![Figure 3. Probability of occurrence of the number concentration, n, of homogeneously frozen ice crystals for the color-coded cases shown in Figure 2. The simulation results (stair steps) with homogeneous freezing temperatures near 218K include the effect of preferential freezing. We include the instantaneous statistics without preferential freezing (dashed curves) for each simulation case. Both stochastic and deterministic distributions are compared to the statistic (black curve) derived from aircraft measurements of midlatitude cirrus (Midlatitude Cirrus Properties Experiment [MACPEX]). Total nucleated number concentrations of ice crystals with all sizes (typically > 1𝜇m), averaged over 1 cm/s ≤ w0 ≤ 𝜇w in each simulation case (or calculated analytically from the instantaneous statistics) and from MACPEX for ice crystal sizes > 15𝜇m taken in the temperature range 215–221K, are given in the left legend.](/figures/figure-3-probability-of-occurrence-of-the-number-3825j8ah.webp)
The Impact of Mesoscale Gravity Waves on Homogeneous
Ice Nucleation in Cirrus Clouds
B. Kärcher
1
, E. J. Jensen
2
, and U. Lohmann
3
1
DLR Oberpfaffenhofen, Institut für Physik der Atmosphäre, Wessling, Germany,
2
National Aeronautics and Space
Administration, Ames Research Center, Mountain View, CA, USA,
3
ETH Zürich, Institute of Atmospheric and Climate
Science, Zurich, Switzerland
Abstract Effects of a spectrum of mesoscale gravity waves on homogeneous aerosol freezing in
midlatitude cirrus are studied by means of parcel model simulations that are driven by random vertical
wind speeds constrained by balloon measurements. Stochastic wave forcing with mean updraft speeds of
5–20 cm/s leads to substantial nucleated ice crystal number concentrations (ICNC) of 0.1–1 cm
−3
in
situations with slow large-scale cooling, which by itself would generate fewer ice crystals. The stochastic
nature of wave-driven air parcel temperatures enhances ICNC even further, but the times required to reach
freezing conditions unsupported by large-scale cooling may vary widely. In the presence of wave forcing,
ice crystals with low ICNC (<1–10 L
−1
) are also generated by homogeneous freezing, albeit only rarely.
Comparisons with aircraft measurements suggest significant influences of heterogeneous ice-nucleating
particles and ice crystal sedimentation on ICNC, but quantifying their individual contributions remains
elusive.
Plain Language Summary Spontaneous freezing of airborne, water-containing particles below
−38
◦
C is a fundamental pathway to form ice crystals in high-altitude cirrus clouds. This ice formation
process has been well researched and was the first represented in weather forecast and climate models to
advance cirrus predictions. One key characteristic is its strong dependence of the number of ice crystals
formed on the cooling rate of air. Recent observations show that rapid cooling rates are generated by ubiq-
uitous gravity waves. Here, we explore the rich suite of phenomena taking place during cirrus formation
caused by a spectrum of gravity waves. We find that wave effects should be considered in future model sim-
ulations, when comparing model results with observations, and in parameterizations of cloud ice crystal
formation.
1. Introduction
Mesoscale air motion variability is crucial for the nucleation of ice crystals in cirrus (see Kärcher, 2017a,
and references therein). We refer to cirrus as upper tropospheric ice clouds that form in situ, for exam-
ple, in frontal systems. Superpressure balloon (SPB) measurements at altitudes of 18–21 km directly link
mesoscale vertical wind speed and the associated temperature variability to gravity waves and quantified
spectral properties (Podglajen et al., 2016; Schoeberl et al., 2017). Since these airborne measurement plat-
forms are advected by the wind field, properties of fluctuations derived from them are highly useful for
Lagrangian cloud studies. Dinh et al. (2016) and Jensen et al. (2016) used SPB temperature time series to
drive detailed ice nucleation simulations, focussing on the tropical tropopause layer. Kienast-Sjögren et al.
(2015)—using a Lagrangian microphysical aerosol-cloud model—investigated cirrus over a midlatitude site
combining small-scale vertical wind speeds and temperature fluctuations inferred from radiosonde sound-
ings and a high-resolution weather prediction model. Haag and Kärcher (2004) used results from a global
weather prediction model with superimposed fluctuations to study hemispheric properties of midlatitude
cirrus.
