Relationship of electric field and charged particle density fluctuations to air turbulence in the mesosphere

Authors


Abstract

[1] A theoretical model is developed that is applicable to the electric field fluctuations that arise in the polar summer mesosphere as a result of the coupling of the charged species to the neutral air turbulence. The motions of electrons, ions, and charged aerosol particles are described as harmonic oscillators both driven and damped by the drag force exerted by the neutral air. The relative fluctuations in the ion density are found to be nearly the same as those in the neutral air as a consequence of the ions' high-momentum transfer relaxation frequency. The aerosol density fluctuations follow those of the neutral air at frequencies below their relaxation frequency, which is in the acoustic range. The electrons move primarily in response to the electric force to partially cancel the net charge density of ions and aerosol particles except at wavelengths shorter than the Debye length. Electric field and charge density fluctuations are calculated for several sets of conditions. In “bite-out” regions in which the electron density is reduced as a consequence of attachment to the aerosol particles the electric field fluctuations are found to be enhanced, which is consistent with observations.

1. Introduction

[2] Rocket-borne probes are launched into the polar mesosphere most often in the summer when noctilucent clouds (NLC) and polar mesosphere summer radar echoes (PMSE) are observed. Attachment of electrons to the cloud particles reduces the number density of free electrons and creates a new population of charged aerosol particles. Probes have measured fluctuations in the neutral air density [Thrane and Grandal, 1981; Thrane et al., 1985; Blix et al., 1990; Lübken et al., 2002; Müllemann et al., 2003], fluctuations in the density of electrons, ions and charged aerosol particles [Lübken et al., 1993; Blix, 1999; Mitchell and Croskey, 2001; Rapp et al., 2003a; Smiley et al., 2003, 2006], and fluctuations in the electric field [Pfaff et al., 2001; Holzworth et al., 2001; Holzworth and Goldberg, 2004]. Rocket payloads have become increasingly sophisticated and many of these measurements can now be made simultaneously from a single payload [Blix et al., 2003a, Croskey et al., 2004]. PMSE originate from fluctuations in the electron density that may have their origin in air turbulence [Cho and Röttger, 1997; Blix et al., 2003b; Rapp et al., 2003b] that has its origin in the breaking of gravity waves [Fritts and Alexander, 2003]. Fluctuations in the number density of electrons have recently been shown to be coupled to the fluctuations in the density of aerosol particles through the requirement of quasineutrality [Rapp and Lübken, 2003, Lie-Svendsen et al., 2003a, 2003b]. It is likely that the electric field fluctuations observed by rocket-borne probes are also consequence of the coupling of the charged species to the air turbulence.

[3] The goal of this work is to find the expected relationship between the electric field fluctuations, the charge density fluctuations, and the neutral air turbulence. The neutral air velocity fluctuations exert unequal forces on electrons, ions, and charged aerosol particles that lead to charge separation. The resulting electric field also drives charged particle motion and must be found self-consistently through Poisson's equation. A new theoretical approach is developed that treats the fluctuating air velocity as a driver and finds the amplitudes of the forced oscillations in the densities of the charged species and in the electric field. The electrons, ions, and charged aerosols are modeled as harmonic oscillators, damped by collisions, coupled by the self-consistent electric field, and driven by the neutral air turbulence. It may be possible to test the validity of these equations in dusty plasma experiments in the laboratory.

[4] The model is developed in section 2 below, and the parameters of the model are evaluated for typical conditions in the polar summer mesosphere. Reduced sets of equations are derived that apply for specific sets of conditions. In section 3, the fluctuations in the charged particle densities and in the electric field are found using several different assumptions for the frequencies and wavenumbers of the neutral air turbulence and for the charge density of aerosol particles. The predictions of the model are discussed in section 4 and are briefly compared with published data on electric field and charge density fluctuations.

