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Interactions between Gravity Waves and Cirrus Clouds: Asymptotic Modeling of Wave-Induced Ice Nucleation

Stamen I. Dolaptchiev aInstitut für Atmosphäre und Umwelt, Goethe-Universität Frankfurt, Frankfurt, Germany

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Peter Spichtinger bJohannes Gutenberg-Universität Mainz, Mainz, Germany

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Manuel Baumgartner bJohannes Gutenberg-Universität Mainz, Mainz, Germany

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Ulrich Achatz aInstitut für Atmosphäre und Umwelt, Goethe-Universität Frankfurt, Frankfurt, Germany

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Online Publication:
08 Dec 2023
Print Publication:
01 Dec 2023
Collections:
Multi-Scale Dynamics of Gravity Waves (MS-GWaves)
DOI:
https://doi.org/10.1175/JAS-D-22-0234.1
Page(s):
2861–2879
Free access

Abstract

We present an asymptotic approach for the systematic investigation of the effect of gravity waves (GWs) on ice clouds formed through homogeneous nucleation. In particular, we consider high- and midfrequency GWs in the tropopause region driving the formation of ice clouds, modeled with a double-moment bulk ice microphysics scheme. The asymptotic approach allows for identifying reduced equations for self-consistent description of the ice dynamics forced by GWs including the effects of diffusional growth and nucleation of ice crystals. Further, corresponding analytical solutions for a monochromatic GW are derived under a single-parcel approximation. The results provide a simple expression for the nucleated number of ice crystals in a nucleation event. It is demonstrated that the asymptotic solutions capture the dynamics of the full ice model and accurately predict the nucleated ice crystal number. The present approach is extended to allow for superposition of GWs, as well as for variable ice crystal mean mass in the deposition. Implications of the results for an improved representation of GW variability in cirrus parameterizations are discussed.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Multi-Scale Dynamics of Gravity Waves (MS-GWaves) Special Collection.

Corresponding author: Stamen Dolaptchiev, dolaptchiev@iau.uni-frankfurt.de

Abstract

We present an asymptotic approach for the systematic investigation of the effect of gravity waves (GWs) on ice clouds formed through homogeneous nucleation. In particular, we consider high- and midfrequency GWs in the tropopause region driving the formation of ice clouds, modeled with a double-moment bulk ice microphysics scheme. The asymptotic approach allows for identifying reduced equations for self-consistent description of the ice dynamics forced by GWs including the effects of diffusional growth and nucleation of ice crystals. Further, corresponding analytical solutions for a monochromatic GW are derived under a single-parcel approximation. The results provide a simple expression for the nucleated number of ice crystals in a nucleation event. It is demonstrated that the asymptotic solutions capture the dynamics of the full ice model and accurately predict the nucleated ice crystal number. The present approach is extended to allow for superposition of GWs, as well as for variable ice crystal mean mass in the deposition. Implications of the results for an improved representation of GW variability in cirrus parameterizations are discussed.

© 2023 American Meteorological Society. This published article is licensed under the terms of the default AMS reuse license. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

This article is included in the Multi-Scale Dynamics of Gravity Waves (MS-GWaves) Special Collection.

Corresponding author: Stamen Dolaptchiev, dolaptchiev@iau.uni-frankfurt.de
Keywords: Gravity waves; Cirrus clouds; Cloud parameterizations

1. Introduction

Clouds consisting exclusively of ice particles, so-called cirrus clouds, account for roughly one-third of the total cloud cover (e.g., Gasparini et al. 2018), yet their net radiative effect is still one major source of uncertainty in the climate system. Since the albedo effect and greenhouse effect are on the same order of magnitude for those clouds, microphysical details of ice crystals (as, e.g., shape or size; see Zhang et al. 1999; Krämer et al. 2020) may determine the net radiative effect. The microphysical properties, however, are strongly influenced by a complex interplay of nuclei composition, microscale cloud processes, and multiscale interactions with the surrounding atmosphere. All those components are poorly understood and, if at all, only crudely represented in climate models.

Cirrus clouds can be subdivided into liquid origin and in situ cirrus (e.g., Krämer et al. 2016). The former class describes clouds originating from cloud droplets, which freeze in upward motions, e.g., in mesoscale convective outflow or warm conveyor belts. In contrast, the ice crystals of in situ cirrus are formed without any preexisting cloud droplets: either by homogeneous freezing of aqueous solution droplets (short: homogeneous nucleation; see, e.g., Koop et al. 2000; Baumgartner et al. 2022), or by heterogeneous nucleation (e.g., Pruppacher and Klett 2010; Hoose and Möhler 2012; Baumgartner et al. 2022) initiated by solid aerosol particles.

Observational studies indicate that cirrus properties and life cycle can be crucially affected by gravity wave (GW) dynamics (e.g., Kärcher and Ström 2003; Kim et al. 2016; Bramberger et al. 2022). GWs are generated to a large fraction in the troposphere and often propagate over considerable horizontal and vertical distances before breaking. During their propagation those waves can generate substantial oscillations in the atmospheric fields and the GW drag, exerted in the breaking region, alters the mean atmospheric state. Because of the importance of small-scale GW dynamics some cirrus studies explicitly resolve the GWs using LES models (e.g., Joos et al. 2009; Kienast-Sjögren et al. 2013), or detailed parcel models (e.g., Haag and Kärcher 2004; Jensen and Pfister 2004; Spichtinger and Krämer 2013).

There are several classes of schemes for modeling the influence of GWs on ice clouds in climate models. In all schemes, a subgrid-scale GW vertical velocity is diagnosed and then directly used in the cirrus cloud scheme. One should keep in mind that climate models diagnose the number concentration of ice crystals in a (homogeneous) nucleation event from the vertical velocity (see, e.g., Kärcher and Lohmann 2002; Ren and Mackenzie 2005; Wang and Penner 2010). First, there are schemes without any physical constraint on GWs. These schemes mostly rely on the use of a turbulent kinetic energy (TKE) scheme in the upper troposphere. This approach is per se questionable since most TKE schemes were developed for parameterizing turbulence in the planetary boundary layer. The TKE approach is known to produce quite high vertical velocities (see, e.g., Joos et al. 2008; Zhou et al. 2016) and the pattern of enhanced vertical velocities usually do not agree with regions of enhanced GW activity (see Fig. 5 in Joos et al. 2008). Second, there are schemes using distributions of temperature fluctuations constructed from measurements. From these subgrid-scale vertical velocities are derived (Kärcher and Burkhardt 2008; Wang and Penner 2010; Podglajen et al. 2016; Kärcher and Podglajen 2019). In this approach it is not possible to isolate solely the contribution from GWs. Rather than this, one is considering the combined effect of GWs and other fluctuations (e.g., turbulence), which might modify or even mask the GW contribution. In addition, there is no direct link to the GW sources, such as mountains, convection, spontaneous imbalance, or others. One should also keep in mind that the measurements are sparse and they are largely extrapolated into other regions without any measurements. Further, it is not clear a priori that the underlying statistical description of temperature fluctuations will remain unaltered under climate change. Finally, there have been two attempts to diagnose the GW vertical velocity using linear theory for mountain waves (Dean et al. 2007; Joos et al. 2008). In both schemes there is a direct connection between the source of GWs and the cloud scheme, which is a clear advantage in comparison to schemes relying on statistical information. However, no other sources than mountain waves are represented up to now.

Most of the current GW parameterizations in climate models rely on the single-column, steady-state approximation. Under this assumption GWs propagate only in the vertical and instantly fast up to the breaking altitude, where they deposit energy and momentum. The limitations of steady-state parameterizations were demonstrated in the study by Bölöni et al. (2016), where a transient approach was proposed. The new transient parameterization was implemented by Bölöni et al. (2021) in the weather-forecast and climate model ICON and Kim et al. (2021) showed that the resulting intermittency patterns of convectively generated GWs are similar to observations. Thus, developing a cirrus scheme to be coupled to a transient GW parameterization is a promising route for more realistic representation of ice clouds in climate models. Such development requires the systematic identification of the dominant interaction processes between GWs and cirrus and their self-consistent description. Baumgartner and Spichtinger (2019, hereafter BS19), utilized a matched-asymptotic approach [e.g., see Holmes (2013) for an introduction to asymptotics] for studying homogeneous nucleation due to constant updraft velocities. The resulting parameterization successfully reproduces the results of the classical scheme of Kärcher and Lohmann (2002). Encouraged by the results of BS19, we extend their asymptotic approach to allow for GW dynamics. We construct a self-consistent simplified model for GW–cirrus interactions and corresponding asymptotic solutions applicable for diagnosing ice crystal numbers in nucleation events forced by passing GWs. An application of our analytical approach would be a direct coupling of the transient GW parameterization (Bölöni et al. 2021; Kim et al. 2021) to our analytical model. The GW parameterization will provide information about the wave amplitudes, frequencies, and wavenumbers, which can directly be used for our approach. The detailed information on the wave spectrum allows one to predict the ice crystal number concentration more realistically than simple diagnostic relations in large-scale models (e.g., Kärcher and Lohmann 2002), which are based on constant vertical updraft motion.

