The formation of ice particle density irregularities with a
meter scale in the mesopause region is explored in this paper by developing
a growth and motion model of ice particles based on the motion equation of a
variable mass object. The growth of particles by water vapor adsorption and
the action of gravity and the neutral drag force on particles are considered in
the model. The evolution of the radius, velocity, and number density of ice
particles is then investigated by solving the growth and motion model
numerically. For certain nucleus radii, it is found that the velocity of
particles can be reversed at a particular height, leading to a local gathering
of particles near the boundary layer, which then forms small-scale ice
particle density structures. The spatial scale of the density structures can
be affected by vertical wind speed, water vapor density, and altitude, and
it remains stable as long as these environmental parameters do not change. The
influence of the stable small-scale structures on electron and ion density
is further calculated by a charging model, which considers the production,
loss, and transport of electrons and ions, along with dynamic particle
charging processes. Results show that the electron density is
anti-correlated to the charged ice particle density and ion density for
particles with radii of 11 nm or less due to plasma attachment by particles
and plasma diffusion. This finding is in accordance with most rocket
observations. The small-scale electron density structures created by
small-scale ice particle density irregularities can produce the polar
mesosphere summer echo (PMSE) phenomenon.
Introduction
Polar mesosphere summer echoes (PMSEs) are strong radar echoes from the polar
mesopause in summer (Rapp and Lübken, 2004). One of the features of
the PMSEs is that the spectral widths of echoes are much narrower than that of
incoherent scatter due to the Brownian movement of
electrons (Röttger, et al., 1988, 1990). It
has been proposed that the PMSEs are caused by radar waves coherently scattered
by irregularities in the refractive index, which are mainly determined by
electron density (Rapp and Lübken, 2004). The efficient scattering
occurs when the spatial scale of electron density structures is half of the
radar wavelength, which is called the Bragg scale. The scale is approximately 3
m for typical VHF radars (Rapp and Lübken, 2004). In the ECT02
rocket campaign (Lübken et al., 1998), a sounding rocket with
electron probes detected electron density irregularities in the order of
meters during a simultaneous observation of the PMSEs, providing a compelling
argument that small-scale electron density structures can indeed create
strong radar echoes.
A large amount of research indicates that small-scale ice particle density
irregularities in the PMSE region play a key role in creating and
maintaining small-scale structures of electron density (Chen and Scales,
2005; Lie-Svendsen et al., 2003; Mahmoudian and Scales, 2013; Rapp and
Lübken, 2003; Scales and Ganguli, 2004). Markus Rapp and Franz-Josef
Lübken investigated electron diffusion in the vicinity of charged
particles (Rapp and Lübken, 2003). They developed coupled
diffusion equations for electrons, charged aerosol particles, and positive
ions subject to the initial condition of anti-correlated perturbations in
charged aerosol and electron distributions. The results illustrate that the
perturbations of electron density are anti-correlated to that of the
negatively charged aerosol particles and positive ions. Lie-Svendsen et al.
studied the plasma response to imposed small-scale aerosol particle density
perturbations (Lie-Svendsen et al., 2003). The results were consistent with
the solution provided by Markus Rapp's model, in which particle density
structures in the order of a few meters could lead to small-scale electron
density perturbations due to electron attachment and ambipolar diffusion.
In the work mentioned above, aerosol particle density profiles were directly
set as specific small-scale structures such as Gaussian, hyperbolic tangent,
or sinusoidal. However, the formation mechanism of the small-scale particle
density structures can contribute to a more comprehensive understanding of
the PMSE phenomenon and is neglected in such studies. Kopnin et al. used
dust acoustic solitons to explain the localized structures of the charged
dust particles in the PMSE region (Kopnin et al., 2004), but the
spatial scales of the obtained structures were much smaller than the
observed scale and wavelength of VHF radar. Therefore, the formation
mechanism of small-scale structures in the PMSE region remains not clear.
In the polar mesopause region, neutral airflow moves upward (Garcia and
Solomon, 1985). The ice particles are subject to upward neutral drag force
and downward gravity, and they grow by absorbing water vapor simultaneously. In
addition, the size of initial condensation nuclei has a certain
distribution. These factors can cause complex trajectories of ice particles
and result in an inhomogeneous distribution of particle number density,
leading to small-scale structures of electron density. This may be an
important mechanism that can produce a PMSE phenomenon. However, few studies
have explored the formation process of small-scale ice particle structures
from the perspective of ice particle growth and movement.
A particle growth and motion model is thus developed in this paper to
describe the evolution of the ice particle radius, velocity, and density
distribution in the mesopause region. The growth of particles is based on
the collision and adsorption process of condensation nuclei and water vapor.
The particle movement is predominantly controlled by the gravity and the
neutral drag force. With the obtained ice particle density structures, the
corresponding electron and ion densities are calculated based on a charging
model, which includes the continuity equations for ice particles with
various charges and ions, the momentum equation for ions and electrons, and
the quasi-neutral condition.
(a) Dependence of the ice particle velocity on radius for different
initial nucleus radii. The black solid line Vd0=1-R0 represents the relation between the initial particle velocity and the
initial particle radius. (b) Movement curves of ice particles near the lower
boundary. (c) Movement curves of ice particles near the upper boundary.