2. Model

2.1. Coupled Equations of Motion

[5] It is assumed that in the absence of turbulence there is a homogeneous mixture of air and tracer species that are electrons, ions,and charged aerosol particles. The tracers are assumed to have such a low density (<1010 m−3) in comparison with the neutral gas (∼1020 m−3) that they do not alter the neutral gas turbulence. The density and velocity fields of the turbulence are decomposed into fluctuations with frequency ω and wavenumber k. The model is a local model that ignores larger-scale phenomena such as the finite extent of the aerosol layers (considered by Lie-Svendsen et al. [2003a, 2003b]) and the mixing of air from altitudes having different tracer concentrations (considered by Thrane and Grandal [1981]).

[6] The velocities of the tracer species are coupled to the air velocity by a drag force that is proportional to the difference between the velocity of the tracer and the velocity of the air. This drag force [Epstein, 1924] is obtained from an average over many collisions; thus the frequency ω being considered must be much lower than the collision frequency. The Fourier-transformed, linearized, one-dimensional fluid momentum equation for tracer species j is

equation image

where nj is the zero-order density of the tracer species j, equation imagej is the fluctuating (first-order) part of the tracer density, mj is the mass of the tracer, equation imagej is the fluctuating part of the tracer fluid velocity, equation image0 is the fluctuating part of the air fluid velocity, κ is Boltzmann's constant, qj is the tracer charge, and equation image is the fluctuating electric field arising from charge separation. The zero-order air fluid velocity is assumed to be zero; that is, there is no wind. The frequency for momentum relaxation between the tracer and the neutral air is νj. All species are assumed to be at the same temperature T. The subscript j refers to electrons, ions, and charged aerosols (e, i, and a, respectively), and the subscript 0 refers to the neutral air. It is assumed that the air, ions, and aerosol particles can be described by an average mass and that the charged aerosols can be described by an average charge value. Each of the species is described by an equation of state with a ratio of specific heats γj. The justification for treating the electrons as unmagnetized is discussed in section 2.3. The other charged species have a collision frequency much greater than their gyration frequency in the magnetic field, which allows magnetic effects to be ignored for these species.

[7] The Fourier-transformed continuity equation is

equation image

where sources and sinks of tracer particles are assumed to cancel and have been omitted. Equation (2) will apply to electrons and ions for fluctuations occurring on a timescale that is shorter than the mean lifetime (∼1000 s) determined by the balance of ionization and recombination [Rapp and Lübken, 2003]. For aerosol particles the charging time is much longer.

[8] The momentum equations for the tracer species can be put into the form of damped harmonic oscillator equations driven by the coupling to the neutral air velocity fluctuations and to the electric field fluctuations:

equation image

where cj = equation image. The electric field is found self-consistently using the Fourier-transformed Poisson's equation:

equation image

The equation of motion for the jth charged species, including the electric force, becomes

equation image

which can be rewritten

equation image

where ωp,j = equation image is the plasma frequency of the jth charged species. The 3 × 3 matrix equation describing the response of the three species may be solved by a matrix inversion to obtain equation imagea, equation imagei and equation imagee in terms of the driver equation image0 given the parameters nj, ωp,j, cj, qj, and νj.

[9] As a simple example of the application of equation (6), consider a driving term that is simply a sound wave with ω = c0k. The neutral air fluid velocity is related to the density fluctuation by

equation image

where ω/k = c0 = equation image is the sound speed and γ0 = 5/3. Equation (6) becomes

equation image

Equation (8) gives the relationship between all the fluctuating densities equation imagej for a driver equation image0 that can be decomposed into sound waves. The fluctuating densities of the tracers may be found in terms of equation image0/n0 and used in Poisson's equation to find the fluctuating electric field as a function of equation image0/n0. A motivation for writing equation (8) is that it could be checked in the laboratory by propagating sound waves into plasma with aerosol particles.

2.2. Evaluation of Parameters

[10] At low pressures, acoustic waves are attenuated for wavelengths comparable to or less than mean free path [Greenspan, 1965]. At NLC altitudes the mean free path is approximately 0.4 cm, thus acoustic wave attenuation is not important for the wavenumbers considered here (k ≤ 25/m). For all species the ratio of specific heats is assumed to be 5/3, which is appropriate if the frequencies of oscillations are sufficiently lower than the collision frequencies for the species to be equilibrated in temperature with the neutral gas. The momentum relaxation frequencies νj and the plasma frequencies ωp,j for the tracer species are evaluated below for each tracer.

