ELECTRICAL DECAY ESTIMATES IN ANVIL CLOUDS

Report No. 1 of 2 under Contract No. CC-90233B

John C. Willett, 12/28/01

[Introduction]{.underline}

The purpose of this investigation is to aid NASA and the USAF in
determining when cirrus anvils do not constitute a triggered-lightning
hazard for either an outgoing launch vehicle or a landing Space Shuttle.
The \"Anvil Clouds\" Rule in the current Launch-Commit Criteria (LCC)
prohibits penetration of non-transparent parts of an anvil cloud while
it remains attached to its parent cloud and for the first 3 hours after
detachment is observed (or for the first 4 hours after the last
lightning discharge occurs in the detached anvil cloud) \[*Krider et
al*., 1999\]. This rule causes significant launch delays and scrubs in
conditions that are probably (but are not *known* to be) safe. One such
condition that occurs at the Kennedy Space Center (KSC) during the early
spring is a translucent cirrus shield formed by blow-off from a distant
mesoscale convective complex over the eastern Gulf of Mexico.

Beginning in the summer of 2000 and extending into the summer of 2001,
an Airborne Field Mill (ABFM) experiment has been conducted in the
vicinity of KSC to measure both the ambient electrostatic fields and the
size/shape distributions of the cloud and precipitation particles in
cirrus anvils (as well as in and near other cloud types). One important
goal of this field experiment was to obtain data on the decay of
electric fields within anvil clouds as they age after detachment or are
advected downstream from their parent cumulonimbus clouds. It was hoped
that these observations, coupled with a means of remotely assessing the
microphysics of these clouds from the ground, would allow a description
of hazardous anvils that was much more restricted than the one in the
existing \"Anvil Clouds\" rule. If the observations could also be used
to validate a simple theory of how electric fields *should* decay in
anvil cirrus, then a relatively small number of observations could yield
sufficient confidence to permit the safe relaxation of this launch
constraint.

The work described herein is intended to articulate the simple
theoretical model mentioned above and to insert preliminary ABFM
microphysical measurements into this model in order to predict the decay
times of electric fields within an actual anvil cloud. These decay times
could then be compared with ABFM ambient-field observations in the same
clouds in an effort to validate the theory.

[Theoretical Model]{.underline}

The simplest possible model of a charged cirrus anvil is a
one-dimensional structure with constant microphysics but time-dependent
electrification. Imagine a horizontally homogeneous cloud of constant
thickness, *H* (m), and of uniform cloud-particle size and shape
distributions that initially contains a layer of positive electric
charge residing on particles at its center and having area density,
*~0~* (C m^-2^). (All units in this report are rationalized MKS. All
references to cloud particles should be taken to connote the whole range
of hydrometeors, from the smallest frozen cloud droplets to the largest
aggregates.) We assume an all-ice cloud with no sedimentation,
aggregation, convection, nor any other active processes that might
generate further electrification.

If *H* is large enough, and if there is no turbulent mixing, screening
layers of negative charge will form rapidly on the upper and lower
surfaces of this cloud, confining the elevated electric fields between
the cloud boundaries \[*Brown et al*., 1971; *Hoppel and Phillips*,
1971; *Klett*, 1972\]. For simplicity we imagine the internal charge
layer and the two screening layers (each of area density, -*~0~*/2) to
be infinitely thin; so that the entire cloud volume is filled with a
uniform electric field intensity, *E~0~* = *~0~*/(2*~0~*) (V m^-1^), of
positive polarity (upward-directed) above the cloud center and negative
(downward) below, where *~0~* (F m^-1^) is the dielectric permittivity
of free space. These are the initial conditions for our simple model of
the time evolution of *E*(*t*). This evolution could apply equally to
the temporal decay of field within a stationary detached anvil or (with
substitution, *t* = *x*/*v*, where *x* is distance in the down-stream
direction and *v* is horizontal advection speed) to the spatial decay of
field in an attached anvil down wind from the parent cloud.

Small ions are constantly created in pairs within the cloud, primarily
by cosmic radiation at the altitudes of typical cirrus anvils, at the
volume production rate, *q* (m^-3^ s^-1^). These ions drift parallel to
the ambient field with electrical mobility, *k* (m^2^ V^-1^ s^-1^),
resulting in a current density, *J*(*t*) = 2*ekn*(*t*)*E*(*t*) (A
m^-2^), that tends to neutralize simultaneously all three charge layers
in the cloud. Here, *e* (C) is the electronic charge and *n*(*t*)
(m^-3^) is the volume density of both positive and negative small ions
(assumed equal). By charge continuity and the above relationships,
*J*(*t*) leads to a decay of the field intensity within the cloud, as
follows:

d*E*/d*t* = -2*ekn*(*t*)*E*(*t*)/*~0~* (1)

Note that a macroscopic, continuum approach has been used to compute
electrical conduction and electric-field decay inside the cloud, in
spite of the fact that nearly all charge actually resides on cloud
particles. The main justification for this approach is that, in order to
eliminate the macroscopic electric fields of concern for launch safety,
we do not care whether or not the charge on individual particles is
neutralized. It is sufficient that the large-scale, net-charge density
be neutralized, and this may be accomplished by macroscopic conduction
currents mediated by a bulk \"conductivity.\" For these purposes the
deposition of opposite charge on other particles within the same
macroscopic cloud volume is equivalent to neutralization of the
initially charged particles.