Mesoscale gravity waves are ubiquitous in the upper troposphere and lower stratosphere. SPB measure-
ments show that power spectra of their properties versus intrinsic wave frequency are typically broad and
continuous, consistent with a superposition of many waves. A spectrum of mesoscale gravity waves gener-
ates broad frequency distributions of ice crystal number concentrations (ICNC) via homogeneous freezing
of supercooled aqueous solution particles (Hoyle et al., 2005; Jensen & Pfister, 2004; Kärcher & Ström, 2003).
RESEARCH LETTER
10.1029/2019GL082437
Key Points:
• We present a systematic process
study of effects of a spectrum of
gravity waves on homogeneous
ice nucleation in cirrus through
ensemble simulations
• High cooling rates have
disproportionately large impact
on nucleated ice crystal number
concentrations at low background
updraft speeds
• Analysis of midlatitude continental
cirrus measurements suggests impact
of heterogeneous nucleation and
sedimentation on total ice numbers
Supporting Information:
• Supporting Information S1
Correspondence to:
B. Kärcher,
bernd.kaercher@dlr.de
Citation:
Kärcher, B., Jensen, E. J., & Lohmann,
U. (2019). The impact of mesoscale
gravity waves on homogeneous ice
nucleation in cirrus clouds.
Geophysical Research Letters, 46.
https://doi.org/10.1029/2019GL082437
Received 12 FEB 2019
Accepted 22 APR 2019
©2019. American Geophysical Union.
All Rights Reserved.
KÄRCHER ET AL. 1
Geophysical Research Letters 10.1029/2019GL082437
The random nature of wave perturbations motivates the use of probabilistic methods to describe them. Pro-
cess models have studied aspects of homogeneous freezing using randomized small-scale dynamical forcing
(Dinh et al., 2016; Hoyle et al., 2005; Murphy, 2014; Shi & Liu, 2016).
Faithful applications of cirrus parameterizations in cloud schemes of large-scale models require sound rep-
resentations of unresolved cooling rates (Lohmann & Kärcher, 2002). While the attribution of ice nucleation
mechanisms in cirrus clouds in global climate models is challenging (Dietlicher et al., 2018; Gasparini et al.,
2018) and such models have already begun to parameterize the impact of unresolved gravity waves on cirrus
formation (Penner et al., 2018), it is important to explore this issue systematically on the process level as a
prerequisite to improving ice nucleation parameterizations (Barahona & Nenes, 2008; Kärcher & Lohmann,
2002).
Before enhancing complexity by considering effects of heterogeneous ice-nucleating particles (INPs), which
are poorly constrained by field observations at cirrus temperatures (<230–235 K; Hoose & Möhler, 2012;
Jensen et al., 2018), we focus on the more basic and much better understood homogeneous freezing process.
Field measurements leave little doubt that homogeneous freezing of supercooled aerosol particles is active
in cirrus and responsible for at least the generation of the rightmost tail (
>0.5 cm
−3
) of probability distribu-
tions of ICNC. The main factor limiting INP effects is their low upper tropospheric number concentration
(
<0.1 cm
−3
; DeMott et al., 2010), which makes them inefficient in suppressing homogeneous freezing in
the presence of sufficiently large vertical wind speeds (DeMott et al., 1997). Nonetheless, potent INPs are
capable of modulating or, if sufficiently abundant, dominating cirrus properties according to global model
simulations (Gettelman et al., 2012; Kuebbeler et al., 2014; Penner et al., 2018; Shi & Liu, 2016; Zhou &
Penner, 2014).
We explore systematically the homogeneous freezing process in midlatitude cirrus clouds due to gravity
wave-driven fluctuations of vertical wind speed consistent with Lagrangian SPB measurements. The pre-
scribed wave perturbations represent conditions away from strong local wave sources such as elevated
mountain ridges or deep convective clouds. We evaluate statistically microphysical parcel model simula-
tions forced with a large number of different random realizations of fluctuation time series. We compare our
results with data taken during an extensive airborne field campaign, allowing us to extend previous findings
on factors controlling midlatitude cirrus cloud formation (Jensen et al., 2013). We describe the stochastic
ice nucleation simulations in section 2, analyze the results of tens of thousands of them and compare them
to aircraft observations in section 3, and conclude our work in section 4.
