NPGNonlinear Processes in GeophysicsNPGNonlin. Processes Geophys.1607-7946Copernicus PublicationsGöttingen, Germany10.5194/npg-24-535-2017On the CCN (de)activation nonlinearitiesArabasSylwestersarabas@chathamfinancial.euhttps://orcid.org/0000-0003-2361-0082ShimaShin-ichiros_shima@sim.u-hyogo.ac.jphttps://orcid.org/0000-0001-5540-713XInstitute of Geophysics, Faculty of Physics, University of Warsaw, Warsaw, PolandChatham Financial Corporation Europe, Cracow, PolandGraduate School of Simulation Studies, University of Hyogo, Kobe, JapanSylwester Arabas (sarabas@chathamfinancial.eu) and
Shin-ichiro Shima (s_shima@sim.u-hyogo.ac.jp)5September20172435355429September20164October201623May201724July2017This work is licensed under the Creative Commons Attribution 3.0 Unported License. To view a copy of this licence, visit https://creativecommons.org/licenses/by/3.0/This article is available from https://npg.copernicus.org/articles/24/535/2017/npg-24-535-2017.htmlThe full text article is available as a PDF file from https://npg.copernicus.org/articles/24/535/2017/npg-24-535-2017.pdf
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.
Background
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
see, e.g., a recent Nonlinear Processes in Geophysics paper byfor a discussion of the aerosol–cloud–precipitation interaction
chain, unconventionally modelled as a predator–prey problem.
CCN activation is thus the linking process
between the microscopic human-alterable atmospheric composition
and the macroscopic climate-relevant cloud properties.
As once aptly stated, “there is something captivating about the idea that
fine particulate matter, suspended almost invisibly in the atmosphere, holds
the key to some of the greatest mysteries of climate
science” .
This has certainly contributed to the wealth of literature on the subject
published since the first studies of the 1940s
. For a thorough list of references see, e.g., chap. 7.
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
see Sect. 1 infor a review of modelling studies.
Both activation and deactivation are particular cases of particle
condensational growth, which, in the context of cloud modelling, is generally
regarded as reversible to contrast the irreversible collisional growth
see, e.g.,.
The reversibility of condensational growth is a sound (and often a constituting)
assumption for cloud models
for which activation and deactivation are subgrid processes, both in terms of
time- and length scales.
Yet, when investigated in short-enough timescales,
condensation and evaporation exhibit a hysteretic behaviour
in an activation–deactivation cycle.
The hysteresis can be associated with the kinetic limitations
in the vapour and heat transfers to/from the droplets
and has been previously depicted in the studies of
discussion of Fig. 1,
and .
As we point out in this paper, the system can exhibit a hysteretic behaviour
also in a close-to-equilibrium regime where the kinetic
limitations do not play a significant role.
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 seefor a historical review.
Instruments in which single CCN undergo activation in
conditions similar to those discussed herein
are routinely used in ground-based and airborne research measurements.
Interestingly, an analogue of CCN activation theory applies to the formation of
nanometre-sized aerosol particles via activation of molecular clusters by
organic vapours .
This paper is structured as follows. Section provides a
brief introduction to the constituting
elements of the CCN activation theory.
In Sects. –, we detail how the dynamics of
cloud droplet growth
can be studied employing the techniques of nonlinear dynamics analysis.
In these sections we do not refrain from using the peculiar yet pertinent
jargon of nonlinear dynamics.
For introduction, we refer the reader to the concise and approachable
introductory chapters in chap. 2.0–2.2, 2.4, 3.0
as well as to sections on specific topics therein
to which references are provided throughout the text.
Sections – deal with the
so-called air parcel cloud model framework.
The framework is used here
to corroborate the results from
nonlinear dynamics analysis of a simplified CCN activation model against
numerical solutions of an equation system providing a more comprehensive
description of the process.
Section provides an additional context for the discussion
by pointing out the congruence of the simplifying assumptions embraced
in the presented analysis with the recently popularised
particle-based techniques for modelling aerosol–cloud interactions.
Section concludes the paper.
Droplet growth laws in a nutshell
The key element in the mathematical description of
CCN activation/deactivation is the equation for the
rate of change of particle radius rw (so-called wet radius)
due to water vapour transfer to/away from the particles.
It is modelled by a diffusion equation in a spherical geometry,
r˙w=1rwDeffρwρv-ρ∘,
where ρw is the liquid water density, ρv is the ambient
vapour density (away from the droplet), ρ∘ is the equilibrium
vapour density at the drop surface and the
Deff=Deff(T,rw) is an effective diffusion coefficient
in which the temperature dependence stems from an approximate combination of Fick's
first law and Fourier's law (latent heat release) into a single particle-growth equation
(the Maxwell–Mason formula), while the radius dependence stems from
corrections limiting the diffusion efficiency for smallest particles.
