ACPAtmospheric Chemistry and PhysicsACPAtmos. Chem. Phys.1680-7324Copernicus PublicationsGöttingen, Germany10.5194/acp-16-35-2016Effect of gravity wave temperature fluctuations on homogeneous ice nucleation in the tropical tropopause layerDinhT.tdinh@princeton.eduhttps://orcid.org/0000-0002-0144-2762PodglajenA.https://orcid.org/0000-0001-9768-3511HertzogA.https://orcid.org/0000-0001-6778-7117LegrasB.https://orcid.org/0000-0002-3756-7794PlougonvenR.https://orcid.org/0000-0003-3310-8280Program in Atmospheric and Oceanic Sciences, Princeton University, Princeton, New Jersey, USALaboratoire de Météorologie Dynamique, École Polytechnique, Palaiseau, FranceLaboratoire de Météorologie Dynamique, École Normale Supérieure, Paris, FranceT. Dinh (tdinh@princeton.edu)14January201616135463March201524March20155November201524November2015This work is licensed under a Creative Commons Attribution 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/by/3.0/This article is available from https://acp.copernicus.org/articles/16/35/2016/acp-16-35-2016.htmlThe full text article is available as a PDF file from https://acp.copernicus.org/articles/16/35/2016/acp-16-35-2016.pdf
The impact of high-frequency fluctuations of temperature on
homogeneous nucleation of ice crystals in the vicinity of the
tropical tropopause is investigated using a bin microphysics scheme
for air parcels. The imposed temperature fluctuations come from
measurements during isopycnic balloon flights near the tropical
tropopause. The balloons collected data at high frequency,
guaranteeing that gravity wave signals are well resolved.
With the observed temperature time series, the numerical simulations
with homogeneous freezing show a full range of ice number
concentration (INC) as previously observed in the tropical upper
troposphere. In particular, a low INC may be obtained if the gravity
wave perturbations produce a non-persistent cooling rate (even with
large magnitude) such that the absolute change in temperature
remains small during nucleation. This result is explained
analytically by a dependence of the INC on the absolute drop in
temperature (and not on the cooling rate). This work suggests that
homogeneous ice nucleation is not necessarily inconsistent
with observations of low INCs.
Introduction
Cirrus clouds have an important impact on the global radiative energy
budget . In the tropical tropopause layer
TTL;, cirrus clouds contribute to the
radiative heating and control the
dehydration of the air before entry into the stratosphere
. For all cirrus clouds, the
radiative and climate impact, ability to modify water vapour, and
cloud evolution are sensitive to the ice number concentration
e.g., which depends strongly on the nucleation
process of ice crystals.
When evaluating the ice number concentration (INC) produced by
nucleation, it has been often assumed that the relevant timescale is
sufficiently short such that the vertical velocity and associated
adiabatic cooling rate remain constant e.g..
For constant cooling rates, homogeneous freezing of aqueous aerosols
produces higher INCs (>1000 L-1) than those commonly
observed (≲100 L-1) in cirrus clouds
. Observations and
calculations of INC based on homogeneous freezing can be reconciled
only if very low vertical speeds (w<0.01 m s-1) are used
in the simulations. This seems at odds with the ubiquitous presence
of atmospheric gravity waves, which typically generate disturbances an order of
magnitude larger in the vertical velocity. Therefore, it
has been suggested that heterogeneous freezing (instead of homogeneous
freezing) is the dominant nucleation mechanism for cirrus clouds in
the upper troposphere . The INC obtained
by heterogeneous freezing is apparently limited by the availability of
suitable ice nuclei (generally less than 100 L-1) in the
upper troposphere .
However, pointed out that high-frequency variations
in temperature and cooling rates can substantially decrease the INC produced
during homogeneous nucleation compared to those obtained with constant
updraft speeds. However, their numerical results are based on ideally constructed temperature time series
and so remain somewhat conceptual. The present work complements their study
by using temperature time series data collected at high temporal resolution
during long-duration balloon flights near the tropical tropopause. The
observed temperatures contain perturbations from a spectrum of atmospheric
waves, with periods ranging from days to minutes. Our numerical simulations
based on these observed temperature time series confirm the earlier results
of .
