, a lower ui means a lower E. Since electrons are retarded by the electric field, a weaker field means higher electron speed, in other words enhanced electron diffusion.
 Not only is the electron diffusivity dependent on the sign of the aerosol particle charge, but the problem is not even a pure diffusion problem. This can be illustrated as follows. We assume all aerosol particles carry a charge q (in units of e), and have a prescribed density na. We regard the particles as being at rest, so that the zero current requirement still implies that neue = niui. Moreover, we neglect gravity and all chemical processes (such as attachment and electron-ion recombination). Combining equations (20), (16) for positive ions, and (1) for electrons, we end up with the following equation for ne:
where DA ≡ 2kT/(miνi) is the (ambipolar) diffusivity without aerosol particles present. This equation illustrates that the diffusion coefficient (the term multiplying ∂2ne/∂z2) indeed increases when q is negative, implying enhanced electron diffusivity, and is reduced when particles are charged positively. But (45) also illustrates that in the presence of charged particles, electron (and ion) motion is no longer a pure diffusion process since the equation contains additional terms, proportional to (∂ne/∂z)2 and ∂2na/∂z2.
 Whether the electron diffusion is enhanced or retarded in the presence of particles is not so important, however. The main point is that ambipolar electron and ion diffusion does not act to smear out the perturbations; rather this process will create an ion perturbation where there was none at the outset, and the asymptotic solution for electrons is a density structure (approximately given by (38)) that is attached to the aerosol particle density perturbation and will remain as long as the particle perturbation exists.
 Regarding the possibility of producing significant PMSE from these structures, is there an “optimum” particle size? Clearly, small particles, with radii ∼1 nm or less, are not very attractive candidates: As we have seen, particle diffusion then becomes important and the structures therefore have a very short lifetime. That means that some process must continuously create these structures fairly rapidly to counteract diffusion. In addition, the slow electron attachment rate for small particles implies that creating the required electron disturbance will take a rather long time. The heaviest particles (ra ≳ 50 nm) produce much larger electron depletions, and the lifetime of the structure is not limited by particle diffusion. On the other hand, the lifetime of the heavy particles can be limited by gravitational settling: They fall rapidly and may therefore quickly reach a warmer region where they evaporate. This can be offset by a strong neutral atmosphere updraft, but for particles larger than 30 nm or so the vertical wind speed must be on the order of 10 cm s−1 to have a significant impact on their fall speed, while modeled time-averaged updraft speeds in the summer polar mesosphere are found to be only ∼3 cm s−1 or less [Garcia and Solomon, 1985]. However, at lower altitudes, around 80 km, the neutral density is significantly higher and here an updraft of this magnitude may be sufficient to keep heavy particles suspended. The large particles can produce deep electron depletions; as we have shown, the drop in electron density can be much larger than the increase in aerosol particle density for large particles (whereas for small particles the electron depletion tends to be smaller than the increase in aerosol density), and this will favor large particles as PMSE candidates provided the density of large particles is not too small. A recent study by Rapp et al.  also indicates that large particles may be a major contributor to PMSE. However, most of the in situ rocket measurements of electron and ion densities made during PMSE conditions find an anticorrelation between ne and ni on small length scales, and this is a strong argument in favor of smaller particles (∼10 nm) being the cause of PMSE: From the model results presented here, whether the density of electrons and positive ions are correlated or anticorrelated on small length scales seems to be closely associated with their size. Hence the degree of correlation found in rocket data can be a useful indicator of the particle size. In order to determine the relative importance of large and small particles for PMSE, more in situ measurements of electron and ion densities on short (a few meters) length scales would be helpful.
 This study, in which we have modeled the plasma response to imposed small-scale aerosol particle density perturbations, is consistent with the claim that such heavy particles can cause the small-scale structures measured in situ by rockets as well as remotely by radars as PMSE. We find that the plasma quickly adjusts itself to the presence of aerosol particles. For small particles, with radii up to 10 nm or so, the dominant processes are electron attachment and ambipolar diffusion. Electron attachment, which may be complete in only a few seconds, creates an electron depletion in the presence of an aerosol particle enhancement, in other words an anticorrelation between the electron and aerosol particle densities. Subsequent ambipolar diffusion, which may take place within a minute or less, pulls ions and electrons into the aerosol layer, reducing the electron depletion and at the same time creating an ion enhancement. Hence ambipolar diffusion predicts an anticorrelation between electron and ion densities. This is in agreement with in situ rocket observations, which almost invariably find such an anticorrelation at small length scales. Once ambipolar diffusion is essentially complete, the electrons and ions will remain in near diffusive equilibrium and their small-scale density perturbation will remain as long as the aerosol particle structure persists, which is determined by aerosol particle diffusion and evaporation. The lifetime (but not the amplitude) of the electron and ion density perturbations will be independent of the aerosol particle number density, so that even a small number of aerosol particles can create a long-lived electron density perturbation. The diffusive lifetime of the layers will be on the order of a few hours for structures that are a few meters wide and consisting of ∼10 nm aerosol particles. The polarization electric field is too weak to significantly affect aerosol particle diffusion.
 We find that aerosol particles with a size on the order of 10–30 nm are probably optimal for causing PMSE. Smaller particles will quickly diffuse and therefore remove the structure, unless some vigorous process can maintain the sharp gradients. For larger particles, on the order of 50- to 100-nm size, we find that ion attachment becomes important. This process counteracts the effect of ambipolar diffusion and thus removes the anticorrelation between electron and ion densities, creating a correlation instead for very large particles. Most in situ rocket measurements during PMSE conditions show anticorrelation on small length scales between electrons and positive ions, thus arguing against large particles. However, in order to settle this issue more in situ measurements on small length scales are needed.
 In a few instances rockets have detected a positive correlation between electrons and ion densities, and always near the bottom of the PMSE layer. This can be explained if the particles are very large, as mentioned above. Alternatively, it can also be caused by rapid evaporation of aerosol particles, provided the particles release their charge as free electrons as they evaporate. For this latter process to produce a correlation, the evaporation must be so rapid that it overcomes ambipolar diffusion.
 It remains to explain why thin aerosol layers, or structures, occur in the first place. In this paper we have only shown that their presence modifies the electron and ion density structures in a way that will produce PMSE. As pointed out in the introduction, several conceivable processes can lead to small-scale aerosol particle structures. Once they are formed, the low aerosol particle diffusivity implies that the structures, and the accompanying electron density perturbation, can live for a long time, on the order of hours or more.
 This work was supported in part by the Research Council of Norway under grant 115980/431.