For derivation and discussion see
Sect. 5.1.4. Introducing two
non-dimensional numbers,
the relative humidity RH=ρv/ρvs
(the ratio of the ambient vapour density to the vapour density at saturation
with respect to plane surface of pure water) and the equilibrium relative
humidity RHeq=ρ∘/ρvs, the drop growth
equation is given by
r˙w=1rwDeffρvsρwRH-RHeq.
The crux of the matter is the dependence of RHeq on rw.
In the context of atmospheric clouds, it is determined primarily by the droplet
curvature and by the presence of dissolved substances.
The theory capturing the interplay between these two effects was formulated
by Köhler in .
Note that the qualitatively similar
interplay between the surface tension and electric charge (as opposed to chemical composition)
results in an analogous particle activation phenomenon which served as the
principle of operation of the Wilson cloud chamber – a key instrument
in the early days of elementary particle physics for references, see.
The Köhler theory provides us with the so-called Köhler curve; the leading terms of
its common κ-Köhler formulation can be approximated with (for
rd≪rw which is a reasonable assumption in context
of activation/deactivation)
RHeq=rw3-rd3rw3-rd3(1-κ)expArw≈1+Arw-κrd3rw3,
where A∼10-3µm is a temperature-dependant coefficient
related to the surface tension of water, while the dry radius rd
and the solubility parameter κin general, 0<κ<1.4; see are proxy
variables depicting the mass and chemical composition of the substance the
CCN are composed of.
The ∂rwRHeq
derivative has an analytically derivable root corresponding to the maximum
of the Köhler curve at (rc,RHc), where
rc=3κrd3/A is the so-called critical radius
and RHc=1+2A3rc is the critical relative humidity.
Saddle-node bifurcation at Köhler curve maximum
Phase portraits of the system discussed in Sect. for different values of the control parameter RH.
Arrows have their heads pointing right (left) if the sign of ξ˙ is positive (negative).
The half-filled circle denotes a half-stable fixed point.
Filled and open circles denote stable and unstable fixed points, respectively.
The dashed line corresponds to RH=1.
Köhler curve for CCN with rd=0.05µm, κ=1.28 (NaCl)
and its Taylor expansions at rc and at infinity.
Rewriting Eq. () in terms of ξ=rw2+C (where
C is an arbitrary constant) gives
ξ˙=2DeffρvsρwRH-RHeq(ξ).
Figure depicts the phase portrait of the dynamical system
defined by Eq. (), for different values of
relative humidity RH which is chosen as the control parameter
in the following fixed-point analysis.
Fixed points correspond to equilibrium conditions defined by ξ˙=0,
which can be geometrically identified as crossings of the -RHeq
curve (a flipped Köhler curve) and the constant function RH.
For RH>RHc, there are no intersections – there are no fixed points,
the time derivative ξc˙ is always positive;
regardless of their size, the CCN grow.
For RH=RHc, there is just one fixed point – it is
half-stable (small variation in ξ can be either damped or amplified
depending on the direction).
For 1<RH<RHc, there are two fixed points in the system:
one stable and one unstable.
The stability depends on how the sign of ξ˙ changes around a fixed
point (note that the arrows on the plot correspond to the sign of ξ˙).
Around a stable fixed point (also called attractor, sink), small variations in ξ are damped, while
in the case of unstable fixed point (also called repeller, source), small variations in ξ are amplified.
Particles smaller in radius than the radius corresponding to the unstable
fixed point will shrink or grow in the direction of the equilibrium
state corresponding to the stable fixed point (as depicted by the directions of the arrows).
Particles larger in radius than the radius corresponding to the unstable
fixed point will grow provided RH>1 – these are the activated CCN.
In the limit of ξ→∞, the Köhler curve approaches
RH=1; hence the unstable fixed point goes to infinity.
For RH<1, there is just one stable fixed point
corresponding to the unactivated CCN equilibrium.
The above analysis portrays a bifurcation in the behaviour of the system at
RH=RHc. Rewriting RHeq in
terms of ξc=rw2-rc2 and
Taylor-expanding it around ξc=0 gives
RHeq(ξc)=c0+c1ξc+c2ξc2+…,
where c0=RHc, c1 is zero as we are expanding around
the root of ∂ξcRHeq and
c2=-A4rc-5 is negative.
Combining Eqs. () and () gives
ξc˙ξc→0∼RH-RHcA/(4rc5)+ξc2,
which is the normal form of the saddle-node bifurcation
Sect. 3.1.