We implicitly allow the positive and negative small-ion densities to
diverge and to have strong gradients near the surfaces of individual
cloud particles when computing the attachment rates of ions to
particles, below. This does not invalidate the macroscopic calculation
in Equation 1, however, as long as the fraction of cloud volume occupied
by particles is negligibly small. We will, in fact, assume that the
particles are far enough apart that the ion-density gradients produced
by one particle do not affect the attachment rates at another.

Were *n*(*t*) constant, as it normally is in the free atmosphere, the
solution of (1) would yield the familiar exponential decay of *E*(*t*)
according to the \"electrical relaxation time,\" *τ~e~* = *~0~*/(2*ekn*)
(s), for the relevant altitude. (*τ~e~* ≈ 19 s at an altitude of 10 km
in the free atmosphere, which is utterly negligible from the perspective
of launch safety.) In addition to the usual ion-loss processes of
recombination and aerosol attachment (neither of which depends
significantly on *E*, and both of which will be neglected here),
however, small ions may both diffuse to and be electrically driven to
the cloud particles, where they are annihilated. The latter process is
field dependent and is always the dominant loss mechanism at
sufficiently high field intensities. Thus, in order to solve (1), we
must first compute *n*(*t*) in the cloud as a function of both the
microphysics and the time-dependent field.

The motivation for this analysis was a suggestion by *Krehbiel*
\[1998\]. Related to earlier work by *Gunn* \[1954, 1956\], *Krehbiel*
\[1967\] had shown that cloud \"conductivity\" (non-Ohmic in this case)
should become independent of *k*, and that *J*(*t*) should become
constant \-- independent of both *k* and *E*(*t*) \-- when the ambient
field is strong enough. The basic reason for this simplification is that
small ions, drifting through the cloud in response to the ambient field,
are \"swept out\" by the effective cross-section area of the particles
at a rate proportional to field intensity and ion mobility. Since the
conduction-current density is also proportional to these two parameters,
their effects cancel out in strong electric fields.

The present starting point is the steady-state small-ion-budget equation
in a population of stationary, non-interacting, mono-disperse,
spherical, electrically-conducting cloud particles \[*Pruppacher and
Klett*, 1978, Eq. 17-40\]. After neglecting small-ion recombination,
this equation has been simplified to consider only uncharged cloud
particles and has been generalized to account for non-spherical (ice)
particles. The simplification is justified for an ensemble of initially
uncharged particles in air of approximately equal, polar, small-ion
densities by the modeling results of *Griffiths et al*. \[1974\]. The
generalization is based on the well-known analogy between the diffusive
flux to an isolated object and its electrical capacitance, *C* (F)
\[*e.g*., *Pruppacher and Klett*, 1978, Section 13.3.1\] \-- *C* =
2*~0~y* for a sphere of diameter, *y* (m). The \"Einstein relation\"
\[*e.g*., *Pruppacher and Klett*, 1978, Eq. 12-21\] has also been used
to replace the ionic diffusivity with *k* in the second term on the
right-hand side:

*q* ≈ *A~e~kNn*(*t*)*E*(*t*) + (*C*/*~0~*)(*kKT*/*e*)*Nn*(*t*), (2)

The steady-state assumption (that d*n*/d*t* ≈ 0 in the small-ion budget
equation) is valid to the extent that the time scales of interest are
long compared to the small-ion lifetime, *τ~i~* = *n*/*q* (s). An upper
bound can be placed on *τ~i~* by setting the (neglected) ion-ion
recombination-loss rate equal to *q*. At an assumed altitude of 10 km,
this results in *τ~i~* ≈ 140 s, which is certainly negligible from the
point of view of launch safety.

The first term on the right of (2) is the small-ion loss rate due to
field-driven attachment of ions to cloud particles, where *A~e~* (m^2^)
is the effective electrical cross section of a particle \-- *A~e~* =
3*y*^2^/4 for a sphere \-- and *N* (m^-3^) is their volume
concentration. The second term is the diffusive loss rate, where *K* (J
Kelvin^-1^) is the Boltzmann constant and *T* (Kelvin) is the absolute
temperature. Since each of these terms is linear in *N*, the particle
concentration can be replaced with a size distribution, *N*(*y*) (m^-3^
m^-1^) (taken to extend from the smallest ice crystals to the largest
aggregates present in the cloud), both *A~e~* and *C* can be made
functions of *y*, and the right-hand side of (2) can then be integrated
over *y* to obtain the total ion-loss rate. For non-spherical particles,
*A~e~* and *C* also become functions of particle shape. If the particles
can be represented by prolate spheroids, then