2.2.1. Aerosol Neutral Momentum Transfer

[11] The force on a spherical aerosol particle moving relative to the surrounding air was found by Epstein [1924]. For a particle with radius much less than the mean free path, the aerosol velocity has the relaxation frequency

equation image

where M is the aerosol mass, c* = equation image, r is the aerosol radius, and it is assumed that the air molecules are accommodated on the surface and reemitted. For the summertime mesosphere at NLC altitude, n0 = 4 × 1020 m−3 and T = 140 K [Rapp et al., 2001; Lübken, 1999]. The relaxation frequencies for particles of water ice are 8,860/s for r = 1 nm and 886/s for r = 10 nm. Larger particles may exist but only with low density as a consequence of the limited amount of water vapor.

2.2.2. Ion Neutral Momentum Transfer

[12] For collisions of light ions with neutrals, a representative cross section is the one for N+ colliding with N2 neutrals [Phelps, 1991]. The momentum transfer cross section varies approximately inversely with velocity at low energies, and thus the collision frequency is approximately independent of temperature. The rate coefficient σu is approximately 1.3 × 10−15 m3/s, where σ is the momentum transfer cross section and u is the collision speed at 0.1 eV. The momentum relaxation frequency for these ions is then νi = n0σu = 5 × 105/s.

[13] The summer polar mesosphere also contains cluster ions such as H+(H2O)5 which can be the dominant positive species [Kopp et al., 1985; Rapp and Lübken, 2003]. The neutral cluster (H2O)5 has a collision cross section with N2 of approximately 3 × 10−18 m2 [Sternovsky et al., 2001]. The corresponding momentum relaxation frequency is 1.4 × 106/s. It is likely that the corresponding charged cluster ion has a larger cross section as a consequence of the charge. Thus light positive ions and the most abundant positive cluster ions have relaxation frequencies that are significantly above the frequencies that are measured for air turbulence.

2.2.3. Electron Neutral Momentum Transfer

[14] For collisions of electrons with neutrals an appropriate cross section is the electron N2 cross section at 0.01 eV, which is 2 × 10−20 m2 [Engelhardt et al., 1964]. The corresponding momentum transfer relaxation frequency is 5 × 105/s. The rate coefficient is near to that for ion neutral collisions because the smaller cross section is offset by the larger electron velocity.

2.2.4. Plasma Frequencies

[15] The number density of positive ions is typically of order 3000 cm−3 at the mesopause. For light ions with a mass of 30 atomic mass units, the ion plasma frequency is ωp,i = 1.3 × 104/s. This frequency is significantly less than the momentum relaxation frequency; thus ion plasma waves are strongly overdamped, and the precise value of ωp,i is inconsequential for calculating the ion response. This conclusion also holds if the positive charge resides on cluster ions. In calculating the electron plasma frequency, it is assumed that half the negative charge is free electrons and half is negatively charged aerosol particles. The electron plasma frequency is then ωp,e = 2 × 106/s, and electron plasma waves are underdamped. If the negatively charged aerosol particles are 10 nm in radius and have unit charge, the aerosol particle plasma frequency is ωp,a = 33/s. Plasma waves in aerosol particles are also strongly overdamped.

2.3. Reduced Equations for an Acoustic Wave Driver

[16] Insights into the behavior of the tracers when driven by an acoustic wave with ω = c0k can be found without a complete solution to the coupled momentum equations by matrix inversion. Consider the response of the electrons without the self-consistent electric field. Equation (3) becomes

equation image

where De = κT/meνe is the electron diffusivity. The diffusive term has its origin in the ce2k2 term of equation (3). For driving fluctuations with ωc0k, the term γeDek2 can easily be larger than ω because ce2c02. Thus fine-scale fluctuations (k2ω/γeDe) in the electron density, including those at the Bragg scale for radar reflection, should be greatly reduced in the absence of electron coupling to other charged species through the intermediary of the electric field. Observations of PMSE led to the conclusion that the electron density fluctuations were not reduced by diffusion and that the electrons were not decoupled from the charged aerosol particles [Kelley et al., 1987; Kelley and Ulwick, 1988].