It is noteworthy that the standard cloud-physics Köhler curve
plot given in Fig. can well serve as a (flipped) phase portrait
of the system facilitating identification of the fixed points by considering
intersections of the Köhler curve with lines of constant RH.
Figure depicts the approximation Eq. ()
alongside
the κ-Köhler curve, confirming that the parabolic approximation
is valid only in the nearest vicinity of (rc,RHc).
Activation timescale estimation
Activation timescale as a function of dry radius and relative
humidity estimated with Eq. () with A∼10-3µm,
κ=1.28, D∼2×10-5 m2 s-1, ρw∼103 kg m-3
and ρvs=10-3 kg m-3.
Interestingly, the analysis of the CCN activation/deactivation in terms
of saddle-node bifurcation provides a way to estimate the
timescale of activation.
Following Sect. 4.3,
the coalescence of the fixed points is associated with a passage through
a bottleneck.
The key observation is that for the parabolic normal form of the saddle-node
bifurcation, the time of the passage through the bottleneck dominates all
other timescales.
Thus, the timescale of the process can be estimated by integrating ξc
from -∞ to ∞:
τact≈∫-∞+∞dξcξc˙=rc5/2Aρw/ρvsDeffπRH-RHc.
The activation timescale τact given by Eq. (),
plotted
as a function of RH and rd (and substituting rc and
RHc by their analytic formulae given in the preceding section)
is presented in Fig. .
It matches remarkably the data obtained through numerical calculations presented in
.
The white region in the plot corresponds to a situation where activation does not
happen.
The range of RH depicted in the plot is chosen to match the one
of Fig. 2 in
, while in principle the presented weakly nonlinear
analysis of the system is applicable only close to the equilibrium (i.e.
close to the edge of the white region in the plot).
Dependence of f defined in Eq. () on the wet radius
and particle concentration (green wireframe surface).
The red line below depicts the zero-crossings of the first derivative of f
with respect to rw.
Values of all constants are as in Fig. .
See Sect. .
Cusp catastrophe of the RH-coupled system
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 ρvs, A and Deff).
The rate of change of the ambient relative humidity RH˙
can be expressed then as a function of the droplet volume concentration N,
RH˙≈ρ˙vρvs=-N4πρw3ρvs︸α3rw2r˙w,
where the form of α stems from defining the density of liquid water
in the system as Nρw43πrw3.
Integrating in time gives
RH=RH0-αNrw3,
which combined with Eq. () and expressed in terms of ξ with
C=0
leads to the following phase portrait of the RH-coupled system (assuming rw≫rd):
ξ˙∼(RH0-1)-Aξ12-κrd3ξ32+αNξ32︸f,
where the group of terms labelled as f can be intuitively
thought of as corresponding to the Köhler curve with an additional term
representing the simplified RH dynamics.
Figure depicts the dependence of f on the droplet
radius rw=ξ and the droplet concentration N.
To facilitate analysis, the zero-crossings of the first derivative of f with respect
to rw are plotted as well using the analytically derived formula
sgn(f′)=sgnκrd3-A3rw+αNrw3.
For N=0, f has the Köhler curve shape depicted in
Fig. ,
which, as discussed in the preceding sections, implies a saddle-node bifurcation.
With N greater than zero but less than ca. 50 cm-3, a second
saddle-node
point appears as the αNξ32 term causes f to
have a local minimum above the critical radius.
At ca. N=50 cm-3, both the first and second derivatives of f
vanish,
implying a cusp point in the f surface.
For larger N, f is monotonic; hence both of the saddle-node bifurcations cease
to exist.
For N>0, this phase portrait reveals a topological equivalence
seeTheorem 4.3 to the normal form of the
cusp bifurcation.
The cusp bifurcation chap. 8.2, an imperfect
supercritical
pitchfork bifurcation chap. 3.6,
features a cusp catastrophe, which makes it possible to envisage
a “catastrophic” jump from one equilibrium to another and
a hysteretic behaviour of the system when approaching (in terms of rw) the
local minimum of f from below (activation) and from
above (deactivation) for small enough N.
Adiabatic vertically displaced air parcel system
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:
p˙dT˙r˙w=-ρdgw(p˙d/ρd-q˙lv)/cpd(Eq.1),
where ρd and pd are the dry-air (background state)
density and pressure,
w is the vertical velocity of the parcel,
q=ρv/ρd is the water vapour mixing ratio,
cpd is the specific heat of dry air,
lv is the latent heat of vaporisation and
g is the acceleration due to gravity.
The sum of water vapour and liquid water densities is conserved in the system
which makes it possible to diagnose ρv and RH from the state variables,
similarly as in Eqs. ()–() but without
the simplifying assumption of constant temperature and pressure.