![](media/image1.wmf){width="6.440972222222222in"
height="0.6368055555555555in"}

![](media/image2.wmf){width="6.433333333333334in"
height="0.5111111111111111in"}

where *y* is now the length of the major axis (assumed to be oriented by
electrical forces to lie parallel to the electric-field direction) and
*r* is the ratio of major- to minor-axis lengths. Further choices for
particle shape can be found in *Richards et al*. \[1981\]. For example,
if the particles are hexagonal plates (*r* = 21.9 is now the ratio of
width to thickness) oriented edge-on to the field direction, then in the
present notation, *A~e~*(*y*) = 1.35*y*^2^ and *C*(*y*) = 4.52*~0~y*.

![](media/image3.wmf){width="6.497222222222222in"
height="4.009722222222222in"}

> Figure 1 \-- Dependence of dimensionless electrical effective area on
> particle shape for prolate spheroids and for selected shapes from
> *Richards et al*. \[1981\]

Notice that *A~e~* varies approximately quadraticly with the particle\'s
long dimension, *y*, whereas *C* is approximately linear in *y*, these
proportionalities being exact for constant *r*. Thus, field-driven
attachment of ions to particles will dominate diffusive attachment at
the larger sizes. Figure 1 shows the dependence of dimensionless
*A~e~*/*y*^2^ on particle shape (expressed by the ratio of long to short
dimension, *r*). Here the curve applies to prolate spheroids (3), while
the points apply, in order of increasing *r*, to the particle shapes
that have been called column1, bullet, column2, needle, plate (see text
above), and dendrite by *Richards et al*. \[1981\]. The value,
*A~e~*/*y*^2^ = 3/4, for *r* = 1 applies to the sphere. Although *A~e~*
decreases rapidly as *r* increases from 1 to 2, the dependence on shape
(or short dimension, for constant *y*) becomes weak for the highly
elongated particles. The prolate shapes of *Richards et al*. \[1981\]
(column1, bullet, column2, and needle), fit the spheroid formula fairly
well; not surprisingly, the oblate (but edge-on to the field) shapes
(plate and dendrite) do not.

Figure 2 shows the dependence of dimensionless *C*/(*~0~y*) on *r*, with
the curve from Equation 4. The value, *C*/(*~0~y*) = 2, for *r* = 1
applies to the sphere. Most of the comments regarding Figure 1 apply
here as well, except that the dependence of *C* on *r* is never very
strong.

![](media/image4.wmf){width="6.497222222222222in"
height="4.009722222222222in"}

> Figure 2 \-- Dependence of dimensionless capacitance on particle shape
> for prolate spheroids and for selected shapes from *Richards et al*.
> \[1981\]

Given the relatively poor electrical conductivity of ice (a well-known
issue for the ice-ice inductive-electrification mechanism \[*e.g*.,
*Latham and Mason*, 1962\]), it is worth pointing out that cloud
particles can still be considered excellent conductors in the present
context. The steady-state surface-charge density on an isolated body
exposed to an ambient field in air will deviate from that on a perfect
conductor of the same shape in proportion to the ratio of the air\'s
conductivity to that of the body. The bulk conductivity of pure ice at
the low temperatures of cirrus anvils (a worst case, since surface
conductivity, enhanced by impurities, is the dominant mechanism on real
cloud particles \[*Gaskell*, 1981; *Caranti and Illingworth*, 1983\])
can be estimated from the data of *Gross* \[1982\], who measured
relaxation times on the order of 5 ms. If we assume a dielectric
constant for ice of order 100, this bulk conductivity is of order 2 X
10^-7^ S/m. In contrast, the conductivity of the free atmosphere at 10
km altitude (again a worst case, since the cloud conductivity will be
less) is of order 5 X 10^-13^ S/m. Thus, the ratio in question must be
infinitesimal.

The condition of approximately equal diffusive and field-driven ion-loss
rates in (2) can be identified by a value of unity for the dimensionless
parameter, *γ* = *A~e~e~0~E*/(*CKT*), which is essentially the same as
Equation 5 of *Klett* \[1971\]. When *γ* » 1, we are in the \"high-field
limit,\" which will be of most concern here because the decay of
electrification can become quite slow. It is easy to see from (1) and
(2) that d*E*/d*t* ≈ -2*eq*/(*A~e0~N*) in this limit, as pointed out by
*Krehbiel* \[LAP meeting, Tucson, AZ, January, 1998\]. When *γ* « 1, on
the other hand, the \"low-field limit\" obtains, and molecular diffusion
is the dominant loss mechanism (unless the cloud is fairly sparse, in
which case the neglected ion-ion recombination process becomes
significant). This limit yields exponential decay of *E*(*t*) with a
\"diffusive relaxation time,\" *τ~D~* = *CKTN*/(2*e*^2^*q*) (s), which
is valid only if it is much longer than *τ~e~*.