In addition to the numerical simulations using realistic temperature time
series (as described above), our contribution is to provide a theoretical
framework for characterising homogeneous nucleation while taking into account
the temperature fluctuations due to gravity waves. The theoretical framework
put forward here complements previous studies see
also, where the effect of
high-frequency temperature fluctuations on ice nucleation has been described
but not explained analytically.
The article is organised as follows. Sections
and describe the balloon data and the technical details of
the model used here to simulate homogeneous ice nucleation.
Section presents the numerical results.
Section provides the theoretical basis explaining how the
fluctuations in time of temperature may affect homogeneous ice nucleation.
Section contains the conclusions.
Balloon data descriptions
The temperature time series used in this study are derived from data
collected by two long-duration, superpressure balloons launched by the
French Space Agency from Seychelles Islands (55.5∘ E,
4.6∘ N) in February 2010 in the framework of the
pre-Concordiasi campaign . The balloons flew at an
altitude of about 19 km and achieved circumterrestrial flights,
therefore sampling the whole equatorial circle. Details on the balloon
trajectories and large-scale atmospheric dynamics during the flights
can be found in . Superpressure balloons are
advected by the wind on isopycnic (constant-density) surfaces and
therefore behave as quasi-Lagrangian tracers of atmospheric motions.
A further remarkable property of superpressure balloons is their
sensitivity to atmospheric gravity waves . The sampling frequency of
the balloon position, atmospheric pressure, and temperature during the
campaign is every 30 s.
Here, we do not use the temperature observations gathered during the
flights to constrain the nucleation simulations; these time series
tend to be both too noisy and warm biased during daytime. Instead, we
infer the temperature disturbances from the balloon vertical
displacements (ζb′).
The isentropic air parcel vertical displacement (ζ′) is
linked to that of the isopycnic balloon through
ζ′=g/cp+∂T‾/∂zg/Ra+∂T‾/∂zζb′, where g is the gravitational acceleration,
cp is the specific heat at constant pressure, Ra is the
gas constant for air, and ∂T‾/∂z is the
vertical gradient of the background temperature. We use the European
Centre for Medium-range Weather Forecasts (ECMWF) operational analyses
to diagnose ∂T‾/∂z at the balloon position
in the above equation. The isentropic vertical displacement is then
converted to the Lagrangian temperature fluctuation (felt by the air parcel)
at the balloon flight level (i.e. in the lower stratosphere) by
TLS′=-gcpζ′.
We must furthermore take into account that the balloons flew in the
lower stratosphere rather than in the upper troposphere where most of
the cirrus form. Because of the difference in stability of these two
regions, the vertical displacements and hence temperature fluctuations
induced by gravity waves are larger in the upper troposphere than in
the lower stratosphere. For conservative wave propagation, it can be shown that
TUT′=NLSNUTexp-Δz2HTLS′,
where NUT and NLS respectively are the buoyancy
frequencies in the upper troposphere and lower stratosphere, Δz is
the difference between the balloon flight and cloud altitudes, H is the
atmospheric scale height (∼6 km in the TTL), and TUT′
is the temperature disturbance in the upper troposphere induced by the gravity wave
packet observed at the balloon altitude. Typically, NLS∼2NUT,
and TUT′∼TLS′ if the cirrus forms 4 km below the
balloon flight level.
The power spectrum of the temperature perturbation (TUT′) time
series derived from the balloon vertical displacements is shown in
Fig. . Notice that the balloon neutral
oscillations due to the flight mechanics have a frequency of
0.25 min-1. Since the spectrum of gravity waves extends up to the
Brunt–Väisälä frequency (typically less than 0.20 min-1 in
the TTL), we expect that the balloon motions do not negatively affect the
quality of gravity waves in the data set. Nevertheless, we applied a
Butterworth band-stop filter to remove
the balloon oscillations from the temperature time series
(Fig. ). We have also experimented filtering the
data using a high cutoff frequency of 0.10 or 0.20 min-1 (not shown).