Model
Equations for the growth and motion model of condensation nuclei and the
charging model of ice particles are detailed in this section.
The simulation is carried out at a summer polar mesopause region between 80 and 90 km, where the water vapor carried by neutral gas is
determined to move upwards at a constant speed (Garcia and Solomon, 1985).
It is assumed that micrometeorites enter the study region at a certain flux
from the upper boundary, and volcanic ash or particles ejected by the
aircraft rise into the region from the lower boundary. These grains serve as
condensation cores. When the temperature is lower than the frost point
(Körner and Sonnemann, 2001), water vapor molecules that touch the
surface of the grains due to thermal motion can easily condense into ice. In
this process, condensation cores become ice particles and continue to grow.
In this article, only the growth, motion, and charging process of particles
inside the condensation layer are discussed, and only the vertical transport
of particles and plasma is considered as the horizontal gradients of
transport parameters are much smaller comparatively (Lie-Svendsen et
al., 2003).
For growing ice particles, the dynamic equation for a variable mass object is
applied.
mdduddt+(ud-u)dmddt=mdg-μdnmd(ud-u)+qdE,
where md, ud, and qd are the mass, velocity,
and charge of ice particles, respectively, u is the
velocity of neutral gas, g is gravitational acceleration,
μdn is the collision frequency between ice particles and gas, and
E is the electric field. The electric force has a trivial
effect on the motion of ice particles because the charge–mass ratio of
particles is usually very small (Jensen and Thomas, 1988; Pfaff et al.,
2001). The inertial term is also negligible as its magnitude is much smaller
than gravity (Garcia and Solomon, 1985).
The water vapor is supersaturated in the polar mesopause region
(Lübken, 1999), and it is assumed that the size of condensation nuclei
is larger than the condensation critical size, so stable growth of ice
particles will continue when water molecules collide with particles during
thermal motion. Ignoring reverse processes such as sublimation, the mass
change rate of ice particles is
dmddt=μwdmw.
The collision frequency between water vapor and ice particles is μwd=nwπrd2vw based on the hard-sphere
collision model (Lieberman and Lichtenberg, 2005), in which mw,
nw, and vw are the mass, number density, and thermal velocity of
water molecules, respectively.
The collision frequency between air molecules and ice particles in the
neutral drag force term is (Schunk, 1977)
μdn=83πnnmnmd+mn2kBTg(md+mn)mdmnπrd+rn2,
where nn, mn, and rn are the number density, mean molecule mass, and
effective radius of neutral molecule, respectively, and Tg is the gas
temperature. The neutral molecule mass mn is assumed to be
28.96 mu, in which mu is the proton mass.
According to Eq. (1) the velocity of ice particles is obtained as
ud=u+mdμdnmd+μwdmwg.
On the basis that nw≪nn (Seele and
Hartogh, 1999), mw≪md, mn≪md, rn≪rd, vn∼vw and taking vertical up to be the positive
direction, the velocity of ice particles is simplified as
ud=u-g/μdn.
Ice particles are composed of condensation nuclei and the attached ice. The
mass of a single ice particle is
md=43πr03ρ0+43π(rd3-r03)ρd,
where r0 and ρ0 are the initial radius and mass density of
condensation nuclei and ρd is the mass density of ice.
Distribution of (a) ice particle density and (b) averaged particle
radius near the lower boundary of the condensation layer.
Based on the expressions of md and μdn, the relation between
ice particle velocity and radius is
ud=u-gnnmnvn[ρdrd+(ρ0-ρd)r03rd2].
At the boundaries of the condensation region rd=r0, the
initial velocity of condensation nuclei is
ud0=u(1-r0/rc),
where rc is the critical radius of
rc=nnmnvnu/(gρ0).
When the radius of condensation nuclei is r0>rc, gravity
is larger than the neutral drag force of vd0<0 and particles
move downwards. Otherwise, particles move upwards.
Based on the relation between md and rd, the change rate of ice
particle radius is
drddt=14nwmwvwρd=c.
It can be clearly observed that the ice particle radius increases linearly
with time.
rd=r0+ct
The particle trajectory can then be obtained by the following integral:
z-z0=∫0tuddt=c-1∫r0rduddrd,
where z0 is the reference height where condensation nuclei enter the
studied region. In this work z0=0 is set at the lower boundary, and
z0=h is set at the upper boundary, where h is the distance between the
two boundaries.
Number density distribution of (a) electrons ne, (b) ions
ni, (c) particles carrying one negative charge n-1, and (d)
particles carrying two negative charges n-2 near the lower boundary of the
condensation layer at t=1000 s.
It is assumed that the condensation nucleus radius ranging from r0min to
r0max has a certain distribution function f(r0). The density of
condensation nuclei with a radius in the small range is r0→r0+dr0 is dn(r0)=f(r0)dr0, and their velocity is ud0.
When these particles arrive at height z, their radius increases to
rd(r0,z), the corresponding number density turns into
dn(r0, z), and the velocity becomes ud(r0, z) =ud[r0, rd(r0,z)]. According to the particle conservation law,
ud0dn(r0)=ud(r0,z)dn(r0,z).