2. Stochastic Simulations
The spectral parcel model prime solves a large set of equations governing the temporal evolution of heat,
water vapor, and supercooled/frozen water during ice nucleation and aqueous aerosol particle and ice crystal
growth due to uptake of water vapor (Kärcher, 2017b). We consider ambient conditions above ice but below
water saturation, where ice crystals can form homogeneously by freezing of liquid solution droplets.
Air parcel temperature, T, is adiabatically perturbed by random vertical wind speed fluctuations, w
′
. These
fluctuations are superimposed onto a constant updraft speed, w
0
≥ 0, driving ice microphysics in the
absence of wave-driven variability. Its prescribed value may be thought of as representative for synoptic
conditions (w
0
< 10 cm/s), orographic forcing (10–100 cm/s and sometimes exceeding 1 m/s), or pertur-
bations induced by convection (
>1 m/s). Values of w
0
greater than 1–10 m/s are found in the detrainment
zones of deep convective clouds, where homogeneous ice formation involves aerosol particles that have
been activated into cloud droplets prior to or along with freezing at temperatures 233–238 K. Presumably,
at these conditions, vertical wind speed fluctuations have different statistical properties (e.g., mean values,
variances, and power spectral densities) than those caused by gravity waves, so the findings reported here
are not directly applicable to such cases. Background wind speeds of (fractions of) millimeters per second
are found in the tropical tropopause layer, where T is lower (<200 K) than in midlatitudes studied here.
A decomposition of updraft speeds has also been employed by Spichtinger and Krämer (2013), represent-
ing fluctuations by a monochromatic sinusodial wave. Investigating the impact of wave phase at which ice
nucleation sets in, Jensen et al. (2010) showed that a spectrum of waves with random phases leads to broad
ICNC distributions. Since the constant updraft w
0
introduced here refers to the large-scale vertical wind
speeds resolved in global models, only the range w
0
< 1–10 c/s is relevant. For higher w
0
values, the separa-
KÄRCHER ET AL. 2
Geophysical Research Letters 10.1029/2019GL082437
tion between constant updraft and wave-driven vertical wind speed fluctuations, w
′
, is somewhat artificial.
Outside of convection, waves are the only source of updraft speeds in excess of
≈10 cm/s and even smaller
values associated with synoptic and planetary waves.
The wave forcing employed here accounts for the observed, double exponential (Laplacian) shape of the
vertical wind speed statistic, L(w
′
)=exp(−|w
′
|∕𝜇
w
)∕(2𝜇
w
). This distribution has zero mean, and 𝜇
w
is
the mean value taken over the one-sided (updraft) statistic, relating to the standard deviation,
𝜎
w
, via
𝜇
w
≡ 2
∫
∞
0
w
′
L(w
′
)dw
′
= 𝜎
w
∕
√
2. The Laplacian approximately fits Lagrangian measurements (Podglajen
et al., 2016) and is used to generate fluctuation time series, w
′
(t), one for each nucleation simulation. The
vertical wind speed fluctuations are autocorrelated over a time t
c
= 2.8 min. Therefore, they are defined
only at discrete times (multiples of t
c
) and were approximated by stair steps for numerical integration. The
power spectrum of w
′
is flat (Figure 1 in Podglajen et al., 2016), meaning that all wave frequencies up to
the Brunt-Väisälä frequency—an upper limit constraining gravity wave propagation—are included in the
forcing with equal weight.
The temperature fluctuations, T
′
, that result from the wind forcing are obtained by advancing the stochastic
differential equation DT
′
∕Dt =−𝛤 w
′
, where D∕Dt denotes the Lagrangian (material) time derivative and
𝛤 ≈ 0.01 K/m is the dry adiabatic lapse rate. Individual w
′
values are sampled randomly from a Laplacian
with prescribed standard deviation
𝜎
w
, or, in terms of adiabatic cooling rates (𝜅), 𝜎
𝜅
= 𝛤𝜎
w
. This approach
replicates the first-order autoregressive model to represent T
′
(t) as proposed by Podglajen et al. (2016) based
on the underlying w
′
measurements. In prime, DT
′
∕Dt is added to adiabatic and diabatic temperature
tendencies, related to w
0
and latent heat exchange, respectively. Based on previous observational evidence,
we choose a cooling rate standard deviation 𝜎
𝜅
= 5 K/hr as a baseline value at cirrus levels. Cooling rate
fluctuations do not vary significantly with altitude within few kilometers around the tropopause (Podglajen
et al., 2016).