As discussed in Sect. , for
a monodisperse population of N particles, the
changes in the mass of liquid water in the system
are proportional to the particle concentration, hence q˙∼N.
Consequently, the analysis of the activation/deactivation dynamics presented in
Sects. – under the assumption of constant RH
corresponds to the behaviour of the air parcel system in the following limits:
w→0 (and hence p˙d≈0),
i.e. slow, close-to-equilibrium evolution of the system relevant to
fixed-point analysis
(by some means pertinent to the formation of non-convective clouds such as
fog);
N→0 (and hence r˙≈0),
i.e. weak coupling between particle size evolution and the ambient
thermodynamics (pertinent to the case of low particle concentration).
Numerical simulations
Results of numerical simulations discussed in Sect. .
Because the system defined by Eq. () is less susceptible to a
simple
analytic analysis, we proceed with numerical integration.
Furthermore, employing numerical integration allows us to evaluate the Köhler
curve in unapproximated form () to corroborate
the findings obtained with the assumption of rd≪rw.
To this end, a numerical solver was implemented using the
libcloudph++ library
and the CVODE adaptive-timestep integrator
.
Numerical integration is carried out for a system equivalent to
Eq. ()
but expressed in terms of the state variables used in libcloudph++:
water vapour mixing ratio, potential temperature and wet radius
see Appendix A in; supersaturation S=RH-1
is diagnosed from the three state variables.
The solver code is free and open-source and is available as a supplement to
this note.
In order to depict an activation–deactivation cycle, the vertical velocity w
was set to a sinusoidal function of time t such that the maximal
displacement is reached at t=thlf and the average velocity is 〈w〉:
w=〈w〉π2sinπtthlf.
Figure summarises results of nine simulations in three types of
coordinates:
displacement vs. supersaturation (the top row),
supersaturation vs. wet radius (the middle row, same coordinates as in Fig. ) and
displacement vs. wet radius (bottom row).
The nine model runs correspond to three sets of aerosol parameters
(left, middle and right columns) and three values of mean vertical velocity
(depicted by line thickness).
The varied aerosol input parameters are the concentration (NSTP of
50 and 500 cm-3, STP subscript corresponding to the values at
standard temperature and pressure)
and the dry radius (rd of 0.1 and 0.05 µm).
In all panels, black lines correspond to air parcel ascent (activation)
and orange lines correspond to the descent (deactivation).
Besides integration results, the panels in the middle row feature the Köhler
curve plotted with thick grey line in the background.
The plots depict that for mean velocities of 100 and
50cms-1
activation and deactivation are not symmetric and happen far from equilibrium
(the Köhler curve).
This type of hysteresis corresponds to the kinetic limitations on the transfer
of water molecules to/from the droplet surface, which
prevents the droplets from attaining equilibrium under rapidly changing ambient
conditions.
At much lower velocity of 0.2cms-1, the processes are
symmetric and
match the equilibrium curve, but only
for the N=500cm-3 and rd=0.1µm (middle column).
A twofold decrease of the dry radius (right column)
as well as a tenfold decrease of particle concentration (left column)
both cause the system to exhibit a hysteretic
behaviour also at the lowest considered velocity.
This hysteresis is characterised by a “jump” in the wet radius that qualitatively
matches the envisioned catastrophic behaviour associated with the cusp bifurcation.
This behaviour is robust to further reduction in the vertical velocity (not
shown),
confirming that a close-to-equilibrium regime was attained.
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 10-6).
The only exception is the symmetric case which does not feature
the “jump” back onto the equilibrium curve.
Monodisperse system: limitations and applicability
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 e.g. in the cooling rate T˙; seediscussion
of Figs. 10–11 and other studies referenced therein.
The excitations
influence the partitioning between activated and unactivated CCN, and
decay when the characteristic timescale (period) of fluctuations
is largely longer or shorter than the activation timescale
discussed in Sect. .
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 see, e.g.,as well as works referred therein.
In these models, in the spirit of the particle-in-cell approach, the liquid
water is represented with computational particles, each representing a
multiplicity of real-world particles with monodisperse size.
In such models, the particles can undergo repeated activation–deactivation cycles,
potentially also at low vertical velocities.
Consequently, the close-to-equilibrium catastrophic hysteresis observed in
the presented simulations, even if of no foreseeable relevance to the
macroscopic behaviour of the large-scale cloud systems modelled with the
particle-based techniques, has to be taken into account
when developing numerical integration schemes.
Concluding remarks
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 Supplement related to this article is available online at https://doi.org/10.5194/npg-24-535-2017-supplement.
The authors declare that they have no
conflict of interest.
Acknowledgements
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
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