# Exoplanet Clouds Chapter v2.1.docx

Clouds and Hazes in Exoplanet
Atmospheres

Mark S. Marley
NASA Ames Research Center

Andrew S. Ackerman

NASA Goddard Institute for Space Studies

Jeffrey N. Cuzzi

NASA Ames Research Center

Daniel Kitzmann

Zentrum für Astronomie und Astrophysik,
Technische Universität Berlin

## Abstract

Clouds and hazes are commonplace in the atmospheres of solar system planets
and are likely ubiquitous in the atmospheres of extrasolar planets as well.
Clouds affect every aspect of a planetary atmosphere, from the transport of
radiation, to atmospheric chemistry, to dynamics and they influence - if not
control - aspects such as surface temperature and habitability. In this review
we aim to provide an introduction to the role and properties of clouds in
exoplanetary atmospheres. We consider the role clouds play in influencing the
spectra of planets as well as their habitability and detectability. We briefly
summarize how clouds are treated in terrestrial climate models and consider
the far simpler approaches that have been taken so far to model exoplanet
clouds, the evidence for which we also review. Since clouds play a major role
in the atmospheres of certain classes of brown dwarfs we briefly discuss brown
dwarf cloud modeling as well. We also review how the scattering and extinction
efficiencies of cloud particles may be approximated in certain limiting cases
of small and large particles in order to facilitate physical understanding.
Since clouds play such important roles in planetary atmospheres, cloud
modeling may well prove to be the limiting factor in our ability to interpret
future observations of extrasolar planets.

##  1 Introduction

Clouds and hazes are found in every substantial solar system atmosphere and
are likely ubiquitous in extrasolar planetary atmospheres as well. They
provide sinks for volatile compounds and influence both the deposition of
incident flux and the propagation of emitted thermal radiation. Consequently
they affect the atmospheric thermal profile, the global climate, the spectra
of scattered and emitted radiation, and the detectability by direct imaging of
a planet. As other chapters in this book attest, clouds and hazes are a
complex and deep subject. In this chapter we will broadly discuss the roles of
clouds and hazes as they relate to the study of exoplanet atmospheres. In
particular we will focus on the challenge of exoplanet cloud modelling and
discuss the impact of condensates on planetary climates.

The terms “clouds” and “hazes” are sometimes used interchangeably. Here we use
the term “cloud” to refer to condensates that grow from an atmospheric
constituent when the partial pressure of the vapor exceeds its saturation
vapor pressure. Such supersaturation is typically produced by atmospheric
cooling, and cloud particles will generally evaporate or sublimate in
unsaturated conditions. A general framework for such clouds in planetary
atmospheres is provided by Sánchez-Lavega et al. (2004). By “haze” we refer to
condensates of vapor produced by photochemistry or other non-equilibrium
chemical processes. This usage is quite different from that of the terrestrial
water cloud microphysics literature where the distinction depends on water
droplet size and atmospheric conditions.

Because exoplanetary atmospheres can plausibly span such a wide range of
compositions as well as temperature and pressure conditions, a large number of
species may form clouds. Depending on conditions, clouds in a solar
composition atmosphere can include exotic refractory species such as Al2O3,
CaTiO3, Mg2SiO4 and Fe at high temperature and Na2S, MnS, and of course H2O at
lower temperatures. Many other species condense as well including CO2 in cold,
Mars-like atmospheres and NH3 in the atmospheres of cool giants, like Jupiter
and Saturn. In Earth-like atmospheres water clouds are likely important
although Venus-like conditions and sulfuric-acid or other clouds are
possibilities as well. Depending on atmospheric temperature, pressure, and
composition the range of possibilities is very large. Furthermore not all
clouds condense directly from the gas to a solid or liquid phase as the same
species. For example in the atmosphere of a gas giant exoplanet solid MnS
cloud particles are expected to form around 1400 K from the net reaction H2S +
Mn$\rightarrow$MnS(s) + H2 (Visscher et al. 2006).