Our results (Sect. ) are not sensitive to the data
filtering method.
Power spectrum of the raw and filtered temperature perturbation time series
derived from the balloon vertical displacements.
Model configurations
We compute homogeneous freezing of aqueous aerosols following
and depositional growth of ice crystals see
e.g. using the bin scheme designed by .
The formula for the water activity has been revised
following . The saturation water vapour pressure
(over ice) is taken from .
Ice crystals and aerosol particles that form ice crystals are assumed to
be spherical. We use 20 bins to resolve the size distribution of ice
crystals with radii up to 10 µm. The time step used in the
simulations is 0.5 s. The numerical results do not change with more bins
or smaller time step, i.e. the stated bin and time resolutions are
sufficient to ensure accuracy.
The number concentration of the aerosol reservoir is
Na=200 cm-3, and aerosol particles are assumed to
be monodispersed in size with a radius of 0.25 µm.
These assumptions are within observed properties of aerosols in the
upper troposphere . Simulations with polydispersed
aerosols up to 1 µm in size do not show qualitative
differences, and so we retain a monodispersed distribution to simplify
the analytical derivation in Sect. .
In addition, we do not consider ice sedimentation in order to focus
solely on the nucleation process. Further, nucleation is
calculated only for initially ice-free air parcels. The effect of
pre-existing ice on nucleation has been discussed elsewhere see.
Currently, there is not yet a well-constrained limit on the deposition
coefficient (also called accommodation coefficient). The deposition
coefficient controls the number of gas molecules that effectively enter
the condensed phase after a collision with the ice surface. Laboratory
measurements of the deposition coefficient vary by as much as 3 orders
of magnitude, between 0.001 and 1 .
Figure illustrates the effect of varying
the deposition coefficient α on the INC calculated using our model.
For the same constant updraft, the INC obtained by homogeneous nucleation
is smaller for larger α. In the following sections, we first present
the simulations for α=0.1 and then discuss the sensitivity to α
in Sects. and .
INC obtained from homogeneous nucleation at 195 K forced by constant
vertical velocity w for different values of the deposition coefficient α.
Numerical simulations
For adiabatic motions, the effect of pressure variations on the water
vapour mixing ratio (r) can be neglected compared with that due to
temperature variations. Assuming constant air pressure, we prescribe
an initial water vapour content for the air parcels such that
nucleation occurs at a chosen temperature T0. This is possible
because the saturation ratio with respect to ice (S) at the threshold
of nucleation (Snuc) is a function of temperature
, and it is related to the initial
water vapour mixing ratio of air parcels by
r0=esatT0SnucT0pRaRv,
where p is air pressure, esat is the saturation water vapour pressure
over ice, and Ra and Rv are respectively the gas constants of
air and water vapour. The notations esat(T0) and
Snuc(T0)≡S0 refer to respectively esat and
Snuc at T0. Note that up to the nucleation time the
vapour mixing ratio r is conserved (r=r0 for t≤t0). As illustrated
in Fig. , every air parcel follows an isoline of
constant water vapour mixing ratio (r=r0) until crossing the
Snuc(T) curve, at which point (t=t0) nucleation begins.
Diagram illustrating the initial conditions of the air
parcels. Prior to nucleation air parcels follow isolines of water
vapour mixing ratio r (shown here in blue) and approach the
curve Snuc(T) from below (as indicated by the arrows).
Nucleation begins at the intersections of the r isolines with
the curve Snuc(T).
The simulations were first carried out for pressure p=100 hPa,
nucleation temperature T0=195 K, and deposition coefficient
α=0.1 (Sect. , ,
and ). A nucleation event may be formally defined to
start when the rate of nucleation J exceeds a threshold Jε
(J≥Jε) and to end when it becomes less than Jε
(J<Jε). For our simulations, choosing a threshold of
Jε=109 L-1 s-1, we have S0=1.553 for T0=195 K.
Sensitivities to nucleation temperature T0 in the range between 180 and
210 K and α in the range between 0.001 and 1 are discussed in
Sect. and .