The number density of ice particles at height z can then be obtained by
nd(z)=∫dn(r0,z)=∫r0minr0maxud0f(r0)ud(r0,z)dr0.
The averaged ice particle radius at height z is
r‾d(z)=∫rd(z)dn(r0,z)nd(z).
By integrating all condensation nucleus radii, a stable distribution of
nd and rd can be obtained. The particles continue to enter and leave
the condensation region, and as long as the external environment does not
change, the distribution of particle density and radius will remain
unchanged. The influence of these stable nd and rd profiles on
electron and ion density is then calculated.
Movement curves of ice particles near the upper boundary. The
particles with initial radius R0Z1 move upward after turning back at the
Z height (the red line), and the particles with initial radius R0Z2 move
downward after turning back at Z (the blue line).
Considering ionization, electron–ion recombination, and ion loss on ice
particles, the continuity equation of ion density can be written as
∂ni∂t+∂(niui)∂z=Q-αnine-D+ni.
Ignoring gravity, the drift velocity of ions ui is determined by
ui=eEmiμin-kBTgmiμin1ni∂ni∂z.
The electric field E is predominantly determined by the electron density gradient
because the diffusion coefficient and mobility of electrons are much larger
than that of ions.
E=-kBTge1ne∂ne∂z
In the typical PMSE layer, there are several kinds of ions carrying one unit
positive charge: N2+, O2+, NO+, and
H+(H2O)n. As specified by Reid (Reid, 1990), the averaged
ion parameters ni, mi, and Tg are applied to describe the
density, mass, and temperature of ions, respectively, and the averaged ion
mass mi is set as 50 mu at 85 km altitude. According to the theory of Hill and
Bowhill (1977), the ion-neutral collision
frequency is
μin=2.6×10-15nn0.7828Mi+281.74Mi+2828Mi+0.2132Mi+321.57Mi+3232Mi+0.0140Mi+401.64Mi+4040Mi,
where Mi=mi/mu.
The production rate for ions and electrons Q is chosen to be 3.6×107 m-3 s-1, and the electron–ion recombination coefficient
α is set as 10-12 m3 s-1 (Lie-Svendsen et al.,
2003). The undisturbed density of ions and electrons is n0=6×109 m-3. The loss coefficient of ions on ice particles
is D+=Σnqνi,q, where nq is the number
density of the q-charged ice particles and νi,q represents
the capture rate of ions by ice particles with q charges. According to the
discrete charging model (Robertson and Sternovsky, 2008),
νi,q≤0=πrd2ci1+Cqe216ε0kBTgrd+Dqe24πε0kBTgrd.
The particle radius rd used here is the averaged
radius [U+F8E5] rd, which is obtained using Eq. (15). The ion thermal
velocity is ci= (8kBTg/πmi); kB is the Boltzmann
constant; and ε0 is the permittivity of a vacuum. The
Cq and Dq are provided in Table 1 in the work of Robertson and
Sternovsky (Robertson and Sternovsky, 2008), and the corresponding
capture rates of electrons by ice particles (Robertson and Sternovsky,
2008) are denoted as
21νe,q≥0=πrd2ce1+Cqe216ε0kBTgrd+Dqe24πε0kBTgrd,22νe,q<0=πrd2γ2ceexp-qe24πε0kBTgrdγ1-12γγ2-1q.
The thermal velocity of electrons ce= (8kBTg/πme) and the value of γ for each q is referenced from Natanson's
paper (Natanson, 1960).
Although the distribution of total particle density nd=Σnq reaches a stable state under the action of gravity and neutral drag
force, the number density of the q-charged ice particles nq is dynamic in
the charging process. The continuity equation of q-charged ice particle
density is
∂nq∂t=nq+1νe,q+1ne+nq-1νi,q-1ni-nqνe,qne+nqνi,qni.
According to previous research (Lie-Svendsen et al., 2003; Rapp and
Lübken, 2001), it is assumed that a single particle carries two negative
charges at the most, i.e., q=-2, -1, 0, and +1 in this study.
According to typical parameters in the PMSE region (Rapp and Lübken,
2001), the plasma Debye length λD is estimated to be
approximately 9 mm, which is much smaller than the vertical spatial scale of
the PMSE layer. Thus, the dusty plasma satisfies the quasi-neutral
condition.
ni+∑qqnq=ne
For simplicity, dimensionless parameters will be used in subsequent
discussion.
Vd=vd/u,ρ=ρd/ρ0,R0=r0/rc,Rd=rd/rcT=t/tc,Z=(z-z0)/zc,
where tc=rc/c, which represents the time it takes for ice
particles to grow from rd to rd+rc, and zc=utc is the distance that neutral wind moves during the time tc.
The distribution of (a) ice particle density and (b) the averaged
particle radius near the upper boundary of the condensation layer.
The number density distribution of (a) electrons, (b) ions, (c)
particles carrying one negative charge, and (d) particles carrying two
negative charges near the upper boundary of the condensation layer at t=1000 s.