Equation 2 is not quite correct when *γ* ≈ 1 because diffusion of small
ions to a particle tends to diminish the field-driven loss rate to that
particle. *Klett* \[1971\] analyzed this situation theoretically,
however, and concluded that (2) provides a good approximation for
practical purposes.

Here we work directly with (2), instead of focusing on the limiting
cases mentioned above. Since both of the loss terms on the right-hand
side are linear in ion density, it is trivial to solve this equation for
*n*(*t*) and substitute the result into (1) to obtain

d*E*/d*t* = -2*eq*{*~0~N*\[*A~e~* + *CKT*/(*~0~eE*(*t*))\]}^-1^ (5)

where the denominator on the right-hand side is to be regarded as an
integral over the particle size and shape distribution, as discussed
above. (If ion-ion recombination had been retained as a third loss term
in (2), we still could have solved the resulting quadratic equation for
*n*(*t*) and inserted it into (1). This additional complexity is not
justified, however, since recombination is only important when the
electric-field decay time approaches *τ~e~* \-- much shorter than
anything relevant to launch safety.)

[Screening Layers and Other Complications]{.underline}

*Klett* \[1972\] has shown that the thickness of the screening layer
that builds up just inside a stationary cloud boundary perpendicular to
the ambient electric field (in the absence of convection, turbulent
mixing, evaporation, etc.) is *h* ≈ 1.85/(*A~e~N*) (m), and that the
time scale for development of this layer is *τ~s~* ≈ 4*τ~e~*. In order
for the present analysis to be valid, we require *h* « *H*, and *τ~s~*
must be small compared to the time scales of interest. The former will
generally be true in thick and/or dense clouds, and the latter will be
true (in high clouds such as anvils) whenever the field-decay time is
long enough to become a cause for concern. If the screening layer
thickness becomes comparable to, or larger than, the cloud thickness,
however, the field-decay time will be reduced. Thus, the neglect of this
condition is conservative.

The effects of turbulent mixing within the cloud are more difficult to
quantify, but this complication is expected always to accelerate the
decay of electrification, so the estimates made herein may be considered
conservative in this regard as well. Similarly, we expect evaporation,
aggregation, and fall-out of the larger particles all to change the size
distribution in the direction of reducing the electrical-decay time
scale.

Gravitational sedimentation of the cloud particles, on the other hand,
introduces a \"ventilation\" correction to the diffusive loss term in
(2) that increases the loss rate, the more significantly the larger
(hence, the more rapidly falling) the particles \[*e.g., Gunn*,
1954\]**.** This is because the air flow around a falling particle tends
to concentrate the ion-density gradient, especially near its lower
surface, thereby increasing the diffusive flux. Assessment of the
effects of this ventilation correction, if any, will be deferred to the
second and final report on this contract.

Particle ventilation also has an effect on the field-driven loss term,
but in the direction of reducing the ion-loss rate. When the terminal
velocity of an uncharged particle exceeds the ion-drift velocity in a
vertical ambient field, the particle collects ions of one polarity on
its lower surface, while ions of the opposite polarity are convected
away before they can be collected on its upper surface. This is the
well-known *Wilson* \[1929\] selective-ion-capture process. The particle
charges up to a steady-state level at which it is collecting ions of
both polarities at an equal rate. *Whipple and Chalmers* \[1944\] showed
that, for a rapidly falling spherical particle in Stokes flow, the ion
loss rate at equilibrium is just 69% of that if the particle were
motionless (as assumed in Equation 2). This complication will be ignored
here both because the Wilson mechanism does not operate on very many
particles at the high fields of interest to launch safety and because
any resulting error will be on the conservative side.

The most serious complication to the present analysis is the possibility
that an active electrification mechanism might be operating within the
cloud. The treatment of electrification mechanisms is beyond the scope
of this contract. Since this would be likely in a mixed-phase cloud,
however, where riming, vapor deposition, and collisions between ice
particles with different temperatures and/or growth histories can occur,
any evidence of liquid water in an anvil cloud should be cause for
alarm.