This paper is organized as follows: The unified asymptotic representation of the GW and ice microphysics can be found in section 2. In section 3 we derive asymptotic solutions, modeling the dynamics during a nucleation event as well as the pre- and postnucleation dynamics. The reduced model for the GW–cirrus interactions and the corresponding asymptotic solutions are summarized in sections 3g and 3h, respectively. The numerical simulations of the full ice physics model and validation of the asymptotic solutions can be found in section 4. In section 5 the present approach is extended to take into account variations of the (ice crystal distribution) mean mass in the deposition. In section 6 the asymptotic solution is extended to the case of multiple GWs driving the ice physics. Concluding discussions are summarized in section 7.

2. Asymptotic approach for studying GW–cirrus interactions

a. Gravity wave dynamics: Governing equations and scalings

We start with the equations governing a compressible flow on an f plane (e.g., Durran 1989), without diabatic and frictional sources
DuDt+fez×u=−cpθ∇hπ,
DwDt=−cpθ∂π∂z−g,
DθDt=0,
DπDt+Rπcυ∇⋅v=0.
Here, the total wind vector v is separated into a horizontal (u) and vertical (w) component, g denotes the gravitational acceleration, f the Coriolis parameter, and D/Dt the material derivative. In addition, cp and cυ are the specific heat capacities of dry air at constant pressure and volume, respectively, and the ideal gas constant is given by R = cp − cυ. Further, θ denotes the potential temperature based on constant cp (Baumgartner et al. 2020). The Exner pressure π is related to the pressure p by π=(p/p00)R/cp, where p00 is some reference pressure. The ideal gas law p/ρ = RT is assumed to be valid, where ρ is density and T temperature. We consider a hydrostatically balanced reference atmosphere at rest with pressure scale height Hp, and potential temperature scale height Hθ, which depend on altitude and are defined by
Hp−1=|1p¯dp¯dz|=gRT¯,Hθ−1=1θ¯dθ¯dz=1T¯(dT¯dz+gcp).
In the last equations variables with an overbar refer to the reference atmospheric fields.

Within the framework of multiscale asymptotics, we have to specify a distinguished limit in order to define the regimes we are interested in. This is carried out in the following way:

First, we allow for weak and moderately strong stratification appropriate for the dynamics in the upper troposphere and lower stratosphere. Following Achatz et al. (2017), the different stratifications can be expressed using the ratio
HpHθ=εα, whereε=O(10−1) and α=0,1.
In the equation above α = 0 corresponds to the strong and α = 1 to the weak stratification case.
Second, we specify the frequency regime of the GWs. In the analyses here the high- as well as midfrequency GWs are considered. Using the Brunt–Väisälä frequency N=(g/θ¯)(dθ¯/dz) the reference GW time scale Tw is defined as
Tw=1εβN,β=0,1,
where β = 0 characterizes the high-frequency and β = 1 the midfrequency GW time scale. The corresponding GW period Pw is given by Pw = 2πTw. For the vertical length scale of the GW (Hw) we assume (Achatz et al. 2010, 2017)
Hw=εHp .
The appropriate horizontal length scale Lw is estimated using the inertial GW dispersion relation (e.g., Achatz 2022)
ω^2=f2m2+N2kh2kh2+m2,
with ω^ the intrinsic GW frequency, m the vertical wavenumber, and kh the magnitude of the horizontal wave vector. By setting f/N=O(ε(5−α)/2), ω^=1/Tw, m = 1/Hw, and kh = 1/Lw, one obtains the estimate
Lw=ε−βHw.
Since the aspect ratio Hw/Lw defines the anisotropy of the GW, the high- and midfrequency GWs correspond to isotropic and moderately anisotropic waves, respectively. The reference quantities for the horizontal wave velocity scale U and for the vertical wave velocity scale W are estimated using the advection velocities
U=LwTw,W=HwTw.
As shown in Achatz et al. (2017) the above expressions are consistent with the polarization relations for GWs if the mean flow entering the Doppler term is not larger than U. By introducing a reference temperature T00 such that
θ¯=O(T00)
and using T¯=O(T00), one arrives at the expressions
Hp=O(RT00g) ,U=O(ε(2+α)/2RT00),W=O(ε(2+α+2β)/2RT00).
Using the above scaling for W, we allow for strong and moderate vertical velocities: these are on the order of U in the case of high-frequency GWs and on the order of εU for the midfrequency GWs. The scalings presented in this section imply the following distinguished limit for the Mach, Froude, and Rossby numbers:
Ma=URT00∼ε(2+α)/2,Fr=UNHp∼ε,Ro=UfLw∼ε(2β+α−5)/2.
The magnitude of the buoyancy GW fluctuations (Bw) is set to the one associated with GWs close to breaking due to static instability (Achatz et al. 2010), namely,
Bw=N2Hw=εα+1g.
From the buoyancy definition b=g(θ−θ¯)/θ¯ one obtains for the magnitude of GW potential temperature fluctuations
Θw=εα+1T00.
Finally, as shown in Achatz et al. (2017) from the polarization GW relations, the GW Exner pressure fluctuations scale as
Πw∼imω^2−N2N2Bwcpθ¯=O(ε2+α) .
Nondimensionalizing the governing Eqs. (1)–(4) with the reference quantities from Table 1 and replacing
(x,y,z,t,∇h)→(Lwxw,Lwyw,Hwzw,Twtw,Lw−1∇h),
(u,w,θ,π,T,p)→(Uu,Ww,T00θ,π,T00T,p00p),
(f,g)→(ε5/2gRT00f,g)
yields
ε2+αDuDtw+ε(9+α)/2−βezf×u=−cpθR∇hπ,
ε2+α+2βDwDtw=−cpθR∂π∂zw−ε,
DθDtw=0,
DπDtw+Rπcυ∇⋅v=0.
Table 1.

Reference quantities for high-frequency, β = 0, and midfrequency, β = 1, gravity wave scaling. Here we choose T00 = 210 K and N = 10−2 s−1 for the troposphere and N = 2 × 10−2 s−1 for the tropopause region. Note, however, that regimes with other values of T00 and N can be considered as well.
Table 1.

b. Ice microphysics: Governing equations and scalings

The cirrus clouds are described by a double-moment bulk microphysics scheme assuming a unimodal ice mass distribution function. The scheme is the same as the one from BS19, except that the sedimentational sinks are included here. A more detailed description of the ice model can be found in Spichtinger and Gierens (2009) and Spreitzer et al. (2017); in this section we only briefly refer to some key properties. As in BS19 we assume spherical shape of ice crystals, which leads to a simpler description of the cloud processes.

The equations governing the ice crystal number concentration n (number of ice crystals per mass dry air; unit: kg−1), ice mixing ratio q (mass of ice per mass dry air; unit: kg kg−1), and vapor mixing ratio qυ (mass of water vapor per mass dry air; unit: kg kg−1) read
DnDt=Nucn+Sedn,
DqDt=Dep+Nucq+Sedq,
DqυDt=−Dep−Nucq,
where Dep describes the ice crystal growth due to the deposition of water vapor, Nuc the generation of new ice crystals through homogeneous nucleation, and Sed the sedimentation of ice crystals under the effect of gravity. The latter sedimentational processes are modeled as
Sedn=1ρ∂∂zρυnn,
Sedq=1ρ∂∂zρυqq ,
where we assume spatially independent sedimentation velocities υn,q=cn,qmref2/3 with constants cn = 5.8 × 105 m s−1 kg−2/3, cq = 1.2 × 106 m s−1 kg−2/3, and a reference mass mref; this simplification is sufficient for estimating typical values of the sedimentation terms in the following asymptotic analysis.
In (25), (26) the deposition term Dep, also referred to as diffusional growth term, can be parameterized as
Dep=C0m¯1/3psip(S−1)Tn,
with C0 = 4.3 × 10−8 kg2/3 s−1 K−1, psi the saturation pressure over flat ice surface, m¯=q/n the mean ice-particle mass, and S the saturation ratio with respect to ice. The latter is defined as
S=pυpsi,
where pυ is the water vapor pressure. Using the definition of qυ = mυ/md = ρυ/ρd and the ideal gas law to express the dry air pressure, pd = RTρd, and the water vapor pressure, pυ = RυTρυ, yields for the saturation ratio
S=qυpdε0psi≈qυpε0psi,
where ε0 = R/Rυ and the water vapor gas constant is given by Rυ = 461 J kg−1 K−1. The saturation pressure psi is highly dependent on temperature and satisfies the approximate Clausius–Clapeyron equation
dpsidT=LiRυT2psi,
with Li = 2.8 × 106 J kg−1. In Eq. (24) the homogeneous nucleation rate of ice crystals is modeled as
Nucn=Jexp⁡[B(S−Sc)]
(following BS19 and Spichtinger et al. 2023), where Sc is some critical saturation ratio Sc(T) ≈ 1.5, J = 4.9 × 104 kg−1 s−1, and B = 337 (see BS19 for further details and the estimation of these values). The nucleation term in the equation for ice mixing ratio (25) is given by
Nucq=m^0Nucn ,
where the reference mass m^0=10−16 kg≪m¯ is used, which represents a typical mass of newly nucleated ice crystals. A summary of the ice physics scheme parameters can be found in Table 2. Next, characteristic numbers for the ice physics variables are chosen. Those values should describe a typical cirrus cloud in the upper troposphere–lower stratosphere region formed due to homogeneous nucleation. The characteristic values for the number concentration, vapor mixing ratio, and ice mixing ratio are denoted by nc, qυc, and qc, respectively. The estimate mref∼m¯∼mc is used, where mc denotes some mean ice crystal mass satisfying qc = ncmc. All reference values can be found in Table 3; they agree with the characteristic values used in BS19.
Table 2.