Figure 1 shows results from parcel model simulations, each initialized at T
0
= 221 K (air pressure 300 hPa)
and ice saturation ratio S
0
= 1.3. These values are slightly above (≈3 K for T
0
) and below (≈0.2 for S
0
) the
values at freezing and were chosen such that mesoscale temperature variations are capable of triggering
homogeneous freezing and within reasonable time scales (few hours). The initial total number density, mean
dry radius, and geometric standard deviation of lognormally distributed aerosol particles were 500 cm
−3
,
20 nm, and 1.5, respectively. We do not vary aerosol parameters, because the dependence of n on them is
much weaker than on w (Kärcher & Lohmann, 2002; Kay & Wood, 2008; Liu & Shi, 2018). Time steps used in
the simulations are variable, 𝛿t [s]=1∕(w
0
+ |w
′
|)[cm∕s], resolving individual freezing events. In addition,
we imposed t
c
as an upper limit time step to capture each vertical wind speed fluctuation.
With n indicating the number concentration of nucleated ice crystals, the circles in Figure 1 represent all
{n, w
0
} data points resulting from an ensemble of 10, 000 simulations. Each simulation was started with a
random choice for w
0
and a different w
′
(t)-realization (given 𝜎
𝜅
= 5 K/hr or 𝜇
w
≈ 10 cm/s) and terminated
once the nucleated ice crystal number mixing ratio assumed a constant value after the first freezing event.
The quenching of supersaturation right after freezing makes subsequent nucleation events unlikely, except
in cases with very low n; therefore, the restriction of our analysis to first nucleation events is not problematic.
Statistics of first freezing times are presented in the supporting information.
The simulations capture the dependence of nucleated ICNC on the homogeneous freezing temperature, T
⋆
,
which is, however, very weak. We note that across all the cases discussed here, T
*
≈ 218 K with very little
scatter (
±0.25 K), corresponding to a narrow range of ice saturation ratios, S
⋆
≈ 1.5 ± 0.02, characteristic for
homogeneous freezing events (Kärcher & Jensen, 2017). Examining the full upper tropospheric temperature
range would reveal significant variations of T
⋆
and nucleated ICNC.
The data points in Figure 1 exhibit considerable scatter for w
0
< 10–20 cm/s, about 15% lie below n =
10
−3
cm
−3
. These nonpersistent cooling events have been termed temperature limited by Dinh et al. (2016).
Most of these low-n events are generated when w
′
changes its sign before the freezing event is completed.
They happen to occur since the wave-driven vertical wind speed fluctuations contain high-frequency con-
tributions associated with time scales of 5–10 min, comparable to the duration of homogeneous freezing
events (minutes; Jensen et al., 2016). The fraction of nonpersistent cooling events would be lower without
restricting the analysis to first freezing times.
KÄRCHER ET AL. 3
Geophysical Research Letters 10.1029/2019GL082437
Figure 1. Number concentration, n, of homogeneously nucleated ice crystals versus background updraft speed, w
0
,
from ensemble simulations (circles). In each case, gravity wave-induced vertical wind speed fluctuations driving ice
microphysics in the parcel simulations were randomly sampled from a Laplacian distribution with a mean value of
updraft speed (standard deviation of cooling rate) fluctuations
𝜇
w
= 10 cm/s (𝜎
𝜅≈
5 K/hr) added to the constant w
0
.
The arrow points to an excessively high concentration generated by an exceptionally large updraft fluctuation. The
curve was obtained from simulations without fluctuations.
The scatter of n in a given narrow w
0
-range decreases progressively with increasing w
0
. The curve obtained
from simulations without fluctuations follows a power law dependence, as expected from cloud physical
theory (Kärcher & Lohmann, 2002): n(w)=n
1
[(w
0
+ w
′
)∕w
1
]
𝛼
, where n
1
is the nucleated ICNC at a given
T and w = w
1
. A good fit at T = 218 K is obtained for n
1
= 0.185 cm
−3
, w
1
= 10 cm/s, and 𝛼 = 3∕2
(Figure 1). The data collapse onto this curve beyond w
0
= 1 m/s, showing that w
0
governs ice formation
for w
0
≫𝜇
w
(deterministic regime). By contrast, ice formation is entirely controlled by the fluctuations
for w
0
≪𝜇
w
(fluctuation-dominated regime). In the deterministic regime, nonpersistent cooling events no
longer occur; we do not discuss this regime any further with
𝜇
w
= 5 − 20 cm/s, as we are concerned with
nonconvective (in situ) ice formation, where values w
0
< 10 cm/s prevail.