[Results from the ABFM June, 2000, Deployment]{.underline}

Equation 5 has been solved numerically in *Mathematica* (TM), after
numerical integration of the denominator over the observed particle-size
distribution, for several passes from the 13 June 2000 flight of the
ABFM. (The *Mathematica* \[TM\] code is available upon request.) The day
to focus on, as well as many of the times chosen here for analysis, were
recommended to the author by Jim Dye \[personal communication, November,
2001\], based on his extensive survey of the experimental program and
his in-depth analysis of this particular storm. The size distributions
used were kindly provided by Jim Dye and Bill Hall of NCAR \[personal
communication, November-December, 2001\]. In the present analysis all
particles are taken to be spherical \-- the most conservative assumption
since it exaggerates both diffusive and electrically driven small-ion
losses, yielding the longest electrical decay time for a given size
spectrum. (This is a reasonable assumption for particles smaller than
about 100 μm, which appear to be mostly frozen cloud droplets \[Jim Dye,
personal communication, November, 2001\], but it could easily be relaxed
as better information on the shapes of the larger particles becomes
available.) The only other required inputs are the ionization rate,
taken as a function of altitude from *Hake et al.* \[1973, Figure 19\],
the absolute temperature from the U.S. Standard Atmosphere, and (for
explicit loss-rate calculations only, *e.g.*, Figures 5 & 10) the
small-ion mobility, scaled in inverse proportion to atmospheric density
from the U.S. Standard Atmosphere. At an altitude of 10 km, for example,
these parameters become *q* = 3.0 X 10^7^ pairs/m^3^/s, *T* = 223 K, and
*k* = 3.3 X 10^-4^ m^2^/V/s.

During June, 2000, the High Volume Particle Sampler (HVPS, which
effectively counts particles larger than roughly 1 mm) was not working
properly, and data from the Cloud Particle Imager (CPI) has yet to be
fully analyzed \[Jim Dye, personal communication, November, 2001\].
Thus, the size distributions used here are restricted to the Forward
Scattering Spectrometer Probe (FSSP \-- covering the smallest particles,
from a few micrometers to about 55 μm) and the Two-Dimensional optical
array Cloud probe (2D-C \-- from about 55 μm to 1 mm, with estimates
from reconstruction of \"cropped\" images to as large as 4 mm, but with
relatively low sensitivity to the largest particles due to limited
sample volume). An example of such a

![](media/image5.wmf){width="6.495833333333334in"
height="4.013888888888889in"}

> Figure 3 \-- Composite Size Distribution from the FSSP and 2D-C
> sensors for the 30 s interval beginning at 210730 on 13 June 2000.

size distribution is given in Figure 3, which applies to a dense anvil
that was penetrated at an altitude of 10.0 km, only about 25 km downwind
from the core of a still-active thunderstorm.

Particles were counted in pre-defined size bins by each instrument, and
this \"bar-graph\" representation was used explicitly in the software
described herein, as reflected in Figure 3. The size spectrum has been
set to zero below the smallest size bin of the FSSP and above the
largest size bin with any counts from the 2D-C. This upper 2D-C cutoff
varied from pass to pass, depending on the concentrations of the largest
particles, since a single count corresponds to 2.3

![](media/image6.png)

> Figure 4 \-- NEXRAD radar \"curtain\" along the flight path during the
> penetration containing the interval, 210730-210800, on 13 June 2000
> \[NCAR ABFM Web page, December, 2001\]. The flight track is shown as a
> near-horizontal black line.

particles/m^3^ in any given bin. (Based on the width of the first empty
bin in Figure 3, this threshold corresponds to a spectral density of 4.5
X 10^3^ particles/m^4^/m centered at a size of 3.2 mm.)

The example in Figure 3 was chosen for illustration because it
represents one of the highest particle concentrations and strongest
radar returns encountered in the massive anvil of 13 June 2000. Thus, it
probably serves as a reasonable \"worst-case scenario\" for the
electrical decay time. Figure 4 shows the NEXRAD radar \"curtain\" along
the flight path during this pass. At the time of the size spectrum shown
in Figure 3, the aircraft was penetrating the upper part of the
maximum-reflectivity region (radar echo greater than 15 dBZ) in a deep
and/or precipitating cloud. The vertical electric-field intensity was
approximately 30 kV/m \[NCAR ABFM Web page, December, 2001\]. (Note that
the values of electric field quoted here and in Table 1 below are not
from the final ABFM calibration matrix and should be regarded as
estimates only.)

![](media/image7.wmf){width="6.498611111111111in"
height="4.013194444444444in"}

> Figure 5 \-- Small-ion loss rates per unit ion density and per unit
> particle size due to field-driven attachment in an assumed field of 1
> kV/m (green) and due to diffusive attachment (red) in the interval,
> 210730-210800, on 13 June 2000. Note that these loss rates have been
> weighted by particle size to compensate for the logarithmic abscissa.

From the particle size distribution in Figure 3 it is easy to calculate
size spectra of the small-ion loss rates that are modeled by the two
terms on the right-hand side of Equation 2. Per unit small-ion density
\-- that is, with *n*(*t*) cancelled out \-- the field-driven-attachment
rate is shown in green, and the diffusive-loss term is shown in red, in
Figure 5. To compute the former, it is necessary to assume an
electric-field intensity, here taken as 1 kV/m. (The only effect of
changing the ambient field is to shift the green curve vertically in
proportion to *E*.) Each of these loss terms has been multiplied by the
particle long dimension, *y*, to compensate for the effect of the
logarithmic size axis. Thus, the units of these size-weighted spectral
loss rates are simply inverse seconds. This weighting is convenient
because larger magnitudes on the graph make larger contributions to the
total ion-loss rate (the integral over particle size).