Parameters of the ice physics scheme.
Table 2.
Table 3.

Reference quantities used for nondimensionalization of the ice physics scheme.
Table 3.
Next, from (29) a characteristic time scale on which the diffusional growth term acts can be introduced (see also Korolev and Mazin 2003; Krämer et al. 2009); it is defined as
Td=(C0mc1/3psi,cp00qυ,cT00nc)−1∼340 s,
if the reference saturation pressure over ice psi,c = 1 Pa is used (see Table 3 for all other values). As can easily be shown, the estimate above for Td allows for deviations of about 10 K from the reference temperature T00 = 210 K. The characteristic vertical scale of the cirrus cloud is set to
Hc∼Hw.
This corresponds to Hc = 600 m for Hw from Table 1. Finally, the ice physics scheme is nondimensionalized using the reference quantities and all arising nondimensional numbers are expressed in terms of ε (distinguished limit), as summarized in Table 4. Applying the replacements
(z,t)→(Hwzw,Tdtd),
(n,q,qυ,psi,ρ)→(ncn,qcq,qυ,cqυ,psi,cpsi,ρ00ρ),
one yields the following nondimensional equations:
DnDtd=J*εexp⁡[B*ε2(S−Sc)]+ε2Sn*ρ∂∂zwρn,
DqDtd=1εD*psip(S−1)Tn+ε3J*exp⁡[B*ε2(S−Sc)]+ε2Sq*ρ∂∂zwρq,
DqυDtd=−D*psip(S−1)Tn−ε4J*exp⁡[B*ε2(S−Sc)] ,
where an asterisk denotes an order-one constant.
Table 4.

Distinguished limits for the nondimensional numbers in the ice scheme. In the rightmost column, an asterisk denotes an order-one constant. For the nondimensionalization, a time scale Tw, with Tw ∼ Td, was used.
Table 4.
One will also make use of the nondimensional form of the Clausius–Clapeyron equation, which reads
dpsidT=L*εT2psi.
Finally, as shown in appendix A the evolution equation for qυ is rewritten in terms of the saturation ratio S. With this, the nondimensional system governing the ice dynamics reads
DnDtd=J*εexp⁡[B*ε2(S−Sc)] +ε2Sn*ρ∂∂zwρn,
DqDtd=1εD*psip(S−1)Tn+ε3J*exp⁡[B*ε2(S−Sc)]+ε2Sq*ρ∂∂zwρq,
DSDtd=−D*(S−1)Tn−SπDπDtd(L*εT−cpR)−ε4ppsiJ*exp⁡[B*ε2(S−Sc)].

c. Asymptotic expansion

The nondimensional coordinates xw, tw, entering in Eqs. (20)–(23), describe variations on the GW spatial and temporal scales. To take into account variations of the reference atmosphere on the large, i.e., synoptic, vertical scale, Hs, with Hs ∼ Hp = Hw/ε, we introduce a compressed coordinate defined as
zs=εzw.
We consider a wave field (denoted by a prime) superimposed on a hydrostatically balanced reference atmosphere (denoted by a bar)
θ=θ¯(zs)+ε1+αθ′(xw,tw),
π=π¯(zs)+ε2+απ′(xw,tw),
v=v′(xw,tw),
where we have used the scaling from (15) and (16). Next, in accordance with (6) the potential temperature of the reference atmosphere is expanded as
θ¯(zs)={θ¯(0)(zs)+O(ε), ifα=0,1+εθ¯(1)(zs)+O(ε2), ifα=1,
whereas the corresponding Exner pressure expansion reads
π¯(zs)=∑j=01+αεjπ¯(j)(zs)+O(εα+2).
For the wave part we make a wave ansatz, e.g., for the potential temperature field, it reads
θ′(xw,tw)=Re{θ˜(1+α)exp⁡[i(k⋅xw−ωtw)]}+O(ε) ,
with the wave amplitude θ˜(1+α), the wave vector k = (k, l, m)T, and the frequency ω. Further, consistent with the definitions of the potential temperature and Exner pressure one obtains
(T,ρ)=(T¯,ρ¯)(zs)+ε1+α(T′,ρ′)(xw,tw,zs),
p=p¯(zs)+ε2+αp′(xw,tw,zs).
The following asymptotic expansion for the ice fields χ = (n, q, S) is used:
χ=χ(0)+O​(ε);
in addition Sc=O(1) is assumed. Integrating (40) with the boundary condition psi(1) = 1 [or in dimensional form psi(T00) = psi,c], one obtains for the saturation pressure over ice
psi(T)=exp⁡[L*ε(1−1T¯+ε1+αT′)].
In the next section we will consider the ice physics at the reference height z00; at this level we have
p¯(0)=T¯(0)=ρ¯(0)=θ¯(0)=π¯(0)=1,
and from (54): psi(z00) = O(1).

d. Coupling of the GW and diffusion time scale

We consider the following distinguished limit for the GW time scale Tw and for the diffusion time scale Td:
TdTw=O(1).
Since Td = 340 s, the scaling above is valid for midfrequency GWs in the troposphere and stratosphere, as well, for high-frequency GWs in the troposphere. In the case of midfrequency GWs, one has Tw ∼ 103 s in the troposphere and Tw ∼ 500 s in the tropopause region if the Doppler term in the GW dispersion relation is neglected. In the case of high-frequency GWs in the troposphere one has Tw ∼ 100 s. Note, that the corresponding GW period, Pw, reads Pw = 2πTw. For high-frequency GWs in the tropopause region Tw ∼ 50 s and the resulting scaling is discussed in section 6. For low-frequency GWs in the tropopause region Tw ∼ 5000 s is more appropriate. In this case Td/Tw = O(ε) leads to a weak amplitude GW forcing and the corresponding regime will be presented in an upcoming study. The condition (56) together with (46) yields the transformation
DπDtd=(ε2+αDπ′Dtw+εwdπ¯dzs).
Equation (56) implies that the coordinates tw and td resolve variations on the same time scale; hence, we may identify these two and replace td in the following by tw.

3. Reduced model of GW–cirrus interactions

a. GW dynamics

The asymptotic analysis of Eqs. (20)–(23) gives that the leading-order fields satisfy the GW polarization relations
(u˜(0),υ˜(0),w˜(0),π˜(2+α))=−imωkh2N¯2b˜(1+α)(k,l,−kh2m,ωRcpθ¯(0)),
and dispersion relation
ω2=N¯2k2+l2m2+(1−β)(k2+l2),
where the Brunt–Väisälä background frequency N¯ and the buoyancy amplitude b˜(1+α) are defined in (B11) and (B12), respectively. The complete derivation can be found in appendix B.

b. Single-parcel model approximation and single monochromatic GW

In the following we adopt a Lagrangian framework and consider the ice physics of a single air parcel influenced by GW dynamics. Further, we assume that the leading-order vertical velocity in (57) is solely due to a single GW and can be written as
w(0)[x(tw),tw]≈w(0)[x(t*),tw]=|w˜(0)|cos⁡(ωtw+ϕ),
with real amplitude |w˜(0)|, phase ϕ=x(t*)⋅k+δϕ, and initial position of the parcel x(t*). In section 6 we generalize the approach for the case of superposition of many GWs.

c. The different regimes in the ice dynamics

The gravity wave dynamics change the vertical velocity, pressure, and temperature fields in (43) and hence leads to variations of S and consequently of n. Time series illustrating the qualitative behavior of S and n under GW forcing are shown in Fig. 1 (see the discussion in section 4 for details). A typical situation observed is that S fluctuates until it reaches (or approaches sufficiently) the critical value Sc at time t0. At t0 the nucleation term in (41) leads to an explosive production of ice crystals. The increased number concentration n implies a reduction of S below Sc through the diffusional growth term in (43). After this reduction S continues to fluctuate due to the GW forcing and might again approach Sc. Thus, in some cases we have to consider ice nucleation in the presence of preexisting ice crystals, which might be suppressed under certain conditions.

Fig. 1.
Fig. 1.

Time evolution of number concentration n and saturation ratio S for two different initial conditions: (a) n(0) = 0 and (b) n(0) = 2 × 106 kg−1. Tw = 500 s and initial phase of the wave ϕ=−11π/20.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0234.1

Following the matched asymptotic approach of BS19, three different regimes are considered here. First, the prenucleation regime with S < Sc, where the dynamics takes place on the GW time scale. This is followed by a nucleation regime, centered around time t0 with S(t0) = Sc and dynamics on the much faster nucleation time scale. After the nucleation event the postnucleation regime is entered with S < Sc, characterized again by dynamics on the GW time scale.