Time series of temperature is defined by
T(t)=T‾+T′(t),
where T′(t) are either idealised following temperature
variations associated with constant and time-varying vertical
velocities (Sect. and ),
or taken from the balloon data (Sect. ).
The base temperature T‾ is varied between 180 and
220 K, which allows us to explore a range of saturation ratio with respect
to T‾ (0.8≤S(T‾)≤1.2) for a given nucleation
temperature T0. The results and conclusions presented below, including the
analytical derivation in Sect. , do not depend on the choice
of T‾. The only requirement for T‾ is that it is sufficiently
low to allow for nucleation along the temperature time series, specifically
T‾+T′(t)<T0 for at least some time (t) in the time series.
Constant vertical velocity
Here temperature is set to decrease with time due to adiabatic cooling
at a constant vertical velocity in a hydrostatic background, i.e.
T′(t)=-gcpwt.
For α=0.1, the number of ice crystals nucleated
Ni increases with w if w<1 m s-1 (see Fig. ).
For w≥1 m s-1, all aerosols particles form ice, hence
Ni=Na=200 cm-3. Figure
shows that if the vertical velocity and the cooling rate are constant
during the nucleation events, w must be less than
0.01 m s-1 in order for Ni<100 L-1.
This result is consistent with previous studies e.g.
of homogeneous freezing under constant vertical velocity.
INCs obtained for α=0.1 with constant w
(filled circles), and with w=±0.1 m s-1 (open circles),
see Eq. (). Vapour-limit events are shown in blue
and temperature-limit events are shown in red.
Nonpersistent cooling
Now we vary w with time so that the rate of change of temperature
dTdt is no longer constant with time.
Specifically, we set
w(t)=+0.1ms-1ift-t0≤ts-0.1ms-1ift-t0>ts.
The time ts at which w switches signs is varied by setting
ts={1.10; 2.10; 3.10} min. The dash lines in Fig.
show the evolution of the vertical velocity, temperature, saturation ratio,
and INC during the nucleation events forced by w=±0.1 m s-1
as defined above.
Evolution of vertical velocity (a), temperature (b),
saturation ratio (c), and INC (d) during
nucleation events forced by constant w=0.1 m s-1 (solid)
and by w=±0.1 m s-1 (dash) as defined by Eq. ().
Blue curves show vapour-limit events and red curves show temperature-limit events.
In the event where w switches signs at ts=3.10 min
(blue dash curves in Fig. ), the saturation ratio (S)
reaches a maximum (Smax) at t*=3.05 min, which is before the
minimum temperature (Tmin) is reached (t*<ts).
Here, Smax is controlled by the depletion of water vapour by
depositional growth of ice crystals. The INC in this event is almost the same as
that which would have been obtained if w were kept constant at
0.1 m s-1 (see also Fig. ).
We refer to this event and all cases with constant w as “vapour limit”,
indicating that Ni is limited by the depletion of water vapour.
For the other two events in which w switches signs earlier at ts=1.10
and 2.10 min (red curves in Fig. and red circles
in Fig. ), Ni is significantly smaller than that obtained
for the vapour-limit event described above. For these two events,
Smax and Tmin occur at the same time (t*=ts).
After Smax is reached, S decreases with time because temperature
increases with time. We refer to these events as “temperature limit” because
the minimum temperature determines Smax and hence
Ni. The depletion of water vapour by ice depositional
growth can be neglected because Ni is small.
The numerical results show that homogeneous nucleation may be cut
off if the cooling that initiates nucleation does not persist
sufficiently long into the nucleation events. As a consequence, low
INCs can be obtained for temperature-limit events despite initially high
vertical velocities and cooling rates. The results in this section
are consistent with the simulations with similar setups that have been
carried out previously by .
Balloon temperature time series
In contrast to the previous sections which used theoretically
constructed temperature time series, the numerical simulations
presented in this section were carried out using the balloon data.