As expected from the size dependences of *A~e~* and *C*, diffusive loss
dominates at small particle sizes, whereas field-driven loss dominates
at large sizes. The field-dependent cross-over size can be determined
from the condition, *γ* = 1, as discussed

![](media/image8.wmf){width="6.498611111111111in" height="4.0125in"}

> Figure 6 \-- Ambient-field decay in the \"low-field limit.\" The red
> curve is the model result from solution of Equation 5, using the size
> distribution of Figure 3. The green curve is a true exponential decay
> fitted to the points at *E~0~* and *E~0~*/Exp(1) on the red curve.

earlier. Notice that the total diffusive loss rate (the integral of this
term over size) is dominated by particles in the 10-50 μm range, as is
almost always the case in this data set. (This result might change
somewhat, however, when ventilation effects are included.) The total
field-driven loss rate, on the other hand, is dominated by particles
larger than 200 μm. Fortunately, the 2D-C data extends to large enough
sizes to make it clear that this integral is bounded by a rapid decrease
in concentration of particles larger than 2 mm. This may not always be
the case, so HVPS data may be necessary for certain passes.

Returning to Equation 5, we can integrate the two loss terms in the
denominator over particle size and then solve the non-linear,
first-order, ordinary differential equation for *E*(*t*), given the
initial condition, *E~0~*, at *t* = 0. Looking first at the low-field
limit (*E~0~* = 80 V/m), where diffusive loss dominates at all
significant particle sizes, we find the near-exponential decay shown by
the red curve in Figure 6. The time constant of the fitted exponential,
shown in green, is *τ~D~* = 70.4 s, which is long enough compared to
*τ~e~* ≈ 19 s that diffusive attachment clearly dominates the neglected
small-ion recombination in this case.

![](media/image9.wmf){width="6.498611111111111in"
height="4.013194444444444in"}

> Figure 7 \-- Ambient-field decay in the \"high-field limit.\" The
> result from solution of Equation 5 using the size distribution of
> Figure 3.

The time scale indicated by Figure 6 is insignificant from the point of
view of launch safety, but in the high-field limit the electrical decay
is very much slower, as shown in Figure 7. Here the numerical solution
to (5) has been started from an initial ambient field of *E~0~* = 50
kV/m at *t* = 0. As expected from *Krehbiel* \[1967\] and the discussion
above, the field decreases almost linearly to relatively low values.
Extrapolation of this linear decay to *E*(*τ~E~*) = 0 yields a more
relevant time scale, *τ~E~* = 5569 s, which is in excess of 1-1/2 hours!
This is clearly comparable to the waiting times prescribed by the
current LCC, but as we shall see later, it is probably an extreme upper
bound for anvil clouds far down wind from the parent storm.

A closer look at the transition between the high-field and low-field
limits is provided by Figure 8, which shows a blow-up of the region of
Figure 7 near *t* = *τ~E~*. Note that the model field decay, now shown
in red, appears approximately linear down to 2 kV/m, although it
deviates significantly from the linear extrapolation, shown in green,
below about 10 kV/m. The model decay becomes roughly exponential below
300 V/m.

![](media/image10.wmf){width="6.497222222222222in"
height="4.017361111111111in"}

> Figure 8 \-- Expanded version of Figure 7 near *τ~E~* = 5569 s. The
> red curve is the model result; the green curve is a true linear decay
> fitted to the points at *E~0~* and *E~0~*/2 on the model decay.

Seven additional cases from 13 June 2000 have been analyzed in an effort
to span the observed conditions in this anvil cloud. All eight cases are
summarized in Table 1, which lists the following observations at the
aircraft: estimated NEXRAD radar reflectivity and approximate vertical
electric-field magnitude, from the NCAR ABFM Web page \[December,
2001\], and total particle concentrations (given in more conventional
units in this table only) from the FSSP between 3.01 and 54.56 μm, from
the 2D-C between 54.56 μm and 4 mm, and counting only those particles
larger than 960 μm from the 2D-C. Also listed are the model-calculated
electrical-relaxation time constant in the low-field limit, *τ~D~*, and
linear electrical-decay time scale in the high-field limit, *τ~E~*.

Table 1 \-- Summary of observations and calculations for 30 s intervals
starting at the indicated times on 13 June 2000, listed in descending
order of *τ~E~* (last column). The values of electric field quoted here
are not from the final ABFM calibration matrix and should be regarded as
estimates only. The parentheses around most values of *τ~D~* signify
that these numbers are under-estimates because small-ion recombination
losses have been neglected. The asterisk indicates that this value of
*τ~E~* is under-estimated because the 2D-C data do not extend to large
enough particle sizes.