We observe that due to the assumption m¯=mc the evolution of n and S is decoupled from the one of q. Because of this we first consider Eqs. (41) and (43). Once n and S are known, q can be found from (42); see also the discussion in section 3h. The case where all three equations, Eqs. (41)–(43), are coupled is discussed in section 5.

d. Pre- and postnucleation regime

Because the dynamics in the pre- and postnucleation regime takes place at the same characteristic time scale, we treat them simultaneously here. In both regimes S is below the critical value, S < Sc, and S − Sc = O(1) even in the limit ε → 0. This implies that the nucleation term in (41) is transcendentally small:
J*εexp⁡[B*ε2(S−Sc)]→0forε→0.
Next, we substitute in (41), (43) the expansion (53) for n and S and collect the leading-order terms. Evaluating the resulting equations at z00 gives
dn(0)dtw=0 ,
dS(0)dtw=−D*(S(0)−1)n(0)+S(0)F*(tw),
where (55), (57), (60), and (B3) are used and the GW forcing term is defined as
F*(tw)=RL*|w˜(0)|cpcos⁡(ωtw+ϕ).
From (62) one obtains that the number concentration does not change with time:
n(0)={Npre intheprenucleationregime,Npost inthepostnucleationregime. 
Whereas the constant Npre is typically given by the initial condition, the number concentration after the nucleation event Npost is at this stage unknown. By integrating (63) from the initial time t* up to tw, one obtains an integral representation for S(0)
S(0)(tw)=S*Sh(tw,t*)+∫t*twdt′D*n(0)Sh(tw,t′),
where the propagator Sh(tw,t*) is defined as
Sh(tw,t*)=exp⁡[−D*n(0)(tw−t*)+∫ttwdtF*(t)]
and the constant S*=S(t*) is given by the initial condition for the saturation ratio.

e. Nucleation regime

The nucleation regime is around the (unknown) time t0, when the prenucleation S(0) reaches the critical value Sc
S(0)(t0)=Sc ,
and it is characterized by the condition S − Sc = O(ε2) > 0. The ε−2 scaling in the exponent of the nucleation term in (41) indicates that the dynamics during nucleation event evolves on a fast time scale. This motivates one to introduce a rescaled time coordinate:
τ=tw−t0ε2 .
Equations (41) and (43) are expressed in terms of τ giving
dndτ=εJ*exp⁡[B*ε2(S−Sc)]+ε4Sn*ρ∂∂zwρn,
dSdτ=ε2{−D*(S−1)Tn−Sπ(ε2+αdπ′dtw+εwdπ¯dzs)(L*εT−cpR)−ε4ppsiJ*exp⁡[B*ε2(S−Sc)]}.
The last two equations can be written in compact form as
dndτ=εJ*exp⁡[B*ε2(S−Sc)]+O(ε4),
dSdτ=ε2[−D*(S−1)Tn−L*Swπ¯dπ¯dzs]+O(ε3) ,
where the nucleation term in (71) was eliminated using (70). From the leading order of (73) one obtains that the saturation ratio S(0) does not change during the nucleation event
dS(0)dτ=0→S(0)(τ)=Sc,
consistent with (68). As shown in appendix C the leading-order number concentration, n(0), satisfies
dn(0)dτ+δ(n(0))2−γn(0)=μ,
where the following constants are introduced:
δ=12B*D*(Sc−1),
γ=F*(t0)B*Sc,
and μ is some constant of integration. Integrating (75) from 0 up to some time τ, one obtains for the number concentration in the nucleation regime
n(0)(τ)=ns+neCeστ1+Ceστ,
with the constants
σ=γ2+4δμ,
ns=γ−σ2δ,
ne=σ+γ2δ,
C=n0−nsne−n0,
and another constant of integration n0 = n(0)(0).

f. Matching

Next, we find the constants of integration, entering the solution in the nucleation and the postnucleation regime by matching the different solutions in the prenucleation, nucleation, and postnucleation zone. First, we consider the limits τ → ±∞ of the nucleation solution (78):
nnuc(0)(−∞)=ns,
nnuc(0)(∞)=ne,
dnnuc(0)dτ(±∞)=0 .
The nucleation solution for the number concentration, nnuc(0)(τ), should match for τ → −∞ the one from the prenucleation regime, npre(0)(t), for t → t0. Equating (83) and (65) give
ns=Npre .
By considering (75) for τ → −∞ one obtains with the help of (83), (85), and (86):
μ=δNpre2−γNpre .
Matching the saturation ratio in the nucleation regime, Snuc(0) from (74), to the one in the prenucleation regime, Spre(0) from (66), gives the condition
S*Sh(t0,t*)+∫t*t0dt′D*NpreSh(t0,t′)=Sc.
The last equation is an implicit equation for the time of the nucleation event t0, where Sh(t0,t*)=exp⁡[−D*Npre(t0−t*)+∫t*t0dtF*(t)] and S* is given by the initial condition. Next, the nucleation solution for the number concentration, nnuc(τ), should match for τ → ∞ the one from the postnucleation regime, npost(0)(t), for t → t0. Equating (84) and (65) gives
ne=Npost .
From (81), with σ and μ given by (79) and (87), respectively, the postnucleation value for the number concentration can be found:
Npost={2F*(t0)ScD*(Sc−1)−Npre, ifNpre<F*(t0)ScD*(Sc−1),Npre, else.
The two cases in the last solution result from the condition that the expression under the root in (79) is positive. Equation (90) implies that there are newly nucleated ice crystals only for Npre<Nprec=F*(t0)Sc/[D*(Sc−1)]. In the case of nucleation event the larger Npre is, the smaller Npost is; however, the average (Npre + Npost)/2 does not depend on the initial n and is always the same [for fixed F*(t0)]. Interestingly, the threshold Nprec has important implication for the prenucleation dynamics of the saturation ratio. For Npre>Nprec most likely the necessary condition S = Sc for the existence of nucleation event will not be met. To see this consider that S is increasing just before the nucleation event, so one has S˙>0 at time t0 when S = Sc. From (63) this leads again to the condition Npre<Nprec for a nucleation if cos(ωt0 + ϕ) > 0 is assumed. Equation (90) defines further two interesting limits
Npostmax=2Fmax*ScD*(Sc−1)−Npre,
Npremax=Fmax*ScD*(Sc−1),
where Fmax*=max⁡{F*(t):t∈[0,2πTw]}; Npostmax is the maximum possible postnucleation number concentration due to a GW with a given amplitude, and Npremax is the maximum possible prenucleation number concentration that might allow for a nucleation event [see also the discussion on preexisting ice in Gierens (2003)].

g. Summary of the reduced model

The asymptotic analysis presented here allows us to identify a reduced model for the dominant interactions of the ice physics with the GW dynamics. Here we summarize it since the model will be used for the evaluation in the next section. It contains only the dominant terms from the full ice physics model (41)–(43) evaluated at level z00. In dimensional form it reads
dndt=Jexp⁡[B(S−Sc)],
dSdt=−D(S−1)n+SF(t),
dqdt=D(S−1)n,
with F defined in (102).

h. Summary of the asymptotic solution

In appendix D we construct the composite asymptotic solutions (D4) and (D6) valid in all three different regimes. Using the replacements
Tw(tw,t0)→(t,t0),
nc(n,ns,ne)→(n,ns,ne),
the equations are redimensionalized giving
n(t)=ns+neeζ(t−t0)1+eζ(t−t0),
S(t)={S*Sh(t,t*)+∫t*tdt′DnsSh(t,t′)fort≤t0,ScSh(t,t0)+∫t0tdt′DneSh(t,t′)fort>t0,
with the definitions
ζ=BScF(t0)−BD(Sc−1)ns,
Ω=ωTw,D=D*Twnc,
F(t0)=gLiw^cpRυT002cos⁡(Ωt0+ϕ),
and the dimensional GW vertical velocity amplitude w^. Note further that the final postnucleation number concentration ne is given in dimensional form by (90) with all asterisks omitted and the nucleation time t0 is found from the condition S(t0) = Sc.
To account for conservation of total water we supplement the asymptotic system with the equation for the ice mixing ratio (95). By integrating the latter equation with S from (99) and constant n = ns, one has an integral representation for q in the prenucleation regime:
q(t)=∫t*tDns[S(t′)−1]dt′+q(t*).
For initial q(t*)>0, q vanishing at some later time, tevap, implies that all initial crystals evaporated. The solutions (98) and (99) can still be used after tevap, if the prenucleation value ns is set to zero at t*=tevap. The postnucleation solution can be treated in a similar way to account for evaporation events.

4. Numerical experiments and discussion of the asymptotic solution

In this section the asymptotic model is validated against the full ice microphysics model and the reduced model for realistic parameter values taken from BS19 and summarized in Table 2. The full model solves (37)–(39) omitting only sedimentation, the details of it can be found in appendix E. All models are forced with single monochromatic GW from section 3b. The GW vertical velocity amplitude is set to 1 ms−1 corresponding to 0.7Wc, where Wc is the critical vertical velocity amplitude for breaking due to static instability. The GW frequency is Ω=Tw−1=2×10−3s−1 and the initial saturation ratio is set to S(0) = 1.4.

The results for two different initial number concentrations are summarized in Fig. 1. Figure 1a shows a situation where the asymptotic solution reproduces with a high accuracy the time evolution of the ice crystal number concentration and saturation ratio. Figure 1b depicts a case where the nucleation time t0 from the full model is slightly missed by the asymptotic and the reduced models. Although the overall evolution of S is reproduced well, the nucleated ice crystal number is underestimated by roughly 15%. From the asymptotic theory, we expect that the discrepancy will vanish in the limit ε → 0. This asymptotic limit is verified numerically by considering smaller values of ε in the full and in the reduced model, this corresponds to increasing the time scale separation between the different processes in the models [see Dolaptchiev et al. (2013) for another example of this procedure]. The results are summarized in Fig. 2 for ε = 10−1 and ε = 10−2; note that in Fig. 1 ε = 1 implying no increased time scale separation. Figure 2 shows that the models converge quickly to the asymptotic limit already for moderately small values of ε.