[17] Consider electrons again using equation (6) with the self-consistent electric field, and use that νeωωp,e2 and ω2ωp,e2, to obtain

equation image

which can be solved for the fluctuating electron charge density:

equation image

where λD,e = ce/ωp,e is the Debye length for electrons. The denominator on the right hand side of equation (12) expresses the usual result for low frequencies in plasmas: for k2λD,e2 ≪ 1, the electrons move to nearly cancel the space charge of the heavier species. This denominator can be obtained more simply by assuming that the electrons obey the Boltzmann relation. This relation can be satisfied by motion along magnetic field lines. Thus the role of the electrons in the model is not significantly changed by the omission of the magnetic force. Equation (12) is independent of ω and thus is not restricted to the assumption of an acoustic wave as long as the two inequalities involving ω are satisfied.

[18] Poisson's equation and equation (12) may be combined to obtain

equation image

The definition of the Debye length may be used to show that the electric field is

equation image

The electric field may be used to find the relative importance of the drag force and the electric force. For ions, equations (3) and (14) may be combined to obtain

equation image

where we have assumed that k2λD,e2 ≪ 1 and γi = γe. Inspection of the two terms on the right-hand side shows that when the relative fluctuations in the densities of the species are of the same order, the drag force is larger than the electric force for νiω = ≫ ci2k2. This inequality holds for ions because νi ≅ 5 × 105/s is very much greater than acoustic frequencies. Thus the electric force may be ignored for ions when k2λD,e2 ≪ 1, and equation (15) reduces to

equation image

[19] For aerosol particles a momentum equation can be written that is analogous to equation (15) for ions. For these heavier particles, ca2k2νaω because both νa and ω are in the acoustic range, and cac0 as a consequence of the large aerosol particle mass. Thus the electric force may be ignored in the momentum equation for the aerosol particles. It also follows that ca2k2ω2, and the momentum equation reduces to

equation image

The aerosol density fluctuations follow the fluctuations in neutral air density only for frequencies much below νa. At much higher frequencies the response of the aerosols is reduced by the factor νa/ω. An analogous equation would apply to aerosol particles that were positively charged.

[20] Equations (13), (16), and (17) may be combined to obtain the electric field that would be driven by an acoustic wave:

equation image

The corresponding potential fluctuations are

equation image

The first right-hand side of equation (18) shows that for k2λD,e2 ≫ 1, electron Debye shielding is ineffective and the electric field is generated by the “bare” charge of the ions and aerosol particles. For k2λD,e2 ≪ 1, electron Debye shielding of ion and aerosol particle charge is nearly complete and both the electric field and the potential are reduced. The electric field can be larger in a bite-out region because the electron Debye-shielding is reduced as a consequence of the increased the Debye length.

[21] Equations (16) and (17) have demonstrated consequences for the passage of a shock wave created by a rocket payload. A shock wave with a thickness of order 0.1 m traveling at rocket speed (∼500 m/s) will cause shock compression of positive ions with equation imagei/niequation image0/n0, because the ion momentum relaxation frequency, 5 × 105/s, is larger than the inverse of the timescale for the passage of the shock (5 × 103/s). For 10 nm aerosols the momentum relaxation frequency, 886/s, is smaller than the frequency associated with the passage of the shock. Thus equation imagea/naequation imageequation image0/n0, and the aerosol particles have much less compression than the ions. In bite-out regions where the negative charge resides largely upon aerosols, the compression of positive ions without a corresponding compression of the negative aerosols results in an excess density of positive charge in the wake of the shock, which has been seen by rocket-borne electric potential probes [Sternovsky et al., 2004]. Equation (18) suggests that it is not necessary for electrons to be entirely absent for an electric strong electric field to appear in the shock wave, but that it is only necessary for the electron Debye length to be greater than the thickness of the shock wave.