The expression of dimensionless ice particle velocity is
Vd=1-ρRd-(1-ρ)R03Rd2.
The expressions of the dimensionless position coordinate of particles based on
T and Rd are
26Z(R0,T)=T-12ρT(T+2R0)-(1-ρ)R02TT+R0,27Z(R0,Rd)=Rd-R0-12ρ(Rd2-R02)+(1-ρ)R031Rd-1R0.
The dimensionless number density and radius distribution of ice particles
are
28nd(Z)=n0∫R0minR0maxVd0F(R0)Vd[R0,Rd(R0,Z)]dR0,29R‾d(Z)=n0nd(Z)∫R0minR0maxRd(Z)Vd0F(R0)Vd[R0,Rd(R0,Z)]dR0,
where n0 is the density of condensation cores at the boundary, and it is
assumed to be 5×108 m-3 (Bardeen et al., 2008).
The normalized radius distribution function F(R0) satisfies
∫R0minR0maxF(R0)dR0=1.
In subsequent calculations, parameters are taken in the atmospheric
environment at an altitude of 85 km. The number density of neutrals is
nn=2.3×1020 m-3 (Hill et al., 1999); the
number density of water vapor is nw=2.5×1014 m-3 (Seele and Hartogh, 1999); the temperature is Tg=150 K; the
mass density of ice is ρd=1×103 kg m-3; the
velocity of neutral wind is u=3 cm s-1 (Garcia and Solomon, 1985); the mass
density of condensation nucleus is ρ0=2.7×103 kg m-3; and the growth rate of ice particles is c≈7.8×10-4 nm s-1. In this work, we only consider the growth and movement of
condensation nuclei which fall from the upper boundary with the initial radius
r0>rc and rise from the lower boundary with r0≤rc.
The distribution of (a) ice particle density, (b) the averaged
particle radius, (c) electron density, and (d) ion density for various
vertical wind speeds near the lower boundary of the condensation layer.
Results and discussionSpeed and trajectory of ice particles
The relation between Vd and Rd is illustrated in Fig. 1a, which
shows that condensation nuclei with an initial radius R0≤1 rise into
the PMSE region through the lower boundary, while particles with R0>1 fall into the region from the upper boundary. At the
beginning, the upward-moving particles accelerate and the downward particles
decelerate due to ∂Vd/∂Rd=2-3ρ>0 when Rd=R0. Later, with the increase of
Rd, ∂Vd/∂Rd<0, all particles will
move with a downward acceleration, which makes them eventually move downward.
Figure 1b shows the movement curves of ice particles near the lower
boundary. These particles, with an initial radius R0≤1 will rise
into the condensation layer. With the collection of ice, the grains become
larger and heavier, leading to their deceleration. The grains will then
accelerate downward until they leave the condensation layer from the lower
boundary. All particles rising from the lower boundary will retrace in the
range Zm<Z<ZM. Zm is the maximum height that
particles with an initial radius R0=1 can reach, and ZM is the
maximum height that particles with an initial radius R0=R0min=0.5 can reach. Based on above parameters, Zm=0.1512 and ZM=0.7682.
The distribution of (a) ice particle density, (b) the averaged
particle radius, (c) electron density, and (d) ion density for various
vertical wind speeds near the upper boundary of the condensation layer.
Figure 1c shows the movement curves of ice particles near the upper
boundary, which can be sorted by the value of R0. For 1<R0<R01, the neutral drag force increases faster than
gravity as the particles fall. The particles decelerate to zero speed,
retrace upward, and then leave the condensation layer from the upper
boundary. For R0=R01, the particles retrace at the height Z=Z1, then they arrive at Z=0 with exactly zero velocity, and the
particles move back into the condensation layer again. For R01<R0<R02, the particles retrace upward in the range of
Z2<Z<Z1 and move downward again before they
reach the upper boundary. For R0=R02, the particles decelerate
downward until zero speed at Z=Z2. Here, the acceleration happens to
be zero. Then the gravity exceeds the drag force, and the particles
accelerate downward. For R0>R02, the particles continue
moving down after entering the condensation layer. According to the above
parameters, R01 and R02 are solved as 1.1519 and 1.19705,
respectively.
It can be observed in Fig. 1 that the particles with a certain initial radius
will move up and down several times near the boundary; namely, ice particles
will accumulate at that region and form some kind of small-scale density
structure.
The distribution of (a) ice particle density, (b) the averaged
particle radius, (c) electron density, and (d) ion density for various water
vapor densities near the lower boundary of the condensation layer.
The distribution of (a) ice particle density, (b) the averaged
particle radius, (c) electron density, and (d) ion density for various water
vapor densities near the upper boundary of the condensation layer.
Density and radius distribution of ice particles and their
effects on plasmaNear the lower boundary
The density and radius distribution of ice particles near the lower boundary
are first solved. As illustrated in Fig. 1b, all ice particles with an
initial radius R0≤1 will pass the range 0<Z<Zm twice, so they contribute twice to the calculation of particle
density. In the height range Zm<Z<ZM, only the
particles that can reach the Z height will contribute to the density at Z. The
density and mean radius of ice particles near the lower boundary are shown
below.