Fig. 2.
Fig. 2.

As in Fig. 1b, but for (a) ε = 10−1 and (b) ε = 10−2; in Fig. 1b ε = 1.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0234.1

Since the phase of the GW is typically unknown in coarse models, we study the sensitivity of the results with respect to this parameter. For that purpose, we vary the GW phase at the initial time t = 0 and determine the nucleated ice crystals within one wave period for an initial condition n(0) = 0. From Fig. 3 it is visible that the asymptotic solution captures, for all GW phases, the number of nucleated ice crystals in the full model. In addition, the values of n are limited from above by the asymptotic estimate Npostmax.

Fig. 3.
Fig. 3.

Nucleated number concentration n as a function of the initial GW phase. Gray horizontal lines denote the initial condition n(0) and the asymptotic estimates Npremax and Npostmax from Eqs. (91) and (92), respectively.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0234.1

In Fig. 4 the normalized vertical velocity at the nucleation time is displayed. It suggests that nucleation takes place at sufficiently high updrafts but not necessary at the maximal.

Fig. 4.
Fig. 4.

Normalized GW vertical velocity at t0 as a function of the initial GW phase for the simulations from Fig. 3.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0234.1

Figure 5 summarizes the dependence of the nucleated ice crystals on the initial number concentration. The asymptotic solution reproduces nearly exactly n from the reduced model. Both models are very close to the full model for n(0) well below Npremax; however, for n(0) approaching Npremax, they underestimate n for some GW phases. The detailed evolution of the solutions for one such particular case (dashed line in Fig. 5b) was shown in Fig. 1b. As shown in Fig. 2 the discrepancy in the models results from the finite time scale separation between processes and diminishes for ε → 0. Further, one observes in Fig. 5 that the asymptotic solution is able to capture regimes without nucleation events, too.

Fig. 5.
Fig. 5.

As in Fig. 3, but for initial number concentration (a) n(0) = 106 kg−1 and (b) n(0) = 2 × 106 kg−1. In (b), the initial n is the same as the one used in Figs. 1b and 2; the dashed gray vertical line marks the initial phase used for the simulation in Figs. 1b and 2.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0234.1

5. The effect of variable ice crystal mean mass in the deposition

Performing realistic air parcel simulations with a box model and a bulk microphysics scheme, BS19 demonstrated that for a wide variety of environmental conditions the constant mean mass assumption in the deposition term is a reasonable approximation during nucleation. However, right before a nucleation event where the saturation ratio is above one, the mean mass of the ice crystals will grow leading to an increased deposition term; see the m¯ dependence in (29). The increased deposition will influence the saturation ratio, which will affect the number of nucleated ice crystals. Since the mean mass of the ice crystals can be diagnosed from the relation m = q/n for n > 0, such effects can be incorporated in the present model if the substitution m¯=m(t)=q/n is introduced in the diffusional growth term. By considering again only the dominant terms in the prognostic equations, this results in the following system of reduced equations with variable mean mass in the deposition:
dndt=J*εexp⁡[B*ε2(S−Sc)],
dSdt=−D*(qn)1/3(S−1)n+SF*(t),
dqdt=D*ε(qn)1/3(S−1)n.
In Fig. 6 we show simulations of the reduced model with constant and with variable mean mass; in both cases the ice crystal mass is diagnosed using m = q/n. The vertical velocity amplitude is set to 1.25 m s−1 corresponding to 0.9Wc. The initial conditions for all models are set to S(0) = 1.45, n(0) = 106 kg−1, and ϕ = 0.
Fig. 6.
Fig. 6.

Time evolution of number concentration n, saturation ratio S, and mean ice mass m computed for different models: the reduced model with constant mass, (93)–(95); the reduced model with variable mass, (104)–(106); the asymptotic solution for constant mass (see section 3h); and the asymptotic solution with variable mass correction, (117). Note that for the latter model, only the pre- and postnucleation values are plotted, resulting in a jump at t0. The mean mass in the reduced models is diagnosed using m = q/n. In the figure’s legend, m0 and m(t) denote models with constant mass and variable mass, respectively.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0234.1

In Fig. 6 an increase of the ice crystal mass is observed before the nucleation event. Taking this into account with the model (104)–(106) results in a larger number of nucleated ice particles, as compared to the constant mass model. The asymptotic solution for the constant mass case (denoted with “asym. m0” in the figure) reproduces the behavior of the constant mass reduced model and underestimates n, as well. In the following, we extend the asymptotic approach to allow for variable mean mass effects.

a. Prenucleation regime

The fully coupled system (104)–(106) involves an additional fast time scale in the q equation as compared to the constant mean mass case discussed in section 3. The new time scale will induce in general nontrivial dynamics on longer time scales; however, the corresponding rigorous asymptotic analysis is out of the scope of this paper. As we will show here, the asymptotic prenucleation solution for the constant mass case can still be used for variety of configurations if appropriate corrections are introduced.

First, we observe in Fig. 6 that before the nucleation event the solution for S does not change much if the variable mean mass effect is taken into account. In such situations, one can use the asymptotic solution (66) for S to compute the time, t0, of the nucleation event from (88). For S − Sc = O(1) < 0 the right-hand side of (104) vanishes at leading order, implying the constant solution n(t) = Npre. Using the latter result and integrating (106) from t* up to t0, we obtain at the end of the prenucleation regime
q(t0)2/3=3D*2εNpre2/3∫t*t0dt′[S(t′)−1]+q(t*)2/3,
with S(t) from (66).

b. Nucleation regime

Next, we consider the nucleation regime. On the fast nucleation time τ we have the following system of equations:
dndτ=εJ*exp⁡[B*ε2(S−Sc)],
dSdτ=−ε2D*(qn)1/3(S−1)n+ε2SF*(t0+ε2τ),
dqdτ=εD*(qn)1/3(S−1)n.
From (110) we see that the leading-order ice mixing ratio is constant during the nucleation event:
q(0)(τ)=qpre(0)(t0) ,
with the value of qpre(0)(t0) from (107). Next, by repeating the manipulations in Eqs. (C1)–(C5), one obtains from (108), (109) the following evolution equation for the number concentration:
dn(0)dτ+δ˜(n(0))5/3−γn(0)=μ˜ ,
with the constant
δ˜=35(q(0))1/3B*D*(Sc−1)
and γ defined in (77) [see below (117) for the definition of μ˜]. Equation (112) can be solved numerically to find the number concentration during the nucleation regime. The numerical integration of the ODE can be avoided if only the final n at the end of the nucleation event is of interest. Proceeding as in section 3f we have the following matching conditions for the prenucleation, nucleation, and postnucleation regime:
nnuc(0)(−∞)=Npre ,
nnuc(0)(∞)=Npost ,
dnnuc(0)dτ(±∞)=dnpre(0)dt(t0)=dnpost(0)dt(t0)=0.
Inserting the matching conditions in (112) leads to an algebraic equation for Npost:
δ˜Npost5/3−γNpost=μ˜ ,
where μ˜=δ˜Npre5/3−γNpre. Note that for Npre = 0, (117) implies for the dependence of Npost on the GW vertical velocity: Npost∼w^3/2, which is consistent with the scaling in Kärcher and Lohmann (2002).

c. Numerical results

Equation (117) is used to find an asymptotic approximation of the nucleated number concentration. The comparison with the numerical results is shown in Fig. 6. The current procedure produces a larger number of nucleated ice crystals as compared with the constant mass model; the magnitude of n is close to the one of the variable mean mass model.

The performance of the current approach is systematically evaluated by varying the initial GW phase. The corresponding results are summarized in Fig. 7. The figure suggests that the proposed procedure captures the nucleated number of ice crystals for various initial GW phases.

Fig. 7.
Fig. 7.

Number concentration n as a function of the initial GW phase for the variable mean mass model, (104)–(106), and for the asymptotic solution, (117) from section 5. Initial conditions are as in Fig. 6.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0234.1

A note of caution should be added on the relevance of the variable mean mass model presented in this section. The use of the prenucleation S from (66) in (107) might be invalid on longer time scales: see the first paragraph of section 5a. Further, Fig. 6 suggests that there might be situations with considerable growth of m before nucleation takes place. Obviously, for large m the sedimentation term will provide a sink for the mean mass: see (42). The correct incorporation of the sedimentation effects will be the subject of a future study.