Clouds strongly interact with incident and emitted radiative fluxes. The
clouds of Earth and Venus increase the planetary Bond albedo (the fraction of
all incident flux that is scattered back to space) and consequently decrease
the equilibrium temperature. Clouds can also “trap” infrared radiation and
heat the atmosphere. Hazes, in contrast, because of their usually smaller
particle sizes can scatter incident light away from a planet but not strongly
affect emergent thermal radiation, and thus predominantly result in a net
cooling. The hazes of Titan play such an “anti-greenhouse effect” role in the
energy balance of the atmosphere. For these reasons global atmospheric models
of exoplanets, including those aiming to define the habitable zone given
various assumptions, must consider the effects of clouds. However clouds are
just one ingredient in such planetary atmosphere models. Bulk atmospheric
composition, incident flux, gravity, chemistry, molecular and atomic
opacities, and more must all be integrated along with the effect of clouds in
order to construct realistic models. Introductory reviews by Burrows & Orton
(2010) and Seager & Deming (2010) cover the important fundamentals atmospheric
modeling and place cloud models in their broader context.

In the remainder of this chapter we discuss the importance of clouds to
exoplanet atmospheres, particularly considering their impact on habitability,
discuss cloud modeling in general and the types of models developed for
exoplanet studies, and finally briefly review observations of exoplanet
clouds. Because clouds have played such a large role in efforts to understand
the atmospheres of brown dwarfs, we also briefly review the findings of this
field. We conclude with an overview of how the Mie opacity of particles
behaves in various limits and consider the case of fluffy particles.

##  2 Importance of Clouds and Hazes in Exoplanet Atmospheres

###  2.1 Albedo, Detectability, and Characterization

Before discussing the role on clouds on the spectra of extrasolar planets it
is worthwhile to review the various albedos which enter the discussion. The
Bond or bolometric albedo is the fraction of all incident light, integrated
over the entire stellar spectrum, which is scattered back to space by a
planet. This albedo is a single number and enters into the computation of a
planet’s equilibrium temperature (the temperature an airless planet would have
if its thermal emission were equal to the incident radiation which it
absorbs). It is also useful to know the monochromatic ratio of all scattered
to incident light as a function of wavelength, which is the spherical albedo.
For historical reasons it is more common to discuss the geometric albedo,
which is specifically the wavelength-dependent ratio of the light scattered by
the entire planet in the direction directly back towards its star compared to
that which would be so scattered by a perfectly reflecting Lambert disk of the
same radius as the planet. Care must be taken to distinguish all these albedos
as they can differ markedly from one another even for the same planet and the
literature is rife with confusion between them.

Perhaps the single most important effect of clouds is to brighten the
reflected light spectra of exoplanets, particularly at optical wavelengths (
$0.38<\lambda<1$ μm). For planets with appreciable atmospheres, Rayleigh
scattering is most efficient in the blue. At longer wavelengths however
absorption by either the planetary solid surface or oceans (for terrestrial
planets) or by atmospheric gaseous absorbers becomes important in the red.
This is because for common molecules at planetary temperatures, such as H2O,
CO2, or CH4, vibrational-rotational transitions become important at
wavenumbers below about 15,000 cm-1 or wavelength $\lambda>0.6$ μm. Except for
diatomic species (notably O2) planetary atmospheres are generally not warm
enough to exhibit strong electronic absorption features in the optical. Thus
the reflected light or geometric albedo spectrum of a generic cloudless planet
with an atmosphere would be bright at blue wavelengths from Rayleigh
scattering and dark in the red and at longer wavelengths from gaseous
molecular or surface absorption.

Clouds, however, tend to be bright with a fairly gray opacity through the
optical. Thus a thick, scattering cloud can brighten a planet in the far red
by scattering more light back to space than a cloudless planet. As a result
two similar planets, one with and one without cloud cover, will have very
different geometric albedos in the red, and consequently differing brightness
contrasts with their parent stars. The mean Earth water clouds increase the
contrast between the reflection spectrum of an Earth-like planet and its host
star by one order of magnitude (Kitzmann et al. 2011b). This effect, first
noted for giant planets by Marley et al. (1999) and Sudarsky et al. (2000) and
further explored in Cahoy et al. (2010) is illustrated in Figure 1 which plots
geometric albedo as a function of $\lambda$ for giant planets with and without
clouds. A warm, cloudless atmosphere is dark in reflected light while a cooler
atmosphere, sporting water clouds, is much brighter. In this case clouds
affect the model spectra far more than a factor of three difference in the
atmospheric abundance of heavy elements, which is also shown.