30nd(Z)=n0∫0.5R0ZVd0F(R0)1Vd1(R0,Rd1)+1|Vd2(R0,Rd2)|dR0,31R‾d(Z)=n0nd(Z)∫0.5R0ZVd0F(R0)Rd1Vd1(R0,Rd1)+Rd2|Vd2(R0,Rd2)|dR0,
where Rd1 and Rd2 are particle radii when particles pass through
the Z height, Vd1 and Vd2 are the corresponding velocities, and the
upper limit of integral R0Z is determined by
R0Z=1ifZm<Z<ZMsolution ofZR0Z,Rd=Zif0<Z<Zm.
In this study, the radius distribution function of condensation cores is
assumed to be a Gaussian distribution.
FR0=Aexp-R0-R002/Δ2,
where the center of the radius distribution function R00 is chosen to be
0.8, the characteristic width is Δ=0.01, and the corresponding
normalized coefficient is A=56.4.
The distribution of (a) ice particle density, (b) the averaged
particle radius, (c) the relative change of electron density Δne/ne, and (d) the relative change of ion density Δni/ni at various altitudes near the lower boundary of the condensation
layer.
The distribution of (a) ice particle density, (b) the averaged
particle radius, (c) the relative change of electron density Δne/ne, and (d) the relative change of ion density Δni/ni at various altitudes near the upper boundary of the condensation
layer.
The obtained density and mean radius of ice particles near the lower
boundary are presented in Fig. 2a and b, respectively. Figure 2a shows
that a sharp peak appears in the density distribution of ice particles. The
width at the half maximum of the irregularity is about 5 m, which is consistent
with the assumed ice particle density structure scale in theoretical
work (Lie-Svendsen et al., 2003; Rapp and Lübken, 2003) as well as in the
observation by the sounding rocket flight ECT02 in July 1994 (Rapp and
Lübken, 2004). In Fig. 2b, it can be observed that the average radius
of ice particles increases from 7 to 11 nm with height.
According to the obtained density and average radius of ice particles in
Fig. 2a and b, the density distribution of electrons, ions, and charged
ice particles is calculated based on the charging model described by Eqs. (16)–(24). At the initial moment of the charging model, all
ice particles are assumed to be neutral in order to conduct the calculation
more conveniently, as the final distributions of charge are independent from
the initial ice particle charge state (Lie-Svendsen et al., 2003). The
timescale of electrons collected by negatively charged particles with a
radius of 10 nm is approximately 700 s, which is the longest timescale in
the charging process. The quasi-steady state of charging can then be
obtained after this timescale. Therefore, the calculation is terminated
after 1000 s, and the results are illustrated in Fig. 3.
Figure 3a shows that electron density decreases sharply around z=60 m
due to adsorption by particles and the reduction of electron density Δne≈(n-1+2n-2)/2, which corresponds to the
results under diffusion equilibrium approximations in earlier work
(Lie-Svendsen et al., 2003). Ion number density increases sharply
around 60 m due to diffusion under the ambipolar electric field. The
ambipolar diffusion process of electrons and ions has been described in
detail in previous work (Lie-Svendsen et al., 2003). Electron density
is anti-correlated to density irregularities of ions and the charged ice
particles due to the attachment and diffusion processes. The
anti-correlations correspond with rocket observations by the sounding rocket
flight SCT-06 in August 1993 (Lie-Svendsen et al., 2003) and the sounding
rocket flight ECT02 in July 1994 (Rapp and Lübken, 2004). According
to Fig. 3c and d, it can be determined that for particles with radii
ranging from 7 to 11 nm, the proportion of particles carrying one
negative charge ranges from 97.5 % to 85.1 %, and the value for
particles carrying two negative charges is 0.53 %–13.6 %, which is
consistent with observations by Havnes et al. (1996) and
numerical results by Rapp and Lübken (2001). The
density of positively charged particles is less than 1.1×105 m-3 and is insignificant in this study.
Near the upper boundary
The parameters of ice particles and plasma near the upper boundary are
discussed in this subsection, based on the movement curves of ice particles
near the upper boundary, as shown in Fig. 4.
For Z1<Z<0, particles with an initial radius R0Z1
and R0Z2 turn back at Z and move upward and downward separately, as shown
in Fig. 4. The values of R0Z1 and R0Z2 are determined by the equations
Vd(R0Z,Rd)=0 and Z(R0Z,Rd)=Z. The contribution
of ice particles to the density distribution near the upper boundary can be
classified as follows:
R0<R0Z1: ice particles cannot reach Z and make no
contributions to the number density.
R0Z1<R0<R01: ice particles pass
through Z twice and contribute to nd(Z) twice. The radius of particles
when passing through the Z height can be obtained as Rd31 andRd32 based
on Eq. (27). Their corresponding velocities are calculated respectively as
Vd31 and Vd32 based on Eq. (25).
R01<R0<R0Z2: ice particles pass
through Z three times. The corresponding radii and velocities at Z are
defined as Rd41, Rd42, and Rd43 andVd41, Vd42, and Vd43.
R0>R0Z2: ice particles pass through Z only
once, and their radius and velocity are Rd5 and Vd5, respectively.