6. Ice physics forced by superposition of gravity waves

In this section we consider the case when a GW spectrum is forcing the ice physics. The GW forcing is constructed by considering NGW waves with vertical wind amplitudes sampled from a white frequency spectrum in accordance with observations (Podglajen et al. 2016). The wave amplitudes w^ are rescaled such that the total GW momentum flux ρu′w′ equals 5 mPa at 8 km altitude. This value for the momentum flux is within the range given by observational and modeling studies (Hertzog et al. 2012; Kim et al. 2021; Corcos et al. 2021). For high-frequency GWs (with Tw ∼ N) in the tropopause region the ratio
TdTw=O​(ε−1)
is more appropriate, compare the last equation with (56). In this case, one obtains from (57)
DπDtd=(ε1+αDπ′Dtw+εwdπ¯dzs),
implying that the tendency of π′ is on the same order as the vertical advection term for α = 0. It was verified that the magnitude of the wave amplitudes allows one to neglect the nonlinear advection term and the term ∂π′/∂t will have the dominant contribution in the material derivative of π′. To account for effects due to high-frequency waves, we will include the latter term into the GW forcing term. We will also include a next-order correction term by keeping the cp/R term in Eq. (43), since we found that this improves the results for the general case of superposition of GWs. We found from simulations that the magnitudes of the waves are too small to trigger nucleation. Because of this, we include a constant vertical updraft w00 = 2 cm s−1 on which the GWs are superimposed. With the assumptions above, the forcing (64) is generalized to
F*(tw)=(L*−εcpR)×{∑jNGW[Rcp|w˜j(0)|cos⁡(ωjtw+ϕj)+ωj|π˜j(0)|sin⁡(ωjtw+ϕj)]+Rcpw00}.
With the new forcing (120) we compute the ice crystal number concentration from the asymptotic solution in section 3h. We perform 103 realizations, each forced by a superposition of 10 waves with random frequencies uniformly distributed within the range ωmin < ωj < ωmax. A random wave phase increment δϕj=ϕj−x(t*)⋅kj is uniformly distributed within [0, 2π]. The vertical wavelength of all waves is set to 1 km and the initial values n = q = 0 and S = 1.4 are used in the simulations.

The results are summarized in Fig. 8a for the case where ωj is drawn from a narrow range around the frequency Tw−1=2×10−3s−1 for which the asymptotic analysis was performed. The asymptotic solution (red stars in Fig. 8a) captures the ice crystal number concentration, n, from the full model (cyan line in Fig. 8a; see appendix E for the details of the full model) for the majority of realizations. The accuracy increases if large values of n are nucleated. Further, by reordering n from the asymptotic solution in ascending order (the dashed black line in Fig. 8a), we see that the statistics produced by the asymptotic model is very close to the one from the full model (cyan line). The above finding is confirmed by inspecting the corresponding probability density function (PDF) of n: Fig. 9a shows that the PDF of the asymptotic and of the full model are nearly identical. This is an important result for the application of the present asymptotic solutions in climate models. Since the phase of the waves is typically unknown, any parameterization will not be able to reproduce the exact nucleation time t0. However, on average our model produces a distribution of n matching the one from the full model.

Fig. 8.
Fig. 8.

Nucleated number concentration n for 103 different realizations of a superposition of GWs with frequencies: (a) 0.5Tw−1<ω<2Tw−1 and (b) f0 < ω < N. The cyan line shows in ascending order the results from the full model. The corresponding asymptotic solutions are shown as red stars. The dashed black line displays the same asymptotic solutions but after sorting them in ascending order.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0234.1

Fig. 9.
Fig. 9.

Probability density function (PDF) of n from the realizations presented in Fig. 8. The PDF was constructed using kernel density estimation with Gaussian kernel.

Citation: Journal of the Atmospheric Sciences 80, 12; 10.1175/JAS-D-22-0234.1

Next, we consider the full frequency range of GWs. The corresponding results are summarized in Fig. 8b. It has to be stressed that our asymptotic analysis is only valid for the frequency range around Tw−1. However, the asymptotic model can still capture many of the nucleation events of the full model. Again, this is particularly valid for the largest values 106 < n < 107. For smaller values of n, the quality of prediction deteriorates. The corresponding PDFs from Fig. 9b reveal small positive bias and some overestimation of the variance in the asymptotic model. We suppose that these discrepancies might be due to small amplitudes of the GW, inconsistent with our scaling where w^∼Wc was assumed, and the omission of the low-frequency GW in the asymptotic analysis.

7. Conclusions

We present an asymptotic approach allowing us to identify a reduced model for the self-consistent description of ice physics forced by a superposition of GWs including the effect of diffusional growth and homogeneous nucleation of ice crystals. Furthermore, using matched asymptotic techniques analytical solutions are constructed, involving a novel parameterization (90) for the ice crystal number concentration n. The latter has as input parameters the wave amplitudes and phases, and the time of the nucleation event. It allows the derivation of an upper bound for the nucleated n, as well as a threshold for the initial n that would inhibit nucleation. The numerical simulations with a Lagrangian parcel model show that the parameterization reproduces nucleation events triggered by a monochromatic GW for a variety of initial conditions. Furthermore, in the case of superposition of GWs within the midfrequency range the parameterization generates a distribution of n matching the one of the full model. By extending the parameterization to high-frequency GWs, it is shown that the asymptotic solution produces distribution similar to the one of the full model even if the complete GW frequency spectrum is used as forcing. The results presented here demonstrate the potential of our approach for constructing improved cirrus schemes in climate models with realistic GW variability as simulated with transient GW parameterizations (Bölöni et al. 2021; Kim et al. 2021).

When comparing the treatment of the ice physics in our approach with the one from BS19, we observe different scaling in the nucleation term: in the latter work J ∼ B ∼ ε−1 is used, whereas here we apply J ∼ ε−1, B ∼ ε−2. Nevertheless, our parameterization (90) is equivalent to the closure of BS19 for constant updraft velocity if the velocity there is replaced by the GW vertical velocity at the nucleation time t0. This is not surprising, since the GW nearly does not vary on the fast nucleation time scale. The correspondence of the two parameterizations becomes more clear if one takes into account that in BS19 ε = O(10−2) and here ε = O(10−1), implying the same magnitude of the nucleation exponent B under the different scalings. As shown by Spichtinger et al. (2023) the exact value of the nucleation rate J is not crucial as long as it is sufficiently large. Still, we have to stress that the present approach generalizes the framework of BS19 to include wave dynamics and the consistency between the two parameterizations only supports our results. In addition, we derive a novel parameterization for the variable mean mass model, see (117), a threshold for nucleation inhibition, Npremax from (92), and a high-frequency correction to the parameterization, (120).

The present asymptotic solutions are applicable mainly to the midfrequency GW in the troposphere and tropopause region, as well as to high-frequency GWs in the troposphere. For high-frequency GWs in the tropopause region, the ratio between the time scale of the diffusional growth and of the wave is given by Td/Tw ∼ ε−1. The simulations from section 6 show larger values of n if the high-frequency GWs are included. This suggests a new regime dominated by the GW forcing term and we propose some asymptotic corrections to account for it. We expect that this regime corresponds to the temperature-limit events studied in Dinh et al. (2016). For low-frequency GWs, the scaling Td/Tw ∼ ε is appropriate. In this case the GW forcing term becomes weaker by a factor of ε, when compared to the depositional growth term. This regime is relevant for low updraft velocities and will be considered in an upcoming study.

Our asymptotic analysis assumes a reference number concentration nc. However, the results from section 6 suggest that the resulting asymptotic model is valid for a wider range of n. If regimes with other values of nc are of interest, the present asymptotic framework can be adapted for the systematic investigation of these, too.

The models presented here predict for the particular GW forcings investigated, values of n which are within or at the upper range of observations; e.g., see Fig. 8 from Krämer et al. (2020). However, a direct comparison with observational data is hampered for two main reasons. First, most of the measurements lack information on the wave properties, e.g., wave amplitude and frequency, so the GW forcing cannot be determined. Second, nucleation takes place at a very fast time scale and within a confined spatial region. Therefore, the vast majority of measurements of ice crystals are taken probably after the nucleation event happened. However, at later stages of the ice cloud life cycle, other processes such as sedimentation determine the microphysical properties. The latter lead to smearing of the clear nucleation signature and to significantly smaller n values (see, e.g., Spichtinger and Gierens 2009); this effect is enhanced if ice crystals fall into subsaturated air and thus evaporate. This might explain why high number densities are quite rarely observed (see, e.g., Krämer et al. 2009, 2020).

In the present regime the magnitude of the sedimentation effects is determined by the sedimentation time scale Tsed=Hc/cqmc2/3. Substituting the reference quantities gives Tsed ∼ ε−2Td ∼ 11 h, implying that at leading-order sedimentation is negligible compared to the diffusional growth term. Note, however, that Tsed will decrease if regimes with larger ice crystal mass mc or smaller vertical scales Hc are of interest. As shown in Podglajen et al. (2018), sedimentation modulated by GW forcing produces localization effects in cirrus.

In the present study, only cirrus formed by homogeneous nucleation are considered, since this is the dominant formation mechanism in the cold temperature regime with strong updraft velocities (e.g., Heymsfield and Miloshevich 1993). Still, heterogeneous nucleation can considerably alter cirrus formation (see, e.g., Gierens 2003; Spichtinger and Cziczo 2010); however, the important feature is, also in case of competing nucleation pathways, the occurrence of preexisting ice crystals, as in our investigations. In addition, turbulence due to GW breaking is another source of GW-generated variability omitted in the present study (e.g., Atlas and Bretherton 2023).

Acknowledgments.

UA and PS thank the German Research Foundation (DFG) for partial support through the research unit “Multiscale Dynamics of Gravity Waves” (MS-GWaves; Grants AC 71/8-2, AC 71/9-2, AC 71/12-2, and SP 1163/5-2) and CRC 301 “TPChange” (Project-ID 428312742, Projects B06 “Impact of small-scale dynamics on UTLS transport and mixing,” B07 “Impact of cirrus clouds on tropopause structure,” and Z03 “Joint model development”). UA acknowledges DFG for partial support through CRC 181 “Energy transfers in Atmosphere an Ocean” (Project Number 274762653, Projects W01 “Gravity-wave parameterization for the atmosphere” and S02 “Improved Parameterizations and Numerics in Climate Models”).