2.4. Reduced Equations for Bite-Out Conditions

[22] In bite-outs the condition k2λD,e2equation image 1 may not hold because of the reduced electron density, and thus the electric field may not be reduced by electron Debye shielding. In this case both the electric force and the drag force can act on ions. The ion response, from equation (6), becomes

equation image

where equation (17) has been used for equation imagea. The electric field and potential fluctuations can be found from equation (13) using equations (17) and (20).

2.5. Reduced Equations for “Fossil” Turbulence

[23] Inhomogeneities in the aerosol density may persist after active turbulence has ceased. In this case the aerosol density fluctuations equation imagea may be characterized by wavenumbers k and frequencies ω that do not have the acoustic relationship, ω = c0k. Fluctuations of this kind have been called “fossil turbulence” [Cho et al., 1996; Rapp and Lübken, 2003]. This turbulence is assumed here to be characterized by ωc0k. In the limit of zero frequency the momentum equation, equation (1), can be integrated once to obtain the Boltzmann relation, equation imagej = nj exp(−qjequation imageT) for electrons and ions [Lie-Svendsen et al., 2003a, 2003b]. The Boltzmann relation is the equilibrium toward which the ions diffuse, thus it applies for νiωcik2, or equivalently ωk2Di, where Di is the ion diffusivity. The corresponding inequality for electrons is satisfied more easily.

[24] If equation imagea is treated as the given quantity rather than equation image0, the Boltzmann relations for both electrons and ions and Poisson's equation can be combined to obtain

equation image

where

equation image

is the combined Debye length for the electrons and ions. For k2λD,e2 ≪ 1 and k2λD,i2 ≪ 1, the fluctuations satisfy

equation image

for qiequation image/T ≪ 1 and qi = qe. The electron and ion fluctuations are 180 degrees out-of-phase because the signs of the charges are different. This phase relationship is often seen in data from rocket-borne probes. Rapp and Lübken [2003] and Lie-Svendsen et al. [2003a, 2003b] have shown that this relationship is also consequence of ambipolar diffusion.

[25] For fossil turbulence the charge density of the aerosol particles is Debye-shielded by both electrons and ions. The electric field becomes

equation image

where qe = qi has been assumed. This result can be applied to fossil fluctuations in a bite-out in electron density by setting ne = 0. Equation (24) in which the aerosol particle charge is shielded by electrons and ions may be compared with equation (14) in which both the ion and aerosol particle charge is shielded by the electrons. Equation (14) is used when νiωcik2 is not satisfied. A difference is that the ratio of specific heats appears in equation (14) but not in equation (24). This difference arises because the Boltzmann relation used for equation (24) is an isothermal relation and an adiabatic equation of state is used in the derivation of equation (14).

3. Solutions for Typical Cases

[26] Solutions to the coupled momentum equations are found below for five cases of interest. Case A is the mesosphere without charged aerosol particles, case B is the mesosphere with 50% of the negative charge on the aerosol particles, case C is the mesosphere with 90% of the charge on the aerosol particles, and case D is the mesosphere within a bite-out in which all of the negative charge is on aerosol particles. Case E is for fossil turbulence having ωc0k.

3.1. Fluctuations Without Aerosol Particles

[27] In the absence of aerosol particles, equation imagee driven by an acoustic wave is given by equation (12) with naqa = 0, and equation imagei is given by equation (16). These fluctuating densities are plotted in Figure 1 as a function of the wavenumber k and of the frequency f(k) = c0k/2π for an acoustic wave with equation image0/n0. The assumed electron and ion densities are 3000 cm−3. The range of k is chosen so that f(k) spans 0 to 1000 Hz. The relative fluctuations in the tracers are divided by the relative fluctuations in the neutral gas to obtain the response factor Nj, defined as

equation image

If this quantity is unity, a 1% fluctuation in the neutral air density causes a 1% fluctuation in the tracer density. The response factor is a complex number and the absolute value is used in plots.

Figure 1.