Substituting these parameters into Eqs. (28) and (29), the density and mean
radius of ice particles in the range Z1<Z<0 are
deduced as
34nd(Z)=n0∫R0Z1R01|Vd0|F(R0)1|Vd31(R0,Rd31)|+1Vd32(R0,Rd32)dR0+n0∫R01R0Z2|Vd0|F(R0)1|Vd41(R0,Rd41)|+1Vd42(R0,Rd42)+1|Vd43(R0,Rd43)|dR0+n0∫R0Z2R0max|Vd0|F(R0)|Vd5(R0,Rd5)|dR0,35R‾d(Z)=n0nd(Z)∫R0Z1R01|Vd0|F(R0)Rd31|Vd31(R0,Rd31)|+Rd32Vd32(R0,Rd32)dR0+n0nd(Z)∫R01R0Z2|Vd0|F(R0)Rd41|Vd41(R0,Rd41)|+Rd42Vd42(R0,Rd42)+Rd43|Vd43(R0,Rd43)|dR0+n0nd(Z)∫R0Z2R0maxRd5|Vd0|F(R0)|Vd5(R0,Rd5)|dR0.
The center of the radius distribution function is R00=1.08; the
characteristic width is Δ=0.01; and the corresponding normalized
coefficient is A=56.4.
The ice particle density in the region of Z<Z1 is close to
zero, as only particles with an initial radius R0≥R01 can arrive
at the region, and the number of the particles in this radius range is very
low based on the radius distribution function set above.
At the upper boundary, the number density of condensation cores n0 is set
as 5×108 m-3, and the maximum radius of condensation
cores is R0max=1.3. The number density and mean radius of ice
particles are obtained from Eqs. (34) and (35) and are shown in Fig. 5. The
density distribution of electrons, ions, and charged ice particles is then
calculated further based on the charging model.
Figure 5a shows that there is a meter scale structure in the distribution
of ice particle density, which is consistent with the assumed ice particle
density structure scale in previous theoretical work (Lie-Svendsen et
al., 2003; Rapp and Lübken, 2003) and rocket observations (Rapp and
Lübken, 2004). In addition, the average radius of ice particles is
slightly larger than 5 nm (shown in Fig. 5b).
As illustrated in Fig. 6a, compared with ice particle density, there is a
similar but anti-correlated structure in the electron density profile because of
the adsorption of electrons by particles. Due to ambipolar diffusion, ion
density increases in the perturbed region. The reduction of electron density
Δne and the increment of ion density Δni are
consistent with the results under diffusion equilibrium approximations:
Δne≈Δni≈(n-1+2n-2)/2 (Lie-Svendsen et al., 2003). According to Fig. 6c and
d, 97 % of the particles carry one negative charge, and few particles
carry two negative charges. This is reasonable for particles with a radius
slightly larger than 5 nm.
Influence of the vertical wind speed on the spatial scale of the
irregularities
The vertical wind speed is varied from 3 to 5 cm s-1 to investigate the
influence of the wind speed on the spatial scale of the irregularities. With
all other parameters remaining the same, the numerical results are shown in
Figs. 7 and 8. When the wind speed is increased, the spatial scale of
the irregularities increases as higher wind speed corresponds to a larger
critical particle radius rc in the growth model (see Eq. 9). This
leads to a longer timescale (tc) and larger spatial scale (zc) of ice
particle growth and movement. In addition, as shown in Fig. 7c and d,
with the increase of wind speed, the variation amplitude of electron density
and ion density near the lower boundary obviously increases. This is because
the averaged radius of the ice particles increases with the extension of
the particle growth time (see Fig. 7b), and the particles' influence on the
plasma increases. The variation amplitude of electron density and ion
density near the upper boundary does not notably change (see Fig. 8c and
d) because there is little variation in the averaged radii of the ice
particles for different wind speeds, as shown in Fig. 8b.
Influence of the water vapor density on the spatial scale of the
irregularities
Water vapor density can also affect the spatial scale of the particle
density structures by modifying the change rate of the particle radius. As
illustrated in Figs. 9 and 10, the spatial scale of the irregularities
decreases when the water vapor density increases. A larger vapor density
results in a higher change rate of the particle radius (see Eq. 10) and a
shorter timescale (tc) of ice particle growth. The particles can then
reach the inversion condition more quickly, and the reverse position is
closer to the boundary, meaning that the spatial scale of the ice particle
density structures becomes shorter.
