# Scientific Description¶

The simplest tools available for describing the growth and evolution of a cloud droplet spectrum from a given population of aerosols are based on zero-dimensional, adiabatic cloud parcel models. By employing a detailed description of the condensation of ambient water vapor onto the growing droplets, these models can accurately describe the activation of a subset of the aerosol population by predicting how the presence of the aerosols in the updraft modify the maximum supersaturation achieved as the parcel rises. Furthermore, these models serve as the theoretical basis or reference for parameterizations of droplet activation which are included in modern general circulation models ([Ghan2011]) .

The complexity of these models varies with the range of physical processes one wishes to study. At the most complex end of the spectrum, one might wish to accurately resolve chemical transfer between the gas and aqueous phase in addition to physical transformations such as collision/coalescence. One could also add ice-phase processes to such a model.

## Model Formulation¶

The adiabatic cloud parcel model implemented here is based on models described in the literature ([Nenes2001], [SP2006],) with some modifications and improvements. For a full description of the parcel model, please see ([Rothenberg2016]) The conservation of heat in a parcel of air rising at constant velocity $$V$$ without entrainment can be written as

(1)$\frac{dT}{dt} = -\frac{gV}{c_p} - \frac{L}{c_p}\frac{d w_v}{dt}$

where $$T$$ is the parcel air temperature. Assuming adiabaticity and neglecting entrainment is suitable for studying cloud droplet formation near the cloud base, where the majority of droplet activation occurs. Because the mass of water must be conserved as it changes from the vapor to liquid phase, the relationship

(2)$\frac{d w_v}{dt} = - \frac{dw_c}{dt}$

must hold, where $$w_v$$ and $$w_c$$ are the mass mixing ratios of water vapor and liquid water (condensed in droplets) in the parcel. The rate of change of water in the liquid phase in the parcel is governed solely by condensation onto the existing droplet population. For a population of $$N_i$$ droplets of radius $$r_i$$, where $$i=1,\dots,n$$, the total condensation rate is given by

(3)$\frac{dw_c}{dt} = \frac{4\pi \rho_w}{\rho_a}\sum\limits_{i=1}^nN_ir_i^2\frac{dr_i}{dt}$

Here, the particle growth rate, $$\frac{dr_i}{dt}$$ is calculated as

(4)$\frac{dr_i}{dt} = \frac{G}{r_i}(S-S_{eq})$

where $$G$$ is a growth coefficient which is a function of the physical and chemical properties of the particle receiving condensate, given by

(5)$G = \left(\frac{\rho_w R T}{e_s D'_v M_w} + \frac{L\rho_w[(LM_w/RT) - 1]}{k'_a T}\right)^{-1}$

Droplet growth via condensation is modulated by the difference between the environmental supersaturation, $$S$$, and the droplet equilibrium supersaturation, $$S_{eq}$$, predicted from Kohler theory. To account for differences in aerosol chemical properties which could affect the ability for particles to uptake water, the $$\kappa$$-Köhler theory parameterization ([PK2007]) is employed in the model. $$\kappa$$-Kohler theory utilizes a single parameter to describe aerosol hygroscopicity, and is widely employed in modeling of aerosol processes. The hygroscopicity parameter $$\kappa$$ is related to the water activity of an aqueous aerosol solution by

$\frac{1}{a_w} = 1 + \kappa\frac{V_s}{V_w}$

where $$V_s$$ and $$V_w$$ are the volumes of dy particulate matter and water in the aerosol solution. With this parameter, the full $$\kappa$$-Kohler theory may be expressed as

(6)$S_{eq} = \frac{r_i^3 - r_{d,i}^3}{r_i^3 - r_{d,i}^3(1-\kappa_i)}\exp\left( \frac{2M_w\sigma_w}{RT\rho_w r_i} \right) - 1$

where $$r_d$$ and $$r$$ are the dry aerosol particle size and the total radius of the wetted aerosol. The surface tension of water, $$\sigma_w$$, is dependent on the temperature of the parcel such that $$\sigma_w = 0.0761 - 1.55\times 10^{-4}(T-273.15)$$ J/m$$^2$$ . Both the diffusivity and thermal conductivity of air have been modified in the growth coefficient equation to account for non-continuum effects as droplets grow, and are given by the expressions

$D'_v = D_v\bigg/\left(1 + \frac{D_v}{a_c r}\sqrt{\frac{2\pi M_w}{RT}}\right)$

and

$k'_a = k_a\bigg/\left(1 + \frac{k_a}{a_T r \rho_a c_p}\sqrt{\frac{2\pi M_a}{RT}} \right)$

In these expressions, the thermal accommodation coefficient, $$a_T$$, is assumed to be $$0.96$$ and the condensation coefficient, $$a_c$$ is taken as unity (see Constants). Under the adiabatic assumption, the evolution of the parcel’s supersaturation is governed by the balance between condensational heating as water vapor condenses onto droplets and cooling induced by the parcel’s vertical motion,

(7)$\frac{dS}{dt} = \alpha V - \gamma\frac{w_c}{dt}$

where $$\alpha$$ and $$\gamma$$ are functions which are weakly dependent on temperature and pressure :

$\alpha = \frac{gM_wL}{c_pRT^2} - \frac{gM_a}{RT}$
$\gamma = \frac{PM_a}{e_sM_w} + \frac{M_wL^2}{c_pRT^2}$

The parcel’s pressure is predicted using the hydrostatic relationship, accounting for moisture by using virtual temperature (which can always be diagnosed as the model tracks the specific humidity through the mass mixing ratio of water vapor),

(8)$\frac{dP}{dt} = \frac{-g P V}{R_d T_v}$

The equations (8), (7), (3), (2), and (1) provide a simple, closed system of ordinary differential equations which can be numerically integrated forward in time. Furthermore, this model formulation allows the simulation of an arbitrary configuration of initial aerosols, in terms of size, number concentration, and hygroscopicity. Adding additional aerosol size bins is simply accomplished by tracking one additional size bin in the system of ODE’s. The important application of this feature is that the model can be configured to simulate both internal or external mixtures of aerosols, or some combination thereof.