Data availability statement.

The Python script used to generate all figures in the paper is available upon request.

APPENDIX A

Time Evolution of S

Here we derive from (39) an evolution equation for the saturation ratio S. Since, qυ,c = ε0psi,c/p00 is used for the scaling of qυ, the definition of S expressed using nondimensional variables reads
S=qυppsi .
Applying D/Dtd to (A1) yields
DSDtd=ppsiDqυDtd+SpDpDtd−SpsiDpsiDtd
=ppsiDqυDtd+cpSRπDπDtd−SL*εT2DTDtd
=ppsiDqυDtd+cpSRπDπDtd−SL*εT2(πDθDtd︸=0+θDπDtd)
=ppsiDqυDtd−SπDπDtd(L*εT−cpR),
where (40) was used to obtain (A3), (22) in (A4), and the definition of potential temperature θ = T/π for Eq. (A5). With this, one obtains (43).

APPENDIX B

GW Dispersion Relation and Polarization Relations

With v = v(0) + O(ε) the leading-order continuity equation reads
∇⋅v(0)=0.
Using the wave ansatz for v(0), this gives the solenoidality condition
k⋅v˜(0)=0,
implying that the wavevector k and v˜(0) are orthogonal. The latter property will be used to eliminate nonlinear advection terms in the equations.
From the leading-order vertical momentum equation we obtain hydrostatic balance between π¯(0) and θ¯(0):
dπ¯(0)dzs=−Rcpθ¯(0).
Similarly, the next-order vertical momentum balance reads
dπ¯(1)dzs=Rθ¯(1)cpθ¯(0)2,
where we have used (B3). Note, that in the case α = β = 0 there is no advection term appearing in the latter equation due to (B2).
The projection of the leading-order equations onto the GW field reads
∂u(0)∂tw=−cpθ¯(0)R∇hπ(2+α),
(1−β)∂w(0)∂tw=−cpθ¯(0)R∂π(2+α)∂zw+θ(1+α)θ¯(0),
∇w⋅v(0)=0,
∂θ(1+α)∂tw+w(0)dθ¯(α)dzs=0,
where again (B2) was utilized. Inserting in Eqs. (B5)–(B8) a wave ansatz for the solution, one obtains the following system of linear equations for the wave amplitudes:
Mz=0,
where
M=[−iω000ik0−iω00il00−i(1−β)ω−N¯im00N¯−iω0ikilim00],
with z=[u˜(0),υ˜(0),w˜(0),b˜(1+α)/N¯,(cp/R)θ¯(0)π˜(2+α)] and
N¯2=1θ¯(0)dθ¯(α)dz,
b˜(1+α)=θ˜(1+α)θ¯(0).
Looking for nontrivial solutions of (B9), one derives the dispersion relation (59) and the polarization relations (58) for the GW amplitudes.

APPENDIX C

Evolution Equation for n in the Nucleation Regime

Here, the equation for n is derived: first (72) is differentiated with respect to τ and after this the exponential function is replaced using (72) giving
d2ndτ2=B*ε2dSdτdndτ+O​(ε2).
Inserting (73) yields
d2ndτ2=B*dndτ[−D*(S−1)Tn−L*Swπ¯dπ¯dzs]+O​(ε).
The evaluation of the leading-order equation takes the form
d2n(0)dτ2=B*dn(0)dτ[−D*(S(0)−1)T¯(0)n(0)−L*S(0)w(0)π¯(0)dπ¯(0)dzs].
After substituting (55), (60), (B3), and (64) in the last equation yields
d2n(0)dτ2=B*dn(0)dτ[−D*(S(0)−1)n(0)+F*(t0)S(0)],
where the expansion F*(tw)=F*(t0)+O(ε2) was used. Taking into account that S(0) = Sc, Eq. (C4) can be written as
ddτ[dn(0)dτ+δ(n(0))2−γn(0)]=0,
with the definitions introduced in (76) and (77). Integrating (C5) over time finally gives (75).

APPENDIX D

Composite Solution

Despite the fact that Npost is determined from (90), the integration constant n0 entering (78) through (82) is still unknown. It will, however, not affect the value of Npost. Moreover, since in the present asymptotic analysis t0 can be found up to some higher-order corrections, n0 is undetermined. To see this, we introduce another constant τ0 defined as C=e−στ0, we can write (78) as
n(0)(τ)=ns+nee(σ/ε2)(t−t0−ε2τ0)1+e(σ/ε2)(t−t0−ε2τ0).
Note that from (88) t0 is determined up to O(ε) corrections, which will result in modifications of the constant n0. One way of setting the value for n0 is by requiring that at τ = 0 the saturation ratio S should reach a maximum. At the end of the prenucleation regime, we have S˙>0; on the other hand, at the beginning of the postnucleation regime S˙<0; thus, S should have a maximum within the nucleation regime. We define n0 by requiring that in (73) S˙=0 at time τ = 0 up to O(ε3) corrections. After using (55), (60), (B3), (64), and (74) this implies
n0=F*(t0)ScD*(Sc−1)⇒C=1.
The requirement of having a maximum in the saturation ratio upon the nucleation is physically meaningful and may be interpreted as the defining feature of a nucleation event.
It remains to construct the composite solution valid in all three regimes. For the number concentration the nucleation regime represents an interior layer (e.g., Holmes 2013), enclosed by the outer layers of the prenucleation and postnucleation regime. In this case the composite solution reads
n(tw)=npre(0)(tw)+nnuc(0)(τ)+npost(0)(tw)−nnuc(0)(−∞)−nnuc(0)(∞)+O​(ε).
Substituting (65), (78), (83), and (84) in (D3) gives for the number concentration
n(tw)=ns+nee(σ/ε2)(tw−t0)1+e(σ/ε2)(tw−t0)+O​(ε).
Since the time derivative of the saturation ratio has a jump from the prenucleation to the postnucleation value, the nucleation regime represents a corner layer for S (Holmes 2013). For such a layer two cases depending on the sign of t − t0 has to be considered when constructing the composite solution
S(tw)={Spre(0)(tw)+Snuc(0)(τ)−Snuc(0)(−∞)fortw≤t0Spost(0)(tw)+Snuc(0)(τ)−Snuc(0)(∞)fortw>t0+O​(ε).
Substituting (66) and (74) in (D5), the solution for S takes the form
S(tw)={S*Sh(tw,t*)+∫t*twdt′D*nsSh(tw,t′)fortw≤t0ScSh(tw,t0)+∫t0twdt′D*neSh(tw,t′)fortw>t0+O​(ε),
with Sh defined in (67).

APPENDIX E

Description Lagrangian Parcel Model

The Lagrangian parcel model describes the evolution of n, q and qυ in an air parcel oscillating in the x–z plane under the gravity wave forcing. It solves (37)–(39) with sedimentation switched off, i.e., Sn*=Sq*=0. The parcel position vector, x(t) = (x, z)T, is determined from
dxdt=v,
where the velocity field is given in general by a superposition of GWs with a possibility to include a constant updraft velocity w00:
v=∑jNGWv˜jcos⁡(ωjt+kjx+mjz+δϕj)+w00ez.
Each frequency ωj and amplitude v˜j is satisfying the general inertial GW dispersion relation and polarization relation (e.g., Achatz 2022) with N = 0.02 s−1 for the tropopause region. A vertical wavelength of 1 km is assumed, comparable to the value of 3 km used in the study of Corcos et al. (2023).

From (E1) with initial condition z = z00 and x = x00 the parcel position is found. With this the wave fluctuations of Exner pressure π′ and of potential temperature θ′ are determined from the corresponding polarization relations. To those fluctuations one has to add the stationary contributions π¯ and θ¯ from the reference atmosphere in order to determine the full fields. It is assumed that in the vicinity of z00 the reference atmosphere can be represented by an isothermal temperature profile T¯(z)=T00 with corresponding pressure p¯(z)=p00e−(z−z00)/Hp (all in dimensional form). From those the Exner pressure, π¯, and potential temperature, θ¯, of the reference atmosphere are calculated using the definitions T = πθ and π=(p/p00)R/cp. By adding all together, one obtains the total fields π and θ, or equivalently p and T. Finally, the saturation pressure can be determined from (54). In practice, Eqs. (37)–(39) and (58) are simultaneously integrated numerically and we use the above mentioned procedure to find p, psi, and T at each time step.

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    • Search Google Scholar
    • Export Citation

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Journal of the Atmospheric Sciences

  • Achatz, U., 2022: Atmospheric Dynamics. Springer, 554 pp., https://doi.org/10.1007/978-3-662-63941-2.

  • Achatz, U., R. Klein, and F. Senf, 2010: Gravity waves, scale asymptotics and the pseudo-incompressible equations. J. Fluid Mech., 663, 120–147, https://doi.org/10.1017/S0022112010003411.

    • Search Google Scholar
    • Export Citation

  • Achatz, U., B. Ribstein, F. Senf, and R. Klein, 2017: The interaction between synoptic-scale balanced flow and a finite-amplitude mesoscale wave field throughout all atmospheric layers: Weak and moderately strong stratification. Quart. J. Roy. Meteor. Soc., 143, 342–361, https://doi.org/10.1002/qj.2926.

    • Search Google Scholar
    • Export Citation

  • Atlas, R., and C. S. Bretherton, 2023: Aircraft observations of gravity wave activity and turbulence in the tropical tropopause layer: Prevalence, influence on cirrus clouds, and comparison with global storm-resolving models. Atmos. Chem. Phys., 23, 4009–4030, https://doi.org/10.5194/acp-23-4009-2023.

    • Search Google Scholar
    • Export Citation

  • Baumgartner, M., and P. Spichtinger, 2019: Homogeneous nucleation from an asymptotic point of view. Theor. Comput. Fluid Dyn., 33, 83–106, https://doi.org/10.1007/s00162-019-00484-0.

    • Search Google Scholar
    • Export Citation

  • Baumgartner, M., R. Weigel, A. H. Harvey, F. Plöger, U. Achatz, and P. Spichtinger, 2020: Reappraising the appropriate calculation of a common meteorological quantity: Potential temperature. Atmos. Chem. Phys., 20, 15 585–15 616, https://doi.org/10.5194/acp-20-15585-2020.

    • Search Google Scholar
    • Export Citation

  • Baumgartner, M., C. Rolf, J.-U. Grooß, J. Schneider, T. Schorr, O. Möhler, P. Spichtinger, and M. Krämer, 2022: New investigations on homogeneous ice nucleation: The effects of water activity and water saturation formulations. Atmos. Chem. Phys., 22, 65–91, https://doi.org/10.5194/acp-22-65-2022.

    • Search Google Scholar
    • Export Citation

  • Bölöni, G., B. Ribstein, J. Muraschko, C. Sgoff, J. Wei, and U. Achatz, 2016: The interaction between atmospheric gravity waves and large-scale flows: An efficient description beyond the nonacceleration paradigm. J. Atmos. Sci., 73, 4833–4852, https://doi.org/10.1175/JAS-D-16-0069.1.

    • Search Google Scholar
    • Export Citation

  • Bölöni, G., Y.-H. Kim, S. Borchert, and U. Achatz, 2021: Toward transient subgrid-scale gravity wave representation in atmospheric models. Part I: Propagation model including nondissipative wave–mean-flow interactions. J. Atmos. Sci., 78, 1317–1338, https://doi.org/10.1175/JAS-D-20-0065.1.

    • Search Google Scholar
    • Export Citation

  • Bramberger, M., and Coauthors, 2022: First super-pressure balloon-borne fine-vertical-scale profiles in the upper TTL: Impacts of atmospheric waves on cirrus clouds and the QBO. Geophys. Res. Lett., 49, e2021GL097596, https://doi.org/10.1029/2021GL097596.

    • Search Google Scholar
    • Export Citation

  • Corcos, M., A. Hertzog, R. Plougonven, and A. Podglajen, 2021: Observation of gravity waves at the tropical tropopause using superpressure balloons. J. Geophys. Res. Atmos., 126, e2021JD035165, https://doi.org/10.1029/2021JD035165.

    • Search Google Scholar
    • Export Citation

  • Corcos, M., A. Hertzog, R. Plougonven, and A. Podglajen, 2023: A simple model to assess the impact of gravity waves on ice-crystal populations in the tropical tropopause layer. Atmos. Chem. Phys., 23, 6923–6936, https://doi.org/10.5194/acp-23-6923-2023.

    • Search Google Scholar
    • Export Citation

  • Dean, S. M., J. Flowerdew, B. N. Lawrence, and S. D. Eckermann, 2007: Parameterisation of orographic cloud dynamics in a GCM. Climate Dyn., 28, 581–597, https://doi.org/10.1007/s00382-006-0202-0.

    • Search Google Scholar
    • Export Citation

  • Dinh, T., A. Podglajen, A. Hertzog, B. Legras, and R. Plougonven, 2016: Effect of gravity wave temperature fluctuations on homogeneous ice nucleation in the tropical tropopause layer. Atmos. Chem. Phys., 16, 35–46, https://doi.org/10.5194/acp-16-35-2016.

    • Search Google Scholar
    • Export Citation

  • Dolaptchiev, S. I., U. Achatz, and I. Timofeyev, 2013: Stochastic closure for local averages in the finite-difference discretization of the forced Burgers equation. Theor. Comput. Fluid Dyn., 27, 297–317, https://doi.org/10.1007/s00162-012-0270-1.

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  • <sc>Fig</sc>. 1.
    View in gallery
    Fig. 1.

    Time evolution of number concentration n and saturation ratio S for two different initial conditions: (a) n(0) = 0 and (b) n(0) = 2 × 106 kg−1. Tw = 500 s and initial phase of the wave ϕ=−11π/20.

  • <sc>Fig</sc>. 2.
    View in gallery
    Fig. 2.

    As in Fig. 1b, but for (a) ε = 10−1 and (b) ε = 10−2; in Fig. 1b ε = 1.

  • <sc>Fig</sc>. 3.
    View in gallery
    Fig. 3.

    Nucleated number concentration n as a function of the initial GW phase. Gray horizontal lines denote the initial condition n(0) and the asymptotic estimates Npremax and Npostmax from Eqs. (91) and (92), respectively.

  • <sc>Fig</sc>. 4.
    View in gallery
    Fig. 4.

    Normalized GW vertical velocity at t0 as a function of the initial GW phase for the simulations from Fig. 3.

  • <sc>Fig</sc>. 5.
    View in gallery
    Fig. 5.

    As in Fig. 3, but for initial number concentration (a) n(0) = 106 kg−1 and (b) n(0) = 2 × 106 kg−1. In (b), the initial n is the same as the one used in Figs. 1b and 2; the dashed gray vertical line marks the initial phase used for the simulation in Figs. 1b and 2.

  • <sc>Fig</sc>. 6.
    View in gallery
    Fig. 6.

    Time evolution of number concentration n, saturation ratio S, and mean ice mass m computed for different models: the reduced model with constant mass, (93)–(95); the reduced model with variable mass, (104)–(106); the asymptotic solution for constant mass (see section 3h); and the asymptotic solution with variable mass correction, (117). Note that for the latter model, only the pre- and postnucleation values are plotted, resulting in a jump at t0. The mean mass in the reduced models is diagnosed using m = q/n. In the figure’s legend, m0 and m(t) denote models with constant mass and variable mass, respectively.

  • <sc>Fig</sc>. 7.
    View in gallery
    Fig. 7.

    Number concentration n as a function of the initial GW phase for the variable mean mass model, (104)–(106), and for the asymptotic solution, (117) from section 5. Initial conditions are as in Fig. 6.

  • <sc>Fig</sc>. 8.
    View in gallery
    Fig. 8.

    Nucleated number concentration n for 103 different realizations of a superposition of GWs with frequencies: (a) 0.5Tw−1<ω<2Tw−1 and (b) f0 < ω < N. The cyan line shows in ascending order the results from the full model. The corresponding asymptotic solutions are shown as red stars. The dashed black line displays the same asymptotic solutions but after sorting them in ascending order.

  • <sc>Fig</sc>. 9.
    View in gallery
    Fig. 9.

    Probability density function (PDF) of n from the realizations presented in Fig. 8. The PDF was constructed using kernel density estimation with Gaussian kernel.

Fig. 1.

Time evolution of number concentration n and saturation ratio S for two different initial conditions: (a) n(0) = 0 and (b) n(0) = 2 × 106 kg−1. Tw = 500 s and initial phase of the wave ϕ=−11π/20.
Fig. 2.

As in Fig. 1b, but for (a) ε = 10−1 and (b) ε = 10−2; in Fig. 1b ε = 1.
Fig. 3.

Nucleated number concentration n as a function of the initial GW phase. Gray horizontal lines denote the initial condition n(0) and the asymptotic estimates Npremax and Npostmax from Eqs. (91) and (92), respectively.
Fig. 4.

Normalized GW vertical velocity at t0 as a function of the initial GW phase for the simulations from Fig. 3.
Fig. 5.

As in Fig. 3, but for initial number concentration (a) n(0) = 106 kg−1 and (b) n(0) = 2 × 106 kg−1. In (b), the initial n is the same as the one used in Figs. 1b and 2; the dashed gray vertical line marks the initial phase used for the simulation in Figs. 1b and 2.
Fig. 6.

Time evolution of number concentration n, saturation ratio S, and mean ice mass m computed for different models: the reduced model with constant mass, (93)–(95); the reduced model with variable mass, (104)–(106); the asymptotic solution for constant mass (see section 3h); and the asymptotic solution with variable mass correction, (117). Note that for the latter model, only the pre- and postnucleation values are plotted, resulting in a jump at t0. The mean mass in the reduced models is diagnosed using m = q/n. In the figure’s legend, m0 and m(t) denote models with constant mass and variable mass, respectively.
Fig. 7.

Number concentration n as a function of the initial GW phase for the variable mean mass model, (104)–(106), and for the asymptotic solution, (117) from section 5. Initial conditions are as in Fig. 6.
Fig. 8.

Nucleated number concentration n for 103 different realizations of a superposition of GWs with frequencies: (a) 0.5Tw−1<ω<2Tw−1 and (b) f0 < ω < N. The cyan line shows in ascending order the results from the full model. The corresponding asymptotic solutions are shown as red stars. The dashed black line displays the same asymptotic solutions but after sorting them in ascending order.
Fig. 9.

Probability density function (PDF) of n from the realizations presented in Fig. 8. The PDF was constructed using kernel density estimation with Gaussian kernel.
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