Influence of the altitude on the irregularities spatial scale
The effect of the altitude on the spatial scale of the irregularities is
explored in this subsection. The altitude mainly affects the neutral gas
density nn, ion composition, ion mass mi, production rate for plasma
Q, electron–ion recombination coefficient α, and plasma
density n0 without ice particles. In addition to 85 km, altitudes of 82
and 88 km are also included as they are near the lower and upper limits of
the PMSE region (Lie-Svendsen et al., 2003). According to previous work
(Blix, 1999; Lübken, 1999; Rapp and Lübken, 2001), at 82 km, the
positive ions are mainly (H3O)+(H2O)3 cluster ions with
mi=73mu, and other parameters are set as nn=4.2×1020 m-3, Q=6.3×107 m-3 s-1, α=7×10-12 m3 s-1, and
n0=3×109 m-3. At 88 km altitude, the positive
ions are mainly NO+ with mi=30mu, and other parameters
are nn=1.1×1020 m-3, Q=6×107 m-3 s-1, α=6×10-13 m3 s-1, and
n0=1×1010 m-3. The numerical results are
provided in Figs. 11 and 12. As the ambient plasma density n0 varies
dramatically at different altitudes, the electron density relative change
Δne/ne and the ion density relative change Δni/ni are calculated for comparison, where Δne=ne-n0 and Δni=ni-n0. Figures 11a and
12a show that with the increase in altitude, the spatial scale of the
ice particle density irregularities becomes shorter. The reason for this
finding is that higher altitudes correspond to a smaller neutral density
nn and critical particle radius rc (see Eq. 9), which leads to a
shorter timescale (tc) and spatial scale (zc) of ice particle
growth and movement. It is notable that the spatial scale of the electron
density irregularities at lower altitudes is longer than that at higher
altitudes (see Figs. 11c and 12c). This result is consistent with the
explanation provided by Bremer et al. that at lower altitudes the PMSE
signals detected by long-wavelength radar (half wavelength of 54 m) are
stronger than those detected by short-wavelength radar (half wavelength of 2.8 m) (Bremer et al., 1997). In addition, with the increase of
altitude, the relative change amplitude of electron density and ion density
decreases significantly because the averaged radius of the ice particles at
higher altitudes is smaller, and the influence of ice particles on plasma
decreases.
Conclusions
In this paper, a growth and motion model of ice particles was developed
based on the equation of variable mass object motion in order to explain the
formation of ice particle density irregularities with a meter scale in the
polar mesopause region. The density profile of ice particles with height was
investigated according to the conservation of the particle number. Based on the
growth and motion model, small-scale structures of ice particle density were
successfully produced. The density distributions of electrons and ions
corresponding to the ice particle density distribution were then obtained
based on quasi-neutrality and the discrete charging model. The findings are
summarized as follows.
The ice particle radius increases linearly with time. However, a complex
relation occurs between the velocity and radius of particles due to the
variable mass of ice particles and the complicated force operating on them.
For a certain radius of the condensation nucleus, ice particles can bounce
near the boundary layer, which leads to a local gathering phenomenon of ice
particles and the creation of meter scale ice particle density structures.
The spatial scale of the density structures can be affected by vertical wind
speed, water vapor density, and altitude. The spatial scale increases with
the increase of wind speed, and it decreases with the increase of water vapor
density and altitude. Small-scale ice particle density irregularities can
remain stable if these atmospheric conditions do not change. In the ice
particle gathering region, the electron density is anti-correlated to the
charged ice particle density and the ion density because of the plasma
attachment by ice particles and plasma diffusion. To summarize, small-scale
ice particle density irregularities are formed and maintained in the polar
mesopause region based on the growth and motion model, and the corresponding
small-scale electron density structures are in accordance with most rocket
observations.
Data availability
No data sets were used in this article.
Author contributions
JM and JW put forward the idea. JM and RT developed the model. RT created the model. YL created the figures. HL analyzed the data. RT and CY wrote the paper. YJ and ZZ revised the paper.
Competing interests
The authors declare that they have no conflict of interest.
Acknowledgements
The research has been financially supported by the National Natural Science
Foundation of China under (grant nos. 11775062 and 61601419) and the Key
Laboratory Foundation of the National Key Laboratory of Electromagnetic
Environment (grant no. 614240319010303).
Financial support
The research has been supported by the National Natural Science Foundation of China (grant nos. 11775062 and 61601419) and the Key Laboratory Foundation of the National Key Laboratory of Electromagnetic Environment (grant no. 614240319010303).
Review statement
This paper was edited by Andrew J. Kavanagh and reviewed by two anonymous referees.
ReferencesBardeen, C., Toon, O., Jensen, E., Marsh, D., and Harvey, V.: Numerical simulations of the
three-dimensional distribution of meteoric dust in the mesosphere and upper
stratosphere, J. Geophys. Res.-Atmos., 113, D17202, 10.1029/2007JD009515,
2008.
Blix, T.: Small scale plasma and charged aerosol variations and their
importance for polar mesosphere summer echoes, Adv. Space Res.,
24, 537–546, 1999.Bremer, J., Hoffmann, P., Manson, A. H., Meek, C. E., Rüster, R., and Singer, W.: PMSE observations at three different frequencies in northern Europe during summer 1994, Ann. Geophys., 14, 1317–1327, 10.1007/s00585-996-1317-7, 1996.Chen, C. and Scales, W.: Electron temperature enhancement effects on plasma
irregularities associated with charged dust in the Earth's mesosphere,
J. Geophys. Res.-Space, 110, A12313, 10.1029/2005JA011341, 2005.
Garcia, R. R. and Solomon, S.: The effect of breaking gravity waves on the
dynamics and chemical composition of the mesosphere and lower thermosphere,
J. Geophys. Res.-Atmos., 90, 3850–3868, 1985.
Havnes, O., Trøim, J., Blix, T., Mortensen, W., Næsheim, L., Thrane, E., and Tønnesen, T.: First detection of charged dust
particles in the Earth's mesosphere, J. Geophys. Res.-Space, 101, 10839–10847, 1996.