Below, for Sect. we use T0=195 K and
α=0.1 (same as previously in Sect.
and ). In Sect.
and , we vary T0 between 180 and
210 K and α between 0.001 and 1 to explore sensitivities to these parameters.
Evolution of vertical velocity (a), temperature (b),
saturation ratio (c), and INC (d) in the period immediately before
and during representative nucleation events forced by the temperature perturbations
derived from the balloon data. The plots are shown for an extended period of time
before nucleation begins at t0 to illustrate the background condition leading to
nucleation. Blue curves show vapour-limit events and red curves show temperature-limit events.
Control simulations with T0=195 K and α=0.1
The evolution of the vertical velocity, temperature, saturation ratio,
and INC for representative nucleation events simulated using the balloon
data is shown in Fig. . The plots in the figure are shown
for an extended period of time before nucleation begins at t0 to illustrate
the background condition leading to nucleation. However, we will show below that
the INC obtained by nucleation is independent of the background condition
prior to nucleation.
Figure shows that the duration τ of the nucleation
events (as formally defined by the period during which J≥Jε) is typically less than 5 min. However, during this short
time period, the cooling rate is
typically not constant because there are high-frequency fluctuations in the
observed vertical velocity and temperature time series. Moreover, more than
one local maximum and minimum in T and S may occur during one nucleation
event. Nevertheless, it is possible to distinguish between
vapour-limit events, for which the absolute maximum
Smax is obtained before the absolute minimum
Tmin because of substantial vapour depletion – constant
cooling rate is a special case of this type; and
temperature-limit events, for which Smax is obtained
at the same time as Tmin; i.e. temperature controls the
cutoff of nucleation, and vapour
depletion is negligible.
As shown in Fig. , the INCs nucleated during
temperature-limit events are typically smaller than for vapour-limit
events. The numerical results suggest that, for all nucleation
events, Ni increases exponentially with the difference
△S≡Smax-S0
as long as Ni≪Na (Fig. ). For
temperature-limit nucleation events, Ni increases
exponentially with |△T|, where
△T≡Tmin-T0.
In Sect. , we will prove analytically that the INC obtained
by nucleation (Ni) is indeed a function of △S (or a function
of △T for temperature-limit events). These two quantities
(△S and △T) are characterisation of the fluctuations in water
vapour and temperature during nucleation; they are defined exclusively within the
period of nucleation and are independent of the background condition prior to nucleation.
Number of ice crystals nucleated at T0=195 K for α=0.1
using the balloon temperature perturbation time series. Blue circles show
vapour-limit nucleation events. Red circles show temperature-limit
nucleation events. The solid curves are obtained from Eqs. ()–()
with μ=0.05 s-1.
Sensitivity of INC to nucleation temperature
Here, we prescribe the initial vapour content r0 of the air parcels such
that the nucleation temperature T0 is either 180 or 210 K. In
Fig. , this is equivalent to choosing another isoline of
r and displacing accordingly the values of T0 and S0 at
nucleation.
Same as Fig. (α=0.1) but for T0=180 K (filled
circles) and 210 K (empty circles).
The number of ice crystals nucleated for T0=180 and 210 K is
shown in Fig. . The data for T0=195 K
shown previously in Fig. generally lie between the
data points for T0=180 and 210 K; that is, there is
a monotonic relationship between Ni and T0. For the
same △S, Ni is smaller for smaller T0.
Conversely, for the same △T, Ni is smaller for
larger T0.
Sensitivity of INC to deposition coefficient
The number of ice crystals nucleated at T0=195 K for
α=0.001 and α=1 is shown in Fig. .
Notice that the transition from temperature-limit events to vapour-limit
events occurs at lower INC for α=1 than α=0.001. This makes
sense because ice crystals deplete water vapour at a faster rate in
the case α=1, and so the number of ice crystals needed to
significantly deplete water vapour is smaller.
For temperature-limit events, the functional dependence of
Ni on △S (or △T) is invariant for
different values of α, i.e. Ni is independent of
α. However, for vapour-limit events, Ni is smaller
for α=1 than α=0.001 for the same △S (or
△T). The sensitivity of vapour-limit events to the
deposition coefficient is explained in the theory section below.
Theory and discussions
In this section we provide the theoretical basis that explains the
numerical results shown previously in Sect. .