The arrow highlights a very rare data point indicating
≈400 ice crystals per cubic centimeter of air. In this
case, one single freezing event caused by an updraft speed fluctuation of 16.7 m/s (≈10 𝜎
w
) depleted about
80% of the available aerosol particles. Fluctuation amplitudes of comparable magnitude are captured by the
non-Gaussian statistic, L(w
′
). The probability of occurrence of such a value is exp(−10
√
2)≈10
−6
. This
means that in a sample of 10, 000 data points, there is a ≈1% chance of encountering such an event. Filling
the gap between the outlier and the majority of the other data points in Figure 1 requires a much larger
number of simulations.
3. Analysis of Nucleated ICNC
3.1. Expectation Values
We performed two additional sets of simulations halving and doubling the mean vertical wind forcing of
10 cm/s. We calculated in each case moving averages of n over 75 w
0
data points from the full {n, w
0
} data
sets. Figure 2 shows the resulting averaged ICNC,
𝜇
n
(w
0
) (expectation values). As expected from the above,
the nucleated concentrations converge showing decreasing scatter as w
0
enters the deterministic regime.
KÄRCHER ET AL. 4
Geophysical Research Letters 10.1029/2019GL082437
Figure 2. Mean ice crystal number concentration, 𝜇
n
,versusw
0
from ensemble simulations for three cases
𝜎
𝜅
= 2.5, 5, 10 K/hr (red, magenta, and blue), or 𝜇
w
≈ 5, 10, 20 cm/s (𝜎
w
≈ 7.5, 15, 30 cm/s, respectively. The black
curve is obtained from simulations without fluctuations, where ice forms at w
0
. For comparison, the colored smooth
curves mark the mean nucleated concentrations calculated analytically from the updraft speed statistic. The left panel
highlights the transition of simulated
𝜇
n
to vanishing background cooling (w
0
→ 0).
A surprising feature is the increase of 𝜇
n
for low w
0
that is most pronounced for larger forcing. We would
expect the nucleated ICNC to approach a constant value for w
0
≪𝜇
w
, if we just looked at the mean ICNC
calculated directly from the updraft speed fluctuation statistic without accounting for sink processes of the
ICNC. We call those values—shown as smooth colored curves in Figure 2—instantaneous ice numbers
(equation (S4)).
High cooling rates have a greater chance than low cooling rates to reach freezing thresholds,
{T
⋆
, S
⋆
}, giving
them a large statistical weight despite their lower abundance in the updraft distribution. For example, at T
already close close to T
⋆
, while a small cooling rate might not suffice to reach T
⋆
, a high cooling rate might
well trigger freezing. This effect (coined preferential freezing) is stronger the larger
𝜇
w
and significantly
increases
𝜇
n
as w
0
diminishes. With increasing w
0
pushing the parcel faster close to freezing conditions,
low cooling rate fluctuations have an increasing chance to nucleate ice as well. The contribution of the
fluctuations to w is dwarfed by w
0
for w
0
≫𝜇
w
.
This said, we still expect
𝜇
n
to saturate in the limit w
0
→ 0, at which point the air parcels undergo a pure
random temperature (supersaturation) walk unsupported by a constant baseline cooling. The left panel in
Figure 2 extends the simulations to lower w
0
and indeed shows the expected behavior as w
0
→ 0. An imme-
diate implication is that freezing is also possible for slightly negative w
0
causing a background warming.
However, stronger subsidence will make cirrus formation increasingly unlikely due to the rapidly growing
separation between the diminishing ice supersaturation and the homogeneous freezing threshold.
Preferential freezing results from the stochastic nature of the temperature fluctuations. It is absent in esti-
mates of instantaneous values of nucleated ICNC (colored curves), or when the updraft statistic is equal to, or
approaches, a monodisperse distribution. Apart from this, preferential freezing does not depend on the exact
functional form of w
′
statistic, as long as it has a finite second moment. Times at which freezing conditions
are met along a stochastic air parcel trajectory vary widely depending on the actual mesoscale temperature
evolution and hence on how far initial conditions {S
0
, T
0
} are away from {S
⋆
, T
⋆
} (Figure S3). These times
KÄRCHER ET AL. 5