In the case of searches for planets by direct coronagraphic imaging in
reflected light, clouds may even control whether or not a planet is
discovered. Depending on the spectral bandpass used for planet discovery at a
given distance from its primary star a cloudy planet may be brighter and more
detectable than a cloudless one. Discussions of the influence of clouds and
albedo on detectability include those by Tinetti et al. (2006a) and Kitzmann
et al. (2011a).

Figure 1: Model geometric albedo spectra for Jupiter-like planets at 2 AU
(green) and 0.8 AU (red) from their parent star. Solid and dashed lines lines
show models with a solar abundance and a 3-times enhanced abundance of heavy
elements respectively. Prominent absorption features are labeled. The 2 AU
model planets possess a thick water cloud while the 0.8 AU models are warmer
and cloudless and consequently darker in reflected light. Figure modified from
Cahoy et al. (2010).

Once a planet is detected, spectra are needed to characterize atmospheric
abundances of important molecules. For terrestrial extrasolar planets this is
best done by the analysis of the thermal emission spectrum (Selsis, 2004;
Tinetti et al., 2012). Clouds, however, may conceal the thermal emission from
the surface and dampen spectral features of molecules (e.g., the bio-
indicators N2O or O3). Indeed, clouds on Earth have a larger impact on the
emitted infrared flux than the differences between night and day (Hearty et
al., 2009; Tinetti et al., 2006a). Thermal emission spectra are therefore very
sensitive to the types and fractional coverages of clouds present in the
atmosphere. At high spectral resolution the most important terrestrial
molecular spectral features in the mid-infrared, such as O3, remain detectable
even for cloud covered conditions in many cases. Figure 2 shows thermal
emission spectra affected by low-level water droplet and high-level water ice
clouds of an Earth-like planet orbiting different main-sequence dwarf stars
(adopted from Vasquez et al., 2013). With increasing cloud cover of either
cloud type the important 9.6 μm absorption band of ozone is strongly dampened,
along with an overall decrease in the thermal radiation flux. For some cases
presented in Fig. 2 (e.g. for the F-type star and 100% high-level cloud cover)
the ozone band seems to be completely absent or even appears in emission
rather than absorption (F-type star, 100% low-level clouds). At lower spectral
resolution, such as could be obtained for terrestrial exoplanets in the near
future, clouds render the molecular features even less detectable; thus clouds
will strongly affect the determination of their atmospheric composition. For
example, as shown by Kitzmann et al. (2011a) a substantial amount of water
clouds in an Earth-like atmosphere can completely hide the spectral signature
of the bio-indicator ozone in low-resolution thermal emission spectra.

Figure 2: Planetary thermal emission spectra influenced by low-level water
droplet (left panel) and high-level ice clouds (right panel) for an Earth-like
planet orbiting different kinds of main-sequence host stars (adopted from
Vasquez et al. 2013). For each central star the spectra are shown for
different cloud coverages. Note especially the strong impact of the cloud
layers on the 9.6 μm absorption band of ozone.

Clouds can also obscure spectral features originating from the surface of a
planet. In principle, signals of surface vegetation (“vegetation red edge”)
are present in the reflected light spectra of Earth-like planets. However, as
often pointed out (e.g., Arnold et al. (2002) and Hamdani et al. (2006)) this
spectral feature can easily be concealed by clouds. For the detectability of
possible vegetation signatures of terrestrial extrasolar planets Montañés-
Rodríguez et al. (2006) and Tinetti et al. (2006b) concluded that clouds play
a crucial role for these signatures in the reflection spectra. Thus, apart
from the scattering characteristics of different planetary surface types, the
presence of clouds has been found to be one of the most important factors
determining reflection spectra.