## Model Implementation and Procedure¶

The parcel model described in the previous section was implemented using a modern modular and object-oriented software engineering framework. This framework allows the model to be simply configured with myriad initial conditions and aerosol populations. It also enables model components - such as the numerical solver or condensation parameterization - to be swapped and replaced. Most importantly, the use of object-oriented techniques allows the model to be incorporated into frameworks which grossly accelerate the speed at which the model can be evaluated. For instance, although models like the one developed here are relatively cheap to execute, large ensembles of model runs have been limited in scope to several hundred or a thousand runs. However, the framework of this particular parcel model implementation was designed such that it could be run as a black box as part of a massively-parallel ensemble driver.

To run the model, a set of initial conditions needs to be specified, which includes the updraft speed, the parcel’s initial temperature, pressure, and supersaturation, and the aerosol population. Given these parameters, the model calculates an initial equilibrium droplet spectrum by computing the equilibrium wet radii of each aerosol. This is calculated numerically from the Kohler equation for each aerosol/proto-droplet, or numerically by employing the typical Kohler theory approximation

$S \approx \frac{A}{r} - \kappa\frac{r_d^3}{r^3}$

These wet radii are used as the initial droplet radii in the simulation.

Once the initial conditions have been configured, the model is integrated forward in time with a numerical solver (see ParcelModel.run() for more details). The available solvers wrapped here are:

• LSODA(R)
• LSODE
• (C)VODE

During the model integration, the size representing each aerosol bin is allowed to grow via condensation, producing something akin to a moving grid. In the future, a fixed Eulerian grid will likely be implemented in the model for comparison.

## Aerosol Population Specification¶

The model may be supplied with any arbitrary population of aerosols, providing the population can be approximated with a sectional representation. Most commonly, aerosol size distributions are represented with a continuous lognormal distribution,

(9)$n_N(r) = \frac{dN}{d \ln r} = \frac{N_t}{\sqrt{2\pi}\ln \sigma_g}\exp\left(-\frac{ \ln^2(r/\mu_g)}{2\ln^2\sigma_g}\right)$

which can be summarized with the set of three parameters, $$(N_t, \mu_g, \sigma_g)$$ and correspond, respectively, to the total aerosol number concentration, the geometric mean or number mode radius, and the geometric standard deviation. Complicated multi-modal aerosol distributions can often be represented as the sum of several lognormal distributions. Since the parcel model describes the evolution of a discrete aerosol size spectrum, can be broken into $$n$$ bins, and the continuous aerosol size distribution approximated by taking the number concentration and size at the geometric mean value in each bin, such that the discrete approximation to the aerosol size distribution becomes

$n_{N,i}(r_i) = \sum\limits_{i=1}^n\frac{N_i}{\sqrt{2\pi}\ln\sigma_g}\exp\left(-\frac{\ln^2(r_i/\mu_g)}{2\ln^2\sigma_g}\right)$

If no bounds on the size range of $$r_i$$ is specified, then the model pre-computes $$n$$ equally-spaced bins over the logarithm of $$r$$, and covers the size range $$\mu_g/10\sigma_g$$ to $$10\sigma_g\mu_g$$. It is typical to run the model with $$200$$ size bins per aerosol mode. Neither this model nor similar ones exhibit much sensitivity towards the density of the sectional discretization .

Typically, a single value for hygroscopicity, $$\kappa$$ is prescribed for each aerosol mode. However, the model tracks a hygroscopicity parameter for each individual size bin, so size-dependent aerosol composition can be incorporated into the aerosol population. This representation of the aerosol population is similar to the external mixing state assumption. An advantage to using this representation is that complex mixing states can be represented by adding various size bins, each with their own number concentration and hygroscopicity.

## References¶

 [Nenes2001] Nenes, A., Ghan, S., Abdul-Razzak, H., Chuang, P. Y. & Seinfeld, J. H. Kinetic limitations on cloud droplet formation and impact on cloud albedo. Tellus 53, 133–149 (2001).
 [SP2006] Seinfeld, J. H. & Pandis, S. N. Atmospheric Chemistry and Physics: From Air Pollution to Climate Change. Atmos. Chem. Phys. 2nd, 1203 (Wiley, 2006).
 [Rothenberg2016] Daniel Rothenberg and Chien Wang, 2016: Metamodeling of Droplet Activation for Global Climate Models. J. Atmos. Sci., 73, 1255–1272. doi: http://dx.doi.org/10.1175/JAS-D-15-0223.1
 [PK2007] Petters, M. D. & Kreidenweis, S. M. A single parameter representation of hygroscopic growth and cloud condensation nucleus activity. Atmos. Chem. Phys. 7, 1961–1971 (2007).
 [Ghan2011] Ghan, S. J. et al. Droplet nucleation: Physically-based parameterizations and comparative evaluation. J. Adv. Model. Earth Syst. 3, M10001 (2011).