We take into consideration the evolution of particle size in a monodisperse aerosol population during activation and deactivation of cloud condensation nuclei (CCN). Our analysis reveals that the system undergoes a saddle-node bifurcation and a cusp catastrophe. The control parameters chosen for the analysis are the relative humidity and the particle concentration. An analytical estimate of the activation timescale is derived through estimation of the time spent in the saddle-node bifurcation bottleneck. Numerical integration of the system coupled with a simple air-parcel cloud model portrays two types of activation/deactivation hystereses: one associated with the kinetic limitations on droplet growth when the system is far from equilibrium, and one occurring close to equilibrium and associated with the cusp catastrophe. We discuss the presented analyses in context of the development of particle-based models of aerosol–cloud interactions in which activation and deactivation impose stringent time-resolution constraints on numerical integration.

Atmospheric clouds are visible to human eye for they are composed of water
and ice particles that effectively scatter solar radiation.
The multi-micrometre light-scattering cloud droplets form on sub-micrometre
aerosol particles
(cloud condensation nuclei, CCN) in a process referred to as CCN activation or (heterogeneous) nucleation.
The concentration (from tens to thousands per cubic centimetre) and size
(from fractions of to multiple micrometres) of
activated particles can both vary by over an order of magnitude depending on the size
spectrum and composition of CCN.
On the one hand, CCN physicochemical properties are influenced by anthropogenic
emissions of particles into the atmosphere.
On the other hand, the resultant size spectrum of cloud droplets determines how
effectively the clouds interact with solar radiation and how effectively
they produce precipitation

Deactivation is the reverse process in which cloud droplets evaporate
back to aerosol-sized particles.
The process is also referred to as aerosol regeneration,
aerosol recycling, drop-to-particle conversion or simply droplet evaporation

It is worth noting that particle nucleation through condensation is relevant
in a much wider context than formation of atmospheric clouds.
Since the late 19th century, the growth of particles through
condensation up to optically detectable sizes has been the principle
of operation of so-called condensation particle counters

This paper is structured as follows. Section

The key element in the mathematical description of
CCN activation/deactivation is the equation for the
rate of change of particle radius

The crux of the matter is the dependence of

The Köhler theory provides us with the so-called Köhler curve; the leading terms of
its common

Phase portraits of the system discussed in Sect.

Köhler curve for CCN with

Rewriting Eq. (

For

The above analysis portrays a bifurcation in the behaviour of the system at

Combining Eqs. (

It is noteworthy that the standard cloud-physics Köhler curve
plot given in Fig.

Activation timescale as a function of dry radius and relative
humidity estimated with Eq. (

Interestingly, the analysis of the CCN activation/deactivation in terms
of saddle-node bifurcation provides a way to estimate the
timescale of activation.
Following

Dependence of

The key limitation of the preceding analysis is that the evolution of particle size is not coupled with the evolution of ambient heat and moisture content, and hence the relative humidity. Limiting the analysis to a monodisperse population, the coupling efficiency is determined by the total number of particles in the system. The so-far assumed constant RH approximates thus the case of small number of droplets.

To at least partially lift the constant-RH assumption, while still allowing
for a concise analytic description of the system dynamics,
let us consider a simple representation of the moisture budget in the system
under a temporary assumption of constant temperature and pressure
(and hence constant volume, constant

Figure

For

In order to lift the assumptions of constant temperature and pressure,
the system evolution can be formulated by supplementing the drop growth
equation with two equations representing the hydrostatic balance and
the adiabatic heat budget.
This leads to a commonly used so-called air parcel
framework depicting behaviour of a vertically displaced adiabatically
isolated mass of air:

As discussed in Sect.

Results of numerical simulations discussed in Sect.

Because the system defined by Eq. (

In order to depict an activation–deactivation cycle, the vertical velocity

Figure

The plots depict that for mean velocities of

At much lower velocity of

The adaptive-timestep solver statistics (not shown) reveal that regardless of
the chosen accuracy, for all considered input parameters,
there are two instants for which the solver needs to significantly reduce
the timestep: when resolving the supersaturation maximum during activation
and when resolving the “jump” back to equilibrium during deactivation.
It is a robust feature that deactivation
requires roughly an order of magnitude shorter timestep as compared
to activation
(ca. 0.01 s vs. 0.1 s for a relative accuracy of 10

The key advantage of the embraced monodisperse simulation
is simplicity – in terms of model formulation and result analysis, and also integration.
Due to the wide span of aerosol and droplet size spectrum, simulations of the particle
size spectrum evolution during activation are prone
to numerical difficulties – both due to the stiffness of the system
and due to the sensitivity to the size spectrum discretisation

The key inherent limitation for applicability of monodisperse simulations is
the lack of description of the cloud droplet size spectrum shape.
Consequently, the model lacks representation of the phenomena that depend
on simultaneous presence of both activated and unactivated CCN.
Such phenomena include the
noise-induced excitations to which even a bi-disperse system would
be susceptible if subject to fluctuations in the
forcing terms

These limitations certainly restrain the relevance of the presented
calculations to real-world problems.
Yet, let us underline that both the monodisperse spectrum and even
the no-RH-coupling assumption are in fact contemporarily used in atmospheric modelling
in the recently popularised particle-based (Lagrangian, super-droplet)
techniques for representing aerosol, cloud and precipitation particles
in models of atmospheric flows

With this note we intend to bring attention to the presence of nonlinear peculiarities in the equations governing CCN activation and deactivation, namely a saddle-node bifurcation and a cusp catastrophe. We have shown that conceptualisation of the process in terms of bifurcation analysis yields a simple yet practically applicable description of the system allowing analytic estimation of the timescale of activation. Both through weakly nonlinear analysis and through numerical integration, we have depicted the presence of a cusp catastrophe in the system and the corresponding hysteretic behaviour near equilibrium.

The deactivation stage was observed to determine the time-stepping constraints for numerical integration when simulating an activation–deactivation cycle of a monodisperse droplet population. It is a finding of interest for the cloud modelling community since monodisperse activation/deactivation models of the studied type play a constituting role in the more and more widespread particle-based models of aerosol–cloud interactions.

The software code is available in the Supplement.

The authors declare that they have no conflict of interest.

We thank Hanna Pawłowska and Ahmad Farhat as well as the three anonymous reviewers for their comments to the initial version of the manuscript. Sylwester Arabas acknowledges support of the Poland's National Science Centre (Narodowe Centrum Nauki; decision no. 2012/06/M/ST10/00434). This research was supported by JSPS KAKENHI Grant-in-Aid for Scientific Research (B) (proposal number: 26286089), and by the Center for Cooperative Work on Computational Science, University of Hyogo. This study was carried out during a research visit of Sylwester Arabas to Japan supported by the University of Hyogo. Sylwester Arabas extends special thanks to the Asada and Okamoto families.Edited by: Amit Apte Reviewed by: three anonymous referees