Transmission spectra of transiting planets can be used to obtain many
atmospheric properties, such as atmospheric composition and temperature
profiles. Transmission spectroscopy of extrasolar giant planets has already
proven to be a successful method for the characterization of giant exoplanets.
As shown by Pallé et al. (2009) the major atmospheric constituents of a
terrestrial planet remain detectable in transmission spectra even at very low
signal-to-noise ratios. Thus, transmission spectra can provide more
information about the atmospheres of exoplanets than reflection spectra.
Theoretical transmission spectra of Earth-sized transiting planets have been
studied by Ehrenreich et al. (2006) including the effects of optically thick
cloud layers. Their results show that the transmission spectra only contain
information about the atmosphere above the cloud layer and that clouds can
effectively increase the apparent radius of the planet. The impact of clouds
on the transmission spectra therefore depends strongly on their atmospheric
height. If cloud layers are only located in the lower atmosphere, which is
already opaque due to absorption and scattering by gas species, their overall
effect on the spectrum will be small (Kaltenegger & Traub, 2009).

###  2.2 Habitability of Extrasolar Terrestrial Planets

From the terrestrial bodies with an atmosphere in the solar system we know
that clouds are a common phenomenon and should also be expected to occur in
atmospheres of terrestrial extrasolar planets. Apart from the usual well-known
greenhouse gases, clouds have the most important climatic impact in the
atmospheres of terrestrial planets by affecting the energy budget in several
ways. Firstly, clouds can scatter incident stellar radiation back to space
resulting in atmospheric cooling (albedo effect). On the other hand, clouds
can trap thermal radiation within the lower atmosphere by either absorption
and re-emission at their local temperature (classical greenhouse effect) or by
scattering thermal radiation back towards the surface (scattering greenhouse
effect), which heats the lower atmosphere and planetary surface. All these
effects are determined by the wavelength-dependent optical properties of the
cloud particles (absorption and scattering cross-sections, single scattering
albedo, asymmetry parameter, scattering phase function). These properties can
differ considerably for different cloud forming species (owing to their
refractive indices) and atmospheric conditions (composition and temperature
structure). Note that the single scattering albedo, yet a fourth type of
albedo (see Section 2.1 for the others), measures the fraction of all incident
light scattered by a single cloud particle.

Life as we know it requires the presence of liquid water to form and survive.
In the context of terrestrial exoplanets we are primarily concerned with
habitable conditions on the planetary surface. Lifeforms may of course also
exist in other environments, such as deep under the planetary surface or
within a subsurface ocean (Lammer et al., 2009). Since there is no possibility
of detecting the presence of such habitats by remote observations our current
definition of a habitable terrestrial planet assumes a reservoir of liquid
water somewhere on its surface. Liquid surface water implies that these
planets would also have abundant water vapor in their atmospheres. We can
therefore safely assume that water (and water ice) clouds will naturally occur
in the atmospheres of these exoplanets throughout the classical habitable
zone.

As such, H2O and CO2 clouds are the prime focus for habitable exoplanets.
Other possible condensing species also found in our solar system include C2H6
and CH4 (e.g., in the atmosphere of Titan), N2 (Triton), or H2SO4 (Venus,
Earth). Any chemical species in liquid form on a planetary surface can in
principle be considered as a potential cloud-forming species in the atmosphere
if its atmospheric partial pressure is high enough and the temperature low
enough.

Given these considerations it is clear that clouds are important for the
determination of the boundaries of the classical habitable zone around
different kinds of stars. In the next two subsections we discuss some of the
ways that clouds influence the habitable zone boundaries.

####  2.2.1 Inner Edge of the Habitable Zone

The closer a terrestrial planet orbits its host star the greater stellar
insolation and consequently surface temperature. This leads to enhanced
evaporation of surface water and increases the amount of water vapor in the
atmosphere. The strong greenhouse effect by this additional water vapor
further increases the surface temperature and therefore the evaporation of
surface water, resulting in a positive feedback cycle (see Covey et al., this
volume).