Ion and electron density response factors Ni and Ne for a mesosphere without aerosol particles; the electric field fluctuations equation image; and the potential fluctuations equation image as a function of frequency f or wavenumber k for an acoustic wave with equation image0/n0 = 0.01.

[28] Also plotted in Figure 1 are the fluctuations in the electric field and in the potential, from equations (18) and (19), respectively, with naqa = 0. The fluctuations in potential are

equation image

where it is assumed that ni = ne and qi = qe. Equation (26) shows that a 1% fluctuation in air density drives a potential fluctuation with amplitude that is 1% of κT/qi in the limit k2λD,e2 ≪ 1. The Debye length is approximately 2 cm if the electron density is 3000 cm−3, thus D,e = 1 corresponds to k ≈ 300/m. Measurements often extend to about 4000 Hz in the rocket frame, which for a rocket speed of 1000 m/s corresponds to k of approximately 25/m [Pfaff et al., 2001]. Thus k2λD,e2 is small for rocket data except possibly in a bite-out where the electron density is greatly reduced. The temperature of the mesosphere is approximately 0.012 eV; thus a 1% fluctuation in air density results in a potential fluctuation of order 0.2 mV. The noise level in rocket-borne instruments for potential fluctuations is typically only a little less than 1 mV; thus potential fluctuations in the mesosphere are likely to be below the level for detection in the absence of aerosol particles for equation image0/n0 = 0.01. The solutions obtained to the full set of coupled momentum equations (equation (8)) in the limit na → 0 do not differ significantly from the solutions to the reduced equations ((12), and (16)(19)).

3.2. Fluctuations With 50% of the Negative Charge on Aerosol Particles

[29] Case B is for a PMSE region with 50% of the negative charge being singly charged aerosol particles and 50% free electrons. Fluctuating quantities are found from the coupled momentum equations, (equations (8) for j = i, a, and e) by matrix inversion. As in case A, the response is found for a driver that is a monochromatic sound wave with ω = c0k. The driven quantities Ne, Na, equation image and equation image for frequencies and wavenumbers spanning 0 to 1000 Hz are shown in Figure 2 for aerosol particles with radii of 1 and 10 nm. The ion density is 3000 cm−3, the electron density is 1500 cm−3, and the negative aerosol particle density is 1500 cm−3. The relative fluctuations in the ion density (not shown) are nearly the same as those of the neutral air for all wavenumbers in agreement with the approximation equation (16). The momentum relaxation frequency for the 10 nm aerosol particles is 886 s−1; thus the aerosol response factor Na falls with increasing frequency. For the 1 nm particles the momentum relaxation frequency is 8860 s−1, and the response factor for these particles remains near unity.

Figure 2.

Ion electron and aerosol particle density response factors Ni, Ne, and Na for a mesosphere with 50% of the negative charge on aerosol particles; the electric field fluctuations equation image; and the potential fluctuations equation image as a function of frequency f or wavenumber k for an acoustic wave with equation image0/n0 = 0.01. The solid line is for 1 nm particles, and the dashed line is for 10 nm particles.

[30] The electron response factor exceeds unity for 10 nm aerosol particles, which can be explained as follows: The self-consistent electric field in the model acts to enforce quasineutrality. The relative fluctuations in the electron density are those needed for the sum of the charge densities to be zero. Because only half the negative charge resides on the electrons, the electrons must be compressed to a greater degree than the positive ions for the sum of the charge densities to be zero. The relative fluctuations in the electrons would approach 2.0 asymptotically for 10 nm particles at large wavenumbers if quasineutrality continued to hold. However, for large k the inequality k2λD2 ≪ 1 does not hold, theelectrons do not enforce quasineutrality, and the electron compression factor approaches zero rather than 2.0.

[31] The electric field is found through Poisson's equation, equation (5), and the sum of the charge densities. The electric field is larger in case B than in case A. This behavior follows from the Boltzmann relation for the electrons. The smaller ne in case B results in the relative fluctuations equation imagee/ne being larger, which implies that qeequation image/T is larger.