  ------------------------ --------------------- ---------------------- ---------------------- --------------------- --------------------- ------------------- -------------------
  Time                     NEXRAD                Field                  FSSP                   2D-C                  \> 960 μm             *τ~D~*              *τ~E~*

  [(HHMMSS)]{.underline}   [(dBZ)]{.underline}   [(kV/m)]{.underline}   [(#/cc)]{.underline}   [(#/l)]{.underline}   [(#/l)]{.underline}   [(s)]{.underline}   [(s)]{.underline}

                                                                                                                                                               

  210730                   15-20                 30                     23.2                   497                   5.67                  70.4                5569

  215500                   15                    30                     25.0                   449                   5.86                  58.7                4940

  214800                   15                    25                     7.89                   205                   3.43                  (15.4)              2329

  235200                   10                    few                    3.05                   49.4                  0.956                 (15.7)              1426

  225100                   15                    50                     3.74                   51.1                  1.26                  (13.4)              1123**\***

  232030                   5-10                  few                    2.21                   39.9                  0.804                 (8.15)              750

  230030                   0                     near 0                 0.643                  14.2                  0.334                 (2.29)              240

  224100                   \< -10                near 0                 0.565                  29.2                  \< .0023              (1.45)              91
  ------------------------ --------------------- ---------------------- ---------------------- --------------------- --------------------- ------------------- -------------------

In Table 1 it is striking that (except for the interval, 225100-225130)
radar reflectivity, ambient field, nearly all the values of particle
concentration in the three size ranges, and nearly all the values of
*τ~D~* decrease monotonically with decreasing *τ~E~*. Strong
correlations among the observed parameters, notably electric field vs.
radar reflectivity and field vs. particle concentrations, have already
been pointed out by *Dye* \[2001\]. In this report attention is directed
instead toward the apparently strong correlations between computed
*τ~E~* and the various observations in the table. The asterisk on the
value of *τ~E~* for the deviant interval, 225100-225130, indicates that
the size-weighted spectrum of small-ion loss rate due to field-driven
attachment (that plotted green in Figure 5 for a different case) does
not decrease toward the upper 2D-C cutoff. Were HVPS data available in
this case, *τ~E~* would probably be found appreciably larger than
indicated. An anomalously large ambient field was also observed in this
case, probably related to a thick layer of unusually high reflectivity
(\> 25 dBZ) below the aircraft.

One other case in Table 1 deserves detailed comment. At the opposite
extreme from the dense, highly-electrified anvil at 210730-210800 (see
Figures 3-8) is the tenuous, un-electrified case at 224100-224130. In
this interval the aircraft was penetrating an anvil over KSC, some 110
km downstream from the decaying core of the parent storm. An observer at
the Range Operations Control Center characterized this cloud as
\"opaque\" \[Jim Dye, personal communication, November, 2001\], although
it showed no detectable radar return on the NEXRAD. The flight level of
10.6 km might have been above the densest part of this cloud, but
particles were nevertheless detected in all size bins below 960 μm.

![](media/image11.wmf){width="6.497222222222222in"
height="4.016666666666667in"}

> Figure 9 \-- Composite Size Distribution from the FSSP and 2D-C
> sensors for the 30 s interval beginning at 224100 on 13 June 2000.

The size distribution at 224100-224130 is shown in Figure 9. Comparison
with Figure 3 reveals a similar shape between 3.01 and about 500 μm,
although the spectral densities are between one and two orders of
magnitude smaller in the present case. Above 500 μm the spectrum falls
much more steeply here, however, probably because this is a
substantially older cloud from which the largest particles have fallen
out.

The effect of the dramatically steeper size spectrum above 500 μm in
Figure 9 upon the small-ion loss rates is evident in Figure 10.
Comparison with Figure 5 reveals that the field-driven-attachment loss
rate effectively cuts off around 600 μm in the present case, whereas
this same loss rate remains significant up to about 2.5 mm in the
previous example. This difference is partially compensated by a
relatively higher particle concentration in the 50 to 300 μm size range
here. Nevertheless, Table 1 shows that the relative reduction between
these two cases of the high-field electrical-decay time scale, *τ~E~*,
is greater than that of the low-field electrical-relaxation time,
*τ~D~*, by about 26%. Whatever the reason, the 91 s decay time modeled
in this tenuous but \"opaque\" anvil is utterly negligible in the
context of launch safety.

![](media/image12.wmf){width="6.498611111111111in"
height="4.013194444444444in"}

> Figure 10 \-- Weighted small-ion loss rates due to field-driven
> attachment (green) and diffusive attachment (red), as in Figure 5.

[Preliminary Conclusions]{.underline}

Much remains to be done before final conclusions can be presented (in my
second contract report). Most importantly, several cases from the 2001
campaigns that include HVPS spectra must be analyzed to make sure that
the 2D-C is not missing significant numbers of large particles that
could appreciably lengthen *τ~E~*. Ventilation effects on the
diffusive-loss rate must also be estimated to verify that diffusion
remains negligible in the context of launch safety. Finally, with Jim
Dye\'s help I hope to follow the decay of electric field in specific
cloud parcels in an attempt to verify the model predictions against *in
situ* observations.