The inner boundary of the classical habitable zone is usually defined either
as the distance from a host star at which an Earth-like planet completely
loses its liquid surface water by evaporation (runaway greenhouse limit), or
as the distance from the host star when water vapor can reach the upper
atmosphere (water loss limit). The latter is the definition favored by Kasting
et al. (1993); the former definition is used by Hart (1979). In the present
Earth atmosphere water escape is limited by the cold trap at the tropopause
which loses its efficiency in a sufficiently warm, moist atmosphere (Kasting
et al., 1993).

Using a one-dimensional atmospheric model, Kasting et al. (1993) estimated
that the inner boundary of the habitable zone for the runaway greenhouse
scenario for an Earth around the Sun would be located at 0.84 AU. This
distance, however, was calculated with the clouds treated as a surface albedo
effect and neglecting any feedbacks. For example once formed clouds increase
the planetary albedo which in turn partially reduces the temperature increase
due to the higher stellar insolation and the cloud greenhouse effect. Kasting
(1988) investigated some feedback effects and concluded that with a single
layer of thick water clouds Earth could be moved as close as 0.5 AU (for a
cloud coverage of 100%) or 0.67 AU (50% cloud coverage) from the Sun before
all liquid surface water would be lost. Goldblatt & Zahnle (2011) explored the
various feedback issues in detail and concluded that more sophisticated
modeling approaches are necessary to explore habitability.

Water clouds also contribute to the greenhouse effect. Depending on the
temperature at which the cloud emits thermal radiation, its greenhouse effect
can match or exceed the albedo effect. Cold clouds, such as cirrus, are net
greenhouse warmers. Low clouds with temperatures near the effective radiating
temperature affect climate almost entirely by their albedo, which depends on
such factors as patchiness, cloud thickness, and the size distributions and
composition of cloud particles. The exact climatic impact of clouds therefore
crucially depends on the balance between their greenhouse and albedo effects.
Whether the greenhouse or albedo effect dominates for clouds forming under
runaway greenhouse conditions cannot be easily determined a priori. A better
understanding of the cloud microphysics and convection processes in moist
atmospheres during a runaway greenhouse process is needed to determine the
cloud properties and their fractional coverage near the inner habitable zone
(HZ) boundary. The effect of clouds under runaway greenhouse conditions
represents one of the most important unresolved issues in planetary climate.

Note that this discussion is relevant to wet planets with moist atmospheres
fed by extensive seas. Dry, land or desert, planets will have unsaturated
atmospheres in the tropics and thus radiate at a higher temperature and cool
more efficiently than planets with a water saturated atmosphere. As a result
the habitable zone for such planets may be larger (Abe et al., 2011).

####  2.2.2 Outer Edge of the Habitable Zone

The outer edge of the habitable zone is set by the point at which there is no
longer liquid water available at the surface as it is locked in ice. With
falling insolation planets found progressively farther away from their central
star have lower atmospheric and surface temperatures. With lower surface
temperatures, the removal of CO2 from the atmosphere due to the carbonate-
silicate cycle, which controls the amount of CO2 in Earth’s atmosphere on time
scales of order a million years, becomes less efficient (see Covey et al.,
this volume). Thus, if the terrestrial planet is still geologically active,
CO2 can accumulate in the atmosphere by volcanic outgassing. With decreasing
atmospheric temperatures, CO2 will condense at some point to form clouds
composed of CO2 ice crystals. Because condensation nuclei can be expected to
be available in atmospheres of terrestrial planets, the most dominant
nucleation process for the formation of CO2 clouds is usually assumed to be
heterogeneous nucleation (Glandorf et al 2002), in which cloud particles form
on existing seed particles.

Just like water clouds, the presence of CO2 clouds will result in an increase
of the planetary albedo by scattering incident stellar radiation back to
space. However, in contrast to water, CO2 ice is almost transparent in the
infrared (Hansen, 1997, 2005) except within some strong absorption bands. Thus
CO2 clouds are unlikely to trigger a substantial classical greenhouse effect
by absorption and re-emission of thermal radiation.