3.3. Fluctuations With 90% of the Negative Charge on Aerosol Particles

[32] The driven quantities Ne, Na, equation image and equation image with 90% of the negative charge on the aerosol particles are shown in Figure 3. The solutions plotted were obtained by matrix inversion and the solutions to the approximate equations ((16)(19)) are not significantly different. The reduced Debye shielding by the diminished electron density results in larger electric field and potential fluctuations. The relative fluctuations equation imagee/ne are increased because the denominator ne is much smaller. The equations that are solved are linearized equations which assume small perturbations in the zero-order densities. In the case of small electron density the number of electrons may be insufficient to approximately cancel the fluctuations in the other charge densities.

Figure 3.

Ion electron and aerosol particle density response factors Ni, Ne, and Na for a mesosphere with 90% of the negative charge on aerosol particles; the electric field fluctuations equation image; and the potential fluctuations equation image as a function of frequency f or wavenumber k for an acoustic wave with equation image0/n0 = 0.01. The solid line is for 1 nm particles, and the dashed line is for 10 nm particles.

3.4. Fluctuations in a Full Bite-Out

[33] The driven quantities Na, Ni, equation image and equation image are plotted in Figure 4 for a bite-out in which there are no free electrons. The solutions are obtained from the full set of equations by matrix inversion, with the electron density made sufficiently small for the electron plasma frequency to be much smaller than all other frequencies. The same results are obtained by inverting the 2 × 2 submatrix that omits terms relating to electrons. The solutions for 10 nm aerosol particles show a small dip in the ion response at small wavenumbers as a result of the ions responding to the electric field. This dip is not seen in the solutions obtained from the reduced equations ((16), (18), and (19)). The reduced equations, however, reproduce the full solution for the case of 1 nm particles. At the smallest wavenumbers and frequencies (ωνa), the acoustic wave causes approximately equal compression of positive ions and negative aerosol particles, the fluctuations in the net charge density are reduced, and the induced electric field is reduced. The electric field is larger for the 10 nm particles than for the 1 nm particles because the difference between the response factors for ions and for aerosol particles is greater for 10 nm particles. The field and potential fluctuations are larger than in any case with electrons because there is no electron Debye shielding of the charges of the other species.

Figure 4.

Ion and aerosol particle density response factors Ni and Na, for a mesosphere with all of the negative charge on aerosol particles; the electric field fluctuations equation image; and the potential fluctuations equation image as a function of frequency f or wavenumber k for an acoustic wave with equation image0/n0 = 0.01. The solid line is for 1 nm particles, and the dashed line is for 10 nm particles.

3.5. Fluctuations in Fossil Turbulence

[34] For fossil turbulence it is assumed that the frequency is sufficiently low for the electrons and ions to move so as to shield the charge fluctuations in the aerosol particles. The electric field and potential fluctuations from equation (24) are plotted in Figure 5 for equation imagea/(ne + ni) = 0.01. The amplitudes of the fluctuations are similar to those in case B; however, the electron and ion density fluctuations are out of phase for case E because the ion density fluctuations are driven by equation image rather than equation image0.

Figure 5.

Electric field fluctuations equation image and the potential fluctuations equation image for a mesosphere with fossil turbulence as a function of wavenumber k for an acoustic wave with equation image0/n0 = 0.01. The aerosol density fluctuations are equation imagea/(ne + ni) = 0.01.

4. Summary and Discussion

[35] Air turbulence in the mesosphere will cause charge imbalance in the mesosphere as a result of unequal accelerations of electrons, ions, and charged aerosol particles by the turbulent fluctuations in air velocity. Equations of motion for these three species have been solved simultaneously with Poisson's equation to find the fluctuations that are induced in the charged particle densities, in the electric field, and in the electric potential. The equations have been applied to the simple case of a monochromatic acoustic wave. Except in bite-outs where electrons are absent, the electron response is determined primarily by the electric field. The electrons move to cancel the net charge density of the ions and aerosol particles for long-wavelength fluctuations with k2λD2 ≪ 1, for which electron Debye shielding is effective. Data are usually collected only for the longer wavelengths satisfying this condition. As a consequence of the electric field being reduced by electrons, the electric force is usually negligible for the ions and their response is determined primarily by the drag force exerted by the neutral air. In this case the ions are compressed to the same degree as the neutral air: equation imagei/ni = equation image0/n0. The electric force may also be ignored in finding the response of the aerosol particles, and the aerosol particle motion is determined by their inertia and by the drag force. The aerosol particles are compressed to the same degree as the neutrals for frequencies below the momentum transfer collision frequency. The corresponding fluctuations in potential are of order (κT/qi)(equation image0/n0) for k2λD2 ≪ 1.