Hill, R. and Bowhill, S.: Collision frequencies for use in the continuum momentum
equations applied to the lower ionosphere, J. Atmos. Terr. Phys., 39, 803–811, 1977.
Hill, R., Gibson-Wilde, D., Werne, J., and Fritts, D.: Turbulence-induced fluctuations in
ionization and application to PMSE, Earth Planet. Space, 51, 499–513,
1999.
Jensen, E. and Thomas, G. E.: A growth-sedimentation model of polar mesospheric
clouds: Comparison with SME measurements, J. Geophys. Res.-Atmos., 93, 2461–2473, 1988.
Kopnin, S., Kosarev, I., Popel, S., and Yu, M.: Localized structures of nanosize
charged dust grains in Earth's middle atmosphere, Planet. Space
Sci., 52, 1187–1194, 2004.
Körner, U. and Sonnemann, G.: Global three-dimensional modeling of the water
vapor concentration of the mesosphere-mesopause region and implications
with respect to the noctilucent cloud region, J. Geophys.
Res.-Atmos., 106, 9639–9651, 2001.Lie-Svendsen, Ø., Blix, T., Hoppe, U. P., and Thrane, E.: Modeling the plasma
response to small-scale aerosol particle perturbations in the mesopause
region, J. Geophys. Res.-Atmos., 108, 8442, 10.1029/2002JD002753, 2003.
Lieberman, M. A. and Lichtenberg, A. J.: Principles of plasma discharges and
materials processing, John Wiley & Sons, 733 pp., 2005.
Lübken, F. J.: Thermal structure of the Arctic summer mesosphere, J. Geophys. Res.-Atmos., 104, 9135–9149, 1999.
Lübken, F. J., Rapp, M., Blix, T., and Thrane, E.: Microphysical and turbulent
measurements of the Schmidt number in the vicinity of polar mesosphere
summer echoes, Geophys. Res. Lett., 25, 893–896, 1998.
Mahmoudian, A. and Scales, W.: On the signature of positively charged dust
particles on plasma irregularities in the mesosphere, J. Atmos.
Sol.-Terr. Phys., 104, 260–269, 2013.
Natanson, G.: On the theory of the charging of amicroscopic aerosol particles
as a result of capture of gas ions, Sov. Phys. Tech. Phys., 30, 573–588,
1960.
Pfaff, R., Holzworth, R., Goldberg, R., Freudenreich, H., Voss, H., Croskey, C., Mitchell, J., Gumbel, J., Bounds, S., and Singer, W.: Rocket probe observations of
electric field irregularities in the polar summer mesosphere, Geophys.
Res. Lett., 28, 1431–1434, 2001.
Rapp, M. and Lübken, F.-J.: Modelling of particle charging in the polar
summer mesosphere: Part 1 – General results, J. Atmos.
Sol.-Terr. Phys., 63, 759–770, 2001.Rapp, M. and Lübken, F. J.: On the nature of PMSE: Electron diffusion in the
vicinity of charged particles revisited, J. Geophys. Res.-Atmos., 108, 8437, 10.1029/2002JD002857, 2003.Rapp, M. and Lübken, F.-J.: Polar mesosphere summer echoes (PMSE): Review of observations and current understanding, Atmos. Chem. Phys., 4, 2601–2633, 10.5194/acp-4-2601-2004, 2004.
Reid, G. C.: Ice particles and electron “bite-outs” at the summer polar
mesopause, J. Geophys. Res.-Atmos., 95, 13891–13896,
1990.Robertson, S. and Sternovsky, Z.: Effect of the induced-dipole force on charging
rates of aerosol particles, Phys. Plasmas, 15, 040702, 10.1063/1.2907162, 2008.
Röttger, J., La Hoz, C., Kelley, M. C., Hoppe, U., and Hall, C.: The structure and dynamics of
polar mesosphere summer echoes observed with the EISCAT 224 MHz radar,
Geophys. Res. Lett., 15, 1353–1356, 1988.
Röttger, J., Rietveld, M., La Hoz, C., Hall, T., Kelley, M., and Swartz, W.: Polar mesosphere summer echoes
observed with the EISCAT 933 MHz radar and the CUPRI 46.9 MHz radar, their
similarity to 224 MHz radar echoes, and their relation to turbulence and
electron density profiles, Radio Sci., 25, 671–687, 1990.
Scales, W. and Ganguli, G.: Investigation of plasma irregularity sources
associated with charged dust in the Earth's mesosphere, Adv. Space
Res., 34, 2402–2408, 2004.Schunk, R.: Mathematical structure of transport equations for multispecies
flows, Rev. Geophys., 15, 429–445, 1977.
Seele, C. and Hartogh, P.: Water vapor of the polar middle atmosphere: Annual
variation and summer mesosphere conditions as observed by ground-based
microwave spectroscopy, Geophys. Res. Lett., 26, 1517–1520, 1999.