On the other hand, CO2 ice particles can efficiently scatter thermal
radiation. This allows for a scattering greenhouse effect in which a fraction
of the outgoing thermal radiation is scattered back towards the planetary
surface (Forget & Pierrehumbert, 1997). Depending on the cloud properties this
scattering greenhouse effect can outweigh the albedo effect and can in
principle increase the surface temperature above the freezing point of water.
However, the scattering greenhouse effect is a complex function of the cloud
optical depth and particle size (Colaprete and Toon, 2003). Furthermore,
because the greenhouse effect of CO2 clouds depends on the scattering
properties of the ice particles, the particle shape (which cannot be expected
as spherical) or particle surface roughness may play an important role.
However, these effects cannot be easily quantified because neither the
particle shapes nor their surface properties are known. These and other
uncertainties in the cloud microphysics of CO2 ice in cool CO2 dominated
atmospheres makes the calculation of the position of the outer HZ boundary
complicated. More details on the formation of CO2 clouds in the present and
early Martian atmosphere can be found in Glandorf et al. (2002), Colaprete &
Toon (2003), and Määttänen et al. (2005), as well as in Esposito et al. (this
volume). Inferences about CO2 cloud particle sizes in the current martian
atmosphere as constrained by a variety of datasets are presented by Hu et al.
(2012).

For a fully cloud-covered early Mars with thick a CO2 dominated atmosphere and
CO2 clouds composed of spherical ice particles Forget & Pierrehumbert (1997)
estimated that the outer boundary of the HZ is located at 2.4 AU, in contrast
to the cloud-free boundary of 1.67 AU by Kasting et al. (1993). This greater
value has been further used by Selsis et al. (2007) to extrapolate the effects
of CO2 clouds on the outer HZ boundary towards other main sequence central
stars (see also Kaltenegger & Sasselov, 2011).

For terrestrial super-Earths it has been suggested by Pierrehumbert & Gaidos
(2011) that these planets could have retained much of their primordial H2
atmosphere owing to their greater mass. According to their model study, the
classical habitable zone might be far larger than expected for a CO2 dominated
atmosphere although this study did not explore the impact of clouds.

##  3 Clouds and Radiation

Clouds are important for planetary atmospheres because they interact with both
incident short wavelength radiation from the parent star and emergent thermal
radiation emitted by the planetary surface, if present, and the atmosphere
itself. Perhaps the simplest example of such interaction occurs for spherical
cloud particles composed of a single constituent. In this case the wavelength
dependent optical properties can be computed from Mie calculations that depend
only on particle size and the wavelength-dependent complex refractive index of
the bulk condensate. In this section we briefly review the basics of the
interaction of cloud particles with radiation and summarize the use of Mie
theory for modeling this interaction as well as point out some useful
simplifications that can be applied in certain limiting cases.

###  3.1 Basic Radiative Properties of Cloud Particles

Cloud opacity ultimately depends upon the radiative properties of the
constituent particles. A particle with radius r has a cross section to scatter
losslessly or to absorb incident radiation at some wavelength λ, given by Cs
or Ca respectively. These cross sections are defined as $C_{s}=Q_{s}\pi r^{2}$
and $C_{a}=Q_{a}\pi r^{2}$, and their sum is the extinction cross section Ce.
The scattering and absorption efficiencies Qs and Qa, and, from them, the
extinction efficiency Qe = Qs \+ Qa is thus defined. The particle single
scattering albedo is $\omega=Q_{s}/Q_{e}$, and the scattering phase function
is $P\left(\theta\right)$. These quantities can be computed by a Mie code,
which computes the electromagnetic wave propagation through spherical
particles, given a tabulation of the optical properties of the cloud material.
Good physically-based introductions to radiative properties of particles can
be found in van de Hulst (1957), Hansen & Travis (1974), and Liou (2002).

Modeling cloud particle radiative properties can seem forbidding; workers
often think a Mie scattering code, with all its exotic predictions, is needed.
However, in many applications, much simpler approaches not only can provide
very good quantitative fidelity - and greater physical insight - but also can
be applied directly to more complex kinds of particles than the spheres for
which Mie theory is derived.