Formula for ice number concentration
The rate of nucleation of ice crystals during a nucleation event is given by
dNdt=Na-NJVa,
where Na is the aerosol particle number concentration,
Va is the volume of each aerosol particle, and J is the homogeneous
nucleation rate given by their Eq. 7. By
integrating Eq. () from the beginning (t=t0) to end
(t=t0+τ) of the nucleation event we obtain
ln1-NiNa=-Va∫t0t0+τJdt=-VaJmax∫t0t0+τexpln(J)-lnJmaxdt,
where Jmax≡J(t*) is the maximum value of J
during the nucleation event (t0<t*<t0+τ), and
Ni≡N(t0+τ) is the INC obtained at the end of the
nucleation event. Following the steepest descent method, we obtain
ln1-NiNa≈-VaJmax∫t0t0+τexp12d2(lnJ)dt2t*t-t*2dt≈-VaJmax∫t0-t*t0+τ-t*exp-μ2t2dt≈-VaJmax∫-∞∞exp-μ2t2dt≈πVaJmaxμ,
where
μ2=-12d2(lnJ)dt2t*=-12Jmaxd2Jdt2t*.
The approximations used to derive Eq. () are appropriate if
t*-t0 and t0+τ-t* are both significantly larger than
the e-folding timescale given by μ-1. These criteria are well
satisfied in our simulations. From Eq. () we obtain
Ni≈Na1-exp-πVaJmaxμ.
Same as Fig. (T0=195 K) but for α=0.001
(filled circles) and α=1 (empty circles).
For homogeneous ice nucleation, J is given by seelog10(J)=P3(S-1)awi,
where P3 denotes a third-order polynomial, and awi is the
water activity of a solution in equilibrium with ice, which is independent of
the nature of the solute . It follows that
log10Jmax=P3Smax-1awiTt*≈P3S0+△S-1awiT0,
where △S is the change in the saturation ratio during the
nucleation event defined in Eq. (). Since awi
and S0 are both functions of temperature, Jmax is
a function of △S and temperature. Therefore,
Eqs. () and () indicate that Ni is
a function of △S, μ, and temperature. However, note
that △S, μ, and temperature are not exclusively
independent variables. In fact, substituting Eqs. ()
and () into Eq. () we obtain
μ2≈f△S,T0d2Sdt2t*+h△S,T0d2Tdt2t*,
where f(△S, T0) and h(△S, T0) are functions
of △S and T0, and we have made the approximation that
T≈T0 because the perturbation T′ is small compared with T and
T0. Equation () indicates that μ is a function
of △S, T0, and the second-order time derivatives of S and
T evaluated at t*.
For the nucleation events at T0=195 K shown in Fig. ,
our calculations indicate that 0.01<μ<0.1 s-1. From
Eq. () we deduce that the large range of Ni
(10-3 to 106 L-1) obtained for these nucleation events must
be due to a large range in Jmax. If the differences in μ among
the nucleation events can be ignored, at a chosen temperature Ni
depends solely on Jmax, which depends solely on △S. In
fact, setting μ=0.05 s-1 and T0=195 K in Eqs. ()
and () we obtain a functional dependence of Ni on
△S (the solid curve in Fig. a) that fits the
numerical data well. The error that results from assuming constant μ is
further discussed in Sect. .
For the special case of a temperature-limit event, the partial pressure of
water vapour can be approximated as constant during the nucleation event for
t0<t<t0+τ, and so
△S≈-S0LsRvT02△T,
where △T is the change in temperature during the nucleation event
defined in Eq. (), Ls is the latent heat of
sublimation, and Rv is the gas constant of water vapour. With
μ=0.05 s-1 and T0=195 K, from
Eqs. ()–() we obtain the solid curve in
Fig. b that captures the dependence of Ni on
△T as suggested by the simulations of temperature-limit events.