Other important investigations that are beyond the scope of this
contract include relating the measured particle-size distributions in
anvil clouds (and the corresponding electrical decay rates predicted by
the model) to visual observations and to radar reflectivities. Such
studies are needed to establish one or more ground-based proxy measures
of the electrical decay rates for use during launch operations.
Initially it was expected that optical observations would be the most
relevant in this regard, since the electrical decay time in the
high-field limit seems closely related to the optical extinction
coefficient at visible wavelengths (not measured in the ABFM campaigns)
\[*Krehbiel*, 1998; *Willett*, 2000\]. Nevertheless, the results in
Table 1 above suggest a good correlation between electrical decay time
and 10 cm radar reflectivity. (Data from the 5 cm radar at Patrick AFB
was not available on 13 June 2000.)

This surprising correlation, if valid, might be due to some
characteristic common to all size distributions in anvil clouds. If, for
example, the concentration of particles in the 200 μm to 2.5 mm size
range (those apparently controlling the field-driven-attachment loss of
small ions \-- see Figures 5 & 10) tends to vary with the concentration
of the largest particles in the cloud (those producing the radar
return), then such a correlation would be expected. Further evidence
that particle concentrations tend to be correlated across broad
intervals of the size spectrum can be found by comparing the particle
counts in the three size ranges listed in Table 1, although there are
notable exceptions. This might be a productive topic for further
research.

The main results of the modeling work to date are summarized in Table 1.
Eight cases have been analyzed from the storm on 13 June 2000, spanning
the range of radar reflectivity from greater than 15 dBZ to less than
-10 dBZ. Based on these results, the following tentative conclusions are
offered:

1\) It is possible to explain long electrical decay times (≥ 1.5 hr.) in
terms of measured particle-size spectra in especially dense anvil
clouds. (This conclusion is subject to model validation against *in
situ* observations of electric-field decay. A more accurate
representation of the shapes of the larger particles may be necessary to
accomplish this.)

2\) Diffusive attachment of small ions to cloud particles is not
important, as it is not capable of producing operationally significant
(*i.e.*, long) electrical decay times in anvil clouds. (This conclusion
is subject to assessment of the ventilation correction.)

3\) Field-driven attachment can apparently produce operationally
significant decay times (≥ 30 min.) only in clouds with appreciable
concentrations (≥ 25/l) of particles larger than 200 μm. (This
conclusion is subject to analysis of additional cases with HVPS data.)

4\) The model electrical decay time appears related to the 10 cm radar
reflectivity in anvil clouds, with a threshold of 10 dBZ possibly
separating safe from potentially hazardous cases. (This conclusion
cannot be validated in the present study, however, but requires the
support of further research, as indicated above. The apparent
relationship will remain essentially statistical in the absence of any
accepted theoretical basis.)

[Acknowledgments]{.underline}

I am especially grateful to Jim Dye for an introduction to the
functioning (and malfunctioning) of the various microphysical
instruments, for recommending both the storm and the individual cloud
passes of greatest interest, for providing the size distributions used
in this study, and for many constructive comments on earlier versions of
the manuscript. This work would not have been possible without the
dedicated field, calibration, and analytical efforts of the entire ABFM
team. It was greatly facilitated by the timely presentation of ABFM
results on the NCAR web site. Thanks are also due to Paul Krehbiel for
pointing out the essential physics of the high-field limit and to Phil
Krider for many helpful discussions and suggestions during model
development. Finally, I appreciate the opportunity provided by NASA/KSC
to work on this project.

[List of Symbols]{.underline}

*A~e~* Electrical effective area of a cloud particle

*C* Electrical capacitance of a cloud particle

*E*(*t*) Time-dependent, vertical, electric-field intensity inside the
cloud

*E~0~* Initial electric-field intensity

*e* Charge on the electron

*H* Vertical cloud thickness

*h* Screening-layer thickness

*J*(*t*) Time-dependent, vertical, electric-current density inside the
cloud

*K* Boltzmann constant

*k* Small-ion electrical mobility

*N*(*y*) Cloud-particle volume concentration per unit size

*n*(*t*) Time-dependent, small-ion volume density inside the cloud

*q* Small-ion volume production rate per unit volume

*r* Ratio of longest to shortest dimension of a cloud particle (major-
to minor-axis ratio for a spheroid)

*T* Absolute temperature

*t* Time

*v* Horizontal cloud-advection speed

*x* Horizontal down-stream distance

*y* Longest dimension of a cloud particle (diameter for a sphere)

*γ* Dimensionless ratio of field-driven to diffusive ion-loss rates

*~0~* Dielectric permittivity of free space

*~0~* Initial charge horizontal-area density

*τ~D~* Diffusion-dominated electrical-relaxation time constant in the
low-field limit

*τ~E~* Time scale for field-driven-attachment-dominated linear
electrical decay in the high-field limit

*τ~e~* Electrical-relaxation time constant in the free atmosphere

*τ~i~* Small-ion lifetime

*τ~s~* Screening-layer time constant

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