The direct beam of radiant flux vertically traversing a cloud layer of
particles composed of some material q is reduced by a factor
$\exp\left(-\tau_{q}\right)$, where the layer optical depth is
$\tau_{q}=nQ_{e}\pi r^{2}H_{q}=nC_{e}H_{q}$, and n is the particle number
density, r the particle radius, and Hq the vertical thickness of the layer
(assuming a uniform vertical distribution for simplicity). Qe and thus
$\tau_{q}$ are functions of the wavelength λ, through the λ–dependent real and
imaginary refractive indices of the material in question (nr , ni; see below,
and Draine & Lee, 1984 or Pollack et al., 1994 or  http://www.astro.uni-
jena.de/Laboratory/Database/databases.html for typical values). The particle
opacity ϰ (in units of length-squared per mass) is the effective particle
cross section per unit mass of either solids or gas.

If the phase function is strongly forward scattering, as can be the case for
particles with $r/\lambda>10$ or so (Hansen & Travis 1974), much of the
“scattering” can be approximated as unattenuated radiation by reducing the
layer optical depth and renormalizing the particle albedo and phase function.
This is done by applying simple corrections to both ω and $\tau_{q}$ known as
similarity relations. Irvine, (1975), van de Hulst (1980), and Liou (2002)
review these, and present a number of ways to solve the radiative transfer
problem in general.

###  3.2 Refractive Indices and Opacity

All calculations of particle radiative properties (Qe, Qs, Qa, $\omega$ and
P(ϑ)) begin with the refractive indices of the material and the particle size.
While Mie codes are available for download (e.g.,
ftp://climate1.gsfc.nasa.gov/wiscombe/Single_Scatt/), analytical
approximations that are valid in all the relevant limits can often be of great
use. Such approximations rely on the fact that realistic clouds have size and
shape distributions that average away the exotic fluctuations shown by Mie
theory for particles of specific sizes (see Hansen & Travis 1974 for
examples). With current computational capabilities, Mie calculations are not
onerous, but if many wavelengths and/or grids of numerous models are of
interest, the burden is compounded, so one needs to understand whether such
detailed predictions are actually needed.

One simple approximation to scattering by equidimensional, but still
irregular, particles was developed by Pollack & Cuzzi (1980). This approach
uses Mie theory for particles with small-to-moderate $r/\lambda$, where shape
irregularities are indiscernible to the waves involved and Mie calculations
converge rapidly. For larger $r/\lambda$ a combination of diffraction,
external reflection, and internal transmission is used as adjusted for shape
and parametrized by laboratory experiments. The approach is easy to apply in
cases where the angular distribution of scattered radiation is important, as
long as a particle size distribution smears away the significant oscillations
in scattering properties which Mie theory predicts for monodisperse spheres
(Hansen & Travis 1974). This method saves on computational effort for
particles with large $r/\lambda$ but does involve a Mie code for the smaller
particles.

Even simpler approaches are feasible. In many cases of interest, only globally
averaged emergent fluxes or reflectivities, or perhaps heating calculations,
are needed; here, the details of the phase function are of less interest than
the optical depth and particle albedo, which are based only on the
efficiencies. For these cases one can do fairly well using asymptotic
expressions for efficiencies in the limiting regimes $r/\lambda{\ll}1$ and
$r/\lambda{\gg}1$, connecting them with a suitable bridging function. These
expressions depend on the refractive indices of the material in question.

The simplest limit is when $r/\lambda{\gg}1$ (geometrical optics); in fact,
many of the cloud models discussed in this chapter fall into this regime at
1-10 μm wavelengths. In this range it is convenient to neglect diffraction,
which is strongly concentrated in the forward direction and can be lumped with
the direct beam as noted above. Then, assuming the particle has density ρ and
is itself opaque, $Q_{e}{\leq}1$ and Ce reduces to the physical cross section.
Thus the solids-based opacity is simply $\kappa=3\pi r^{2}/4\rho\pi
r^{3}=3/4r\rho$. In this regime, particle growth drastically reduces the
opacity because the surface to mass ratio decreases linearly with radius
(Miyake et al., 1993; Pollack et al., 1994).