[36] In conditions approaching a bite-out with 90% of the negative charge on aerosol particles, the electric field is increased because the electrons are less effective at canceling the net charge of the aerosol particles and ions. In full bite-out conditions the electric field and potential fluctuations are the greatest. In this case the ions respond both to the electric field and to the drag force, and the simplified expression equation imagei/ni = equation image0/n0 is no longer applicable.

[37] Equation (13) shows that a condition causing reduction of electric field fluctuations is k2λD,e2 ≪ 1. The absence of this condition in a bite-out (because of the longer Debye length) may explain the enhanced electric field fluctuations observed in a bite-out [Pfaff et al., 2001]. The data show that fluctuations (k ≤ 25/m) in the electric field are absent (below 1 mV/m) except in the bite-out region. This is consistent with the results presented here in case D which show that equation image can be of order 100 mV/m in a bite-out, but in the other cases studied here equation image is below 20 mV/m. Pfaff et al. [2001] note that the spectrum of fluctuations in equation image in a bite-out is similar to that in the neutral density fluctuations. This observation supports electric field fluctuations being driven by the neutral air turbulence. The plots of equation image(k) for equation image0/n0 = 0.01 would be flat if there were a “one-to-one” correspondence between the wavenumber spectra of equation image and of equation image0. In Figure 4 for a bite-out, equation image(k) for the assumed flat spectrum of equation image0 falls by less than an order of magnitude for k from 5/m to 25/m. The spectrum of fluctuations in equation image0 from rocket data falls by about 5 orders of magnitude; thus spectra of equation image will be similar in shape to those of equation image0 but will not mimic equation image0 exactly.

[38] The phase relationship between equation imagea and equation imagei in rocket data has been examined by Rapp and Lübken [2003]. They find both positive and negative correlations between negatively charged aerosol particle densities and electron densities in rocket data from a PMSE region. The negative correlation occurred in the upper part of the PMSE region. This observation is consistent with fossil turbulence in which the electrons reach an equilibrium determined by charge neutrality. The positive correlation occurred only at the lower boundary of the PMSE region. If there is acoustic turbulence in this region, the compressive phase of the wave will compress ions to a greater degree than the heavier aerosol particles, and the electrons will move toward the compressed region in response to the net positive charge density. In this scenario the electrons, ions, and smaller aerosol particles move approximately in phase. (Smaller aerosol particles would lag up to 90 degrees in phase because of inertia.) This suggests that the lower boundary is a region of active turbulence, but there was not evidence for turbulence in the data.

[39] The phase relationship between equation imagee and equation imagei in rocket data has been examined by Lie-Svendsen et al. [2003a, 2003b]. They show data from the SCT-06 rocket payload in which there is a negative correlation between electron density fluctuations and ion density fluctuations. This is consistent with the Boltzmann equilibrium and suggests fossil turbulence rather than active turbulence. These authors suggest that positive correlations may result from ion depletion by attachment to the larger aerosols expected to be at the bottom of the PMSE layer.

[40] Future rocket campaigns may yield simultaneous data for equation image0, equation imagea, equation imagee, equation imagei and equation image (or equivalently equation image). Analysis of data of this type will allow further insights into the relationship between air turbulence, aerosol layers, and radar echoes.

Acknowledgments

[41] The author thanks Zoltan Sternovsky for useful discussions and a critical reading of the manuscript. This research is supported by the National Aeronautics and Space Administration.