Sensitivity of INC to nucleation temperature and deposition coefficient
Using the formulae derived in Sect. we can now explain
the sensitivity of the numerical results to T0 and the deposition
coefficient α. The analytic functions of
Ni-versus-△S and Ni-versus-△T
vary with T0 (because awi depends on T0, recall
Eq. ). If μ is assumed constant, the analytic functions are
independent of the deposition coefficient α. As described further
below, the assumption of constant μ gives consistent result with the
numerical data for temperature-limit events but tends to overestimate INCs
for vapour-limit events at larger T0 and/or larger α.
For α=0.1, Fig. a shows that the analytic
function of Ni-versus-△S is consistent with the
numerical data, except for vapour-limit events at T0=210 K that produce
more than 104 L-1 ice crystals. This error arises because μ
has been assumed to be constant (μ=0.05 s-1) and independent of
△S in the calculation of the analytic curve. The error is larger
for larger temperature.
For α=1, the analytic function of Ni-versus-△S
also overestimates Ni for vapour-limit events
(Fig. a). We again attribute this error to the
assumption that μ is constant over the shown range of △S. The
deposition coefficient governs the growth rate of ice crystals and affects
how the saturation ratio changes with time and how μ changes with
△S (a consequence of Eq. ). Our calculation
indicates that the rate of change of μ with respect to △S
increases with α. For larger values of α, calculation of
Ni (especially for vapour-limit events) must account for the
variations in μ as △S varies.
Let us now study the variation of Ni with respect to △T
(Figs. b and b). Recall
that the function of Ni with respect to △T is derived
by neglecting the depletion of water vapour due to ice depositional growth
(see Eq. ). For all values of T0 and α tested here,
the analytic function with constant μ explains well the pattern of
Ni-versus-△T for temperature-limit events. For
vapour-limit events, the analytic curves in Figs. b
and b overestimate the numerical solution because of
(i) the neglect of water vapour depletion in Eq. () and (ii) the
assumption of constant μ.
Evolution of temperature (a), saturation ratio (b),
and INC (c) for three air parcels with slightly different initial water vapour
mixing ratios: {rA=1.78; rB=1.80; rC=1.82}× 10-5 kg kg-1.
The parcels follow the same temperature time series as shown
in (a), but they begin nucleation at different times (indicated by the dash lines)
and end up with widely different INCs.
Dependence of INC on the initial water vapour mixing ratio
The temperature time series T(t) along the trajectory of an air parcel
(recall Eq. ) and the initial water vapour
content r0 of the parcel are two independent conditions to be specified
for the simulations. The initial water vapour content r0 has a
one-to-one relationship with the temperature at the threshold of nucleation
T0 via Eq. (). In Sect. we have
studied how the INC varies with the various forms of T(t) for a
givenr0 and a corresponding T0. Here,
however, we discuss how the INC
varies as r0 and T0 vary for a givenT(t).
Now, consider air parcels with slightly different initial water vapour
mixing ratios: r0 and r0+δr0. The nucleation
temperatures for these air parcels are respectively T0 and
T0+δT0 (see illustration in Figs.
and ). For constant pressure, δr0 and
δT0 are related by
δr0r0=δesatesat+1S0dS0dT0δT0=LsRvT02δT0+1S0dS0dT0δT0
by Eq. () and the Clausius–Clapeyron relation. The first
term dominates the right-hand side of Eq. (), from which we obtain
dT0dr0≈RvT02Lsr0,
which indicates that T0 increases monotonically with r0. For
a given temperature time series T(t), the minimum temperature
Tmin experienced by the parcels is the same (see
Fig. ). It follows that |△T|=T0-Tmin increases monotonically with r0. For
temperature-limit events, Ni increases exponentially with
|△T| (recall Fig. and Eq. ),
and so it must increase exponentially with r0. As r0 increases,
Ni increases until reaching a limit above which the
nucleation event must be vapour limit (see e.g. Fig. ).
Thus, for a given temperature time series,
r0 controls Ni and also determines whether the
nucleation event is temperature or vapour limit.
For example, consider a temperature time series defined by a cooling rate
associated with w=+0.1 m s-1 between t=0 and ts=5 min,
and a warming rate associated with w=-0.1 m s-1 after ts
(see Fig. ). This temperature time series is similar
to the profiles we have studied earlier in Sect. .