At the opposite extreme, when $r/\lambda{\ll}1$(the Rayleigh limit), simple
analytical expressions also apply (Draine & Lee 1984; Bohren & Huffman 1983).
For simplicity below we give the expressions for $n_{i}{\ll}n_{r}{\sim}1$
(appropriate for silicates, oxides, water, but not iron), but the general
expressions are not much more complicated (Draine & Lee 1984). Thus
$Q_{a}=24xn_{r}n_{i}/\left(n_{r}^{2}+2\right)^{2}$ and
$Q_{s}=8x^{4}\left(n_{r}^{2}-1\right)^{2}/3\left(n_{r}^{2}+2\right)^{2}$,
where $x=2\pi r/\lambda$(see also van de Hulst, 1957, p. 70). Because Qs
decreases much faster than Qa with decreasing $r/\lambda$ in this regime,
small particles become not only less effective at interacting with radiation,
but increasingly absorbers/emitters rather than scatterers. In this limit,
their cross section $C_{a}=Q_{a}\pi r^{2}$ per unit particle mass becomes
constant since Qa is proportional to r.

###  3.3 Shape and Porosity

In the case of terrestrial cloud droplets, in which condensation dominates
growth, and raindrops, in which coagulation dominates growth, spherical
particles of constant density are assumed, provided they fall slowly (though
drag-induced deformation is taken into account when computing terminal fall-
speeds for large raindrops). However, if particles condense from their vapor
phase as solids, tiny initial monomers may instead coagulate by sticking into
porous aggregates, perhaps having some fractal properties where the density
may depend on the size. Non-spherical particle shape adds complexity to the
computation of optical properties, which is commonly the case for solid
particles. For instance, the single-scattering properties of ice particles in
the terrestrial atmosphere depend on not only the geometric shape of the
crystals (which depend on the temperature), but also the microscopic surface
roughness. As noted by Fu et al. (1998), the extinction and absorption cross
sections depend mainly on projected areas and particle volumes (note that
random orientation is typically assumed), while the scattering phase function
are largely determined by the aspect ratio of the crystal components and their
small-scale roughness (Fu, 2007).

In the general exoplanet case for tiny particles in gas of typical densities,
plausible inter-particle collision velocities are very small and lead to
sticking but only minor compaction (see e.g., Cuzzi and Hogan 2003, Dominik
and Tielens 1997) until particles become large enough (10-100 μm) where
sedimentation velocity can exceed several meters per second and compaction or
bouncing arise (section 6.3; see Dominik et al. 2007, Zsom et al. 2010). The
thermodynamic properties of condensates and the temperature-pressure (T-P)
structure of the atmosphere in question determine whether the condensate
appears as a liquid or a solid. Figure 3 shows condensation curves of a number
of important cloud-forming compounds, along with the T-P profiles of a range
of exoplanetary and substellar objects.

Models of growing aggregates have been fairly well developed in the literature
of protoplanetary disks (e.g., Weidenschilling, 1988; Dominik & Tielens, 1997;
Beckwith et al., 2000; Dominik et al., 2007; Blum, 2010). The properties of
clouds made of this rather different kind of particle will differ in their
opacity and vertical distribution from predictions of the simplest models;
these properties in turn will impact, and can in principle be constrained by,
remote observations of reflected and emitted radiation.

Figure 3: Model atmospheric temperature-pressure profiles for brown dwarfs
(solid) and condensation curves (dashed) for a variety of compounds. For each
model effective temperature three curves are shown, corresponding to a
cloudless calculation (blue) and two cloudy models with varying sedimentation
efficiency, fsed = 2 (red) and 4 (orange). Figure modified from Morley et al.
(2012).

###  3.4 Porous Particles and Effective Medium Theory

The most straightforward way of modeling porous aggregates is to model their
effective refractive indices based on their constituent materials and
porosity. If the monomers from which the particles are made are smaller than
the wavelength in question, they act as independent dipoles immersed in an
enveloping medium (the medium can be another material; here we assume it is
vacuum). The aggregate as a whole can then be modeled as having effective
refractive indices which depend only on the porosity of the aggregate and the
refractive indices (but not the size) of the monomers. This is the so-called
effective medium theory (EMT); several variants are discussed by Bohren &
Huffman (1983), Ossenkopf (1991), