Ice clouds, so-called cirrus clouds, occur very frequently in the
tropopause region. A special class are subvisible cirrus clouds
with an optical depth lower than 0.03, associated with very low ice
crystal number concentrations. The dominant pathway for the
formation of these clouds is not known well. It is often assumed
that heterogeneous nucleation on solid aerosol particles is the
preferred mechanism although homogeneous freezing of aqueous
solution droplets might be possible, since these clouds occur in the
low-temperature regime

Clouds consisting exclusively of ice crystals, so-called cirrus clouds, are
frequently found in the tropopause region at low temperatures
(

In the present study we focus on the formation of SVCs by homogeneous
freezing of aqueous solution droplets (hereafter: homogeneous
nucleation). We study the formation and evolution of SVCs in an air
parcel that is lifted in slow vertical upward motions (

For the investigation of subvisible cirrus clouds we develop a parcel
model to which we apply numerical and analytical tools. The model is
developed on the basis of an evolution equation for mass distributions
of ice crystals, including a description of microphysical processes
based on former work

To study the qualitative behaviour of the model we use concepts from
theory of dynamical systems

In Sect.

In this section we describe the development of a reduced ice cloud
model, which is later used for analytical and numerical
investigations. We include the relevant processes for formation and
evolution of ice clouds into the model but we try to avoid too much
complexity, which makes analysis too complicated

An ice cloud is represented by an ensemble of ice particles, which can
be described by a mass distribution

Instead of solving Eq. (

In the following the representation of relevant processes is
described briefly. For more details we refer the reader to
Appendix

Particle formation in terms of ice nucleation is described by the last
term on the right-hand side of Eq. (

We describe homogeneous nucleation as a stochastic process with a
nucleation rate

The growth and evaporation of ice crystals is dominated by diffusion
of water vapour.
With several simplifications of the growth equation
(for details see Appendix

Following

We can compose the general terms for sedimentation in the moment
Eq. (

In order to obtain a consistent but simplified system of ODEs we make the following three assumptions:

Change to Lagrangian point of view and purely vertical motion:The Eulerian time evolution
and advection of a quantity

Closure using an equation for relative humidity with respect to ice:In our study, we will exclusively consider very low vertical
velocities (

As temperature decrease at slow upward motions is only very
small, in a zeroth-order approximation we assume constant
temperature and pressure. In consequence, the parcel's volume
remains constant, too. The resulting error for neglecting density
changes is usually of order

To close the systems of differential equations we introduce an
evolution equation for relative humidity, starting with the total
derivative of

The last term in Eq. (

Approximation of sedimentation:Since we are interested in an analytically treatable model of a
single air parcel, we need to eliminate the partial derivatives
describing sedimentation, which generally lead to a hyperbolic
system of partial differential equations, which is too complicated
for theoretical analysis. For simplification of the equations
we have to consider terms of the form

In summary, the full system of the model equations reads as

We examine the system for a range of parameter values

We investigate the reduced model using analytical tools (see details in
Sect.

The general cloud formation mechanism works as follows: the adiabatic
cooling causes the relative humidity, and thus the nucleation rate, to
rise until ice nucleation occurs. Due to the steepness of

A scenario in state 1 (stable focus regime, damped
oscillation) at

From the numerical simulations we found that the system
exhibits two qualitatively distinct behaviours, depending on values of

A scenario in state 2 (limit cycle regime) is shown at

For a first investigation we discuss the different terms in
Eq. (

For a first analysis of the system we compute the divergence of the
system (i.e. the trace of the Jacobian

The balance of terms in Eq. (

For very small ice crystals, the term including

Real (upper panel) and imaginary part (lower panel) of the
complex eigenvalues

In a first step, the autonomous dynamical system Eq. (

In order to examine the qualitative behaviour of the solution in a
neighbourhood of the equilibrium state, the ODE system is linearised
about the equilibrium state

Real eigenvalue

Complex eigenvalues of the linearised system
indicate oscillatory behaviour, which is prevalent in all
simulations. As can be seen in Fig.

For negative values of the real part (

In this case the equilibrium point (stable focus) corresponds to
state 1 in the numerical simulations. Solutions of the system
Eq. (

Stable focus for state 1 at

For positive values of the real part (

Limit cycle for state 2: orbit in phase space at

Bifurcation diagram for stable focus (state 1) and limit
cycle (state 2) regimes in the

The transition between the two general states of the system (stable
point attractor vs. limit cycle) can be represented in a bifurcation
diagram of the

After discussing the different states of the system qualitatively, we now give an overview of the quantitative cloud properties and relative humidity for the stable focus and the limit cycle, respectively.

In the stable focus regime, i.e.

Ice particle number concentration

Figure

Ice crystal number concentrations at the equilibrium point

As expected from theory

For the limit cycle regime (state 2), we can still derive the values
of mass and number concentrations at the equilibrium state

Ice crystal number concentrations for different temperature
scenarios (

The mass concentration of the ice crystals is largely determined by
the efficiency of diffusional growth. As indicated in the model
description (Sect.

Mean ice crystal mass

For the stable focus regime, we can directly investigate the mean mass of
the ice crystals,

As indicated in Sect.

Oscillation periods for the stable focus regime at

For comparison with observations we first consider in-situ
measurements of ice crystals in subvisible cirrus clouds. Since it is
very difficult to measure low number concentrations, only few
measurement studies are available. We compare our results with
measurements by

In a second step we
expand our comparison to observations from remote sensing. Since SVCs
are optically very thin, we investigate the extinction coefficient for
the visible part of the spectrum. For comparing our results with
measurements, we calculate the extinction

Extinction coefficient at

Overall, we can state that regarding the high spread in the measurements the results from our reduced model agree quite well with in situ measurements.

For comparison with a more detailed model, we carried out simulations
with the box model described by

Henceforth this model is termed “complex
model”. We scan through the

Stable focus case (state 1): comparison between reduced
model (this study) and the complex box model by

We can again identify regimes in the

For the complex model simulations the environmental conditions change; i.e. temperature and pressure are decreasing due to adiabatic expansion. Thus, no steady state can be reached. The values for ice crystal number concentrations and relative humidity are slightly rising with time in the quasi-steady state at the end of the simulation. Ice crystal mass concentration is slightly decreasing.

Limit cycle case (state 2): comparison between reduced
model (this study) and the complex box model by

In Fig.

The bifurcation diagram displayed in Fig.

Transition between stable focus regime (state 1) and limit
cycle regime (state 2): simulation with the complex model by

We also compare our results with the analytical model by

In this study we have developed a reduced model for describing subvisible cirrus clouds formed by homogeneous nucleation in the tropopause region. The model consists of a set of autonomous ordinary differential equations for the variables ice crystal mass and number concentration, and relative humidity with respect to ice. It contains the relevant cloud processes ice nucleation, diffusional growth and sedimentation. The model can be viewed as an externally forced dissipative system. The model is integrated numerically and also investigated using (linear) theory of dynamical systems.

Integration and theoretical analysis show that the system contains two
different states, a stable focus state and a limit cycle state. The
states depend on the environmental parameters vertical updraught,

Ice crystal mass and number concentrations of the cloud in both states depend mostly on the environmental conditions as vertical velocity and temperature. However, for the limit cycle case the spread in ice crystal mass and number concentration is obviously larger than in the case of stable equilibrium. For the stable focus, the mean mass depends only slightly on vertical velocity; thus, we can approximate the mean mass as a function of temperature.

Comparisons with a more detailed box model by

Since there are only few in situ measurements of subvisible cirrus
available, it is quite difficult to carry out solid comparisons.
However, we try to compare with measurements as described by

The major qualitative results can be summarised as follows:

Homogeneous freezing of aqueous solution
droplets at low temperatures (

In unperturbed weak large-scale updraughts subvisible cirrus clouds can exist in two different qualitative states, reaching either a stable equilibrium point (stable focus) in the long-term behaviour or experiencing oscillation behaviour in a limit cycle scenario. The state depends on external parameters as large-scale updraught and temperature, respectively.

The cloud particle properties in the long-term behaviour are very similar for both states. Therefore, we cannot decide from values of mass and/or number concentrations in a certain range in which state the cloud might be. Even if we had more measurements, we probably would not be able to decide between the two states just using the Eulerian measurements without a Lagrangian point of view.

We might derive a minimal model for SVCs from the bifurcation diagram
in the following way. If we assume that SVCs are well approximated by
their attractors, we could express cloud variables and relative
humidity by a simple damped harmonic oscillator of the form

Finally, we can state that we could develop a meaningful reduced model
for describing the main features of subvisible cirrus clouds. Former
investigations using box models indicated that there might be
different regimes in the behaviour of the clouds for longer simulation
times. For instance, in studies by

The observed Hopf bifurcation as a transition between two different
states shows that clouds might exhibit inherent structures, which are
crucially determined by the microphysical cloud processes themselves
in addition to environmental conditions. Similar structure formation
was already seen in analytical cloud models for liquid and mixed-phase
clouds as developed by

The data used in this work are described in Sect. 3.5.

Homogeneous nucleation, i.e. the transformation of a solution droplet
to an ice crystal, can be seen as a stochastic process. The transition
rate

The “advection
velocity”

In this study we make use of the following simplifications:

Latent heat release at the crystal surface is neglected and the temperature of the ice particles is assumed to be equal to temperature of ambient air.

We neglect kinetic corrections, since we are mostly interested
in growth of larger crystals. Kinetic corrections are usually
important for ice crystal growth in regimes with high concentrations
of small crystals. For SVCs we can assume small number
concentrations; thus, crystals will grow fast to sizes larger than

We neglect correction of ventilation, setting

The shape of ice crystals is assumed to be prolate spheroids
with length

The fraction in Eq. (

With these assumptions, Eq. (

The description of sedimentation is based on the concept of
mass- and number-weighted terminal velocities defined by

The dependency of the fall speeds of individual ice crystals on the
crystal mass is approximated by a simple power law

For simplification of the representation of the main system, we
introduced coefficients in Eq. (

For deriving Eq. (

Example of a Poincaré section in the limit cycle
regime. Blue dots indicate intersection points of the trajectory
with

In Fig.

The authors declare that they have no conflict of interest.

We thank Manuel Baumgartner, Martin C. Papke, Lars Grüne and Rupert Klein for
fruitful discussions. We also thank the
three anonymous reviewers; their comments helped to improve the
manuscript significantly. This study was prepared with support by
the German Bundesministerium für Bildung und Forschung (BMBF)
within the HD(CP)