Consider three air parcels following this temperature time series, but for which
r0={1.78; 1.80; 1.82}× 10-5 kg kg-1. All three
air parcels experience nucleation, and in all cases
Tmin=194.71 K occurs during the nucleation periods.
However, our calculations give T0={194.76; 194.83; 194.90} K and
Ni={1.4× 101; 1.7 × 103;
2.7 × 104} L-1 respectively for the three parcels. Moreover, the two drier air
parcels experience temperature-limit nucleation events (red lines in Fig. ),
whereas the moist air parcel experiences a vapour-limit event (blue line in Fig. ).
As illustrated here, small differences in r0 result in many orders of magnitude
changes in Ni. Such a strong dependence of Ni
on r0 could explain the large-amplitude, small-scale
heterogeneities in the INC as observed in cirrus clouds by .
Conclusions
We have simulated homogeneous ice nucleation using temperature time
series data collected at high frequency by long-duration balloon
flights near the tropical tropopause. The simulated nucleation events
can be conceptually categorised as either vapour limit or
temperature limit. For vapour-limit events, nucleation is limited by
the depletion of water vapour. In contrast, for temperature-limit
events, nucleation is controlled by the fluctuations in temperature
(while the depletion in water vapour is negligible). The INC obtained
for temperature-limit events is typically smaller than that obtained for
vapour-limit events.
Our calculations of temperature-limit events confirm the finding by
that high-frequency fluctuations in
temperature may limit the INC obtained by homogeneous freezing.
Indeed, a small INC is obtained if the gravity waves produce large but
non-persistent cooling rates such that the absolute drop in
temperature (i.e. the difference between the temperature at the
threshold of nucleation and the minimum temperature obtained during
nucleation) remains small. This relationship between the INC and
temperature has been illustrated here both numerically and analytically.
In addition to the fluctuations in temperature, small variations in
the initial water vapour content of the air parcels can also lead to
large variations in the INC obtained by nucleation. Moreover,
post-nucleation processes acting during the cirrus life cycle
contribute to modify the cloud original characteristics. Simulations
of cirrus clouds in the TTL by show that
the INC decreases by several orders of magnitude as the cloud ages.
For these reasons, we suggest that homogeneous ice nucleation (even
acting alone in the absence of heterogeneous freezing) is not
inconsistent with recent observations of cirrus clouds in the TTL
that indicate generally low but highly variable INC .
Finally, it is encouraging that the INC for temperature-limit events
does not depend on the deposition coefficient, a parameter still
poorly constrained by theoretical understanding as well as laboratory
measurements and field observations.
Acknowledgements
The data used for simulations in this work were collected during the
project “Concordiasi,” which is supported by the following
agencies: Météo-France, CNES, CNRS/INSU, NSF, NCAR, University
of Wyoming, Purdue University, University of Colorado, Alfred
Wegener Institute, Met Office, and ECMWF. Concordiasi also
benefited from the logistic and financial support of the Institut
polaire français Paul Emile Victor (IPEV), Programma Nazionale
di Ricerche in Antartide (PNRA), United States Antarctic Program (USAP),
British Antarctic Survey (BAS), and from measurements by the
Baseline Surface Radiation Network (BSRN) at Concordia.
Tra Dinh acknowledges support from the NOAA Climate and Global
Change Postdoctoral Fellowship Program and NSF grant AGS-1417659.
This collaborative research emerged from Tra Dinh's visit to the
Laboratoire de Météorologie Dynamique, which was supported
by the “Tropical Cirrus” project of École Polytechnique's
“Chaire pour le Développement Durable”. Aurélien Podglajen,
Albert Hertzog, Bernard Legras, and Riwal Plougonven received support
from the ANR project “Stradyvarius” (ANR-13-BS06-0011-01).
Additional support was provided by the EU 7th Framework Program under grant 603557 (StratoClim).
The authors would like to thank three anonymous reviewers and
Martina Krämer, Bernd Kärcher, and Daniel Knopf for helpful
questions and comments that led to significant improvements of this work.
Edited by: M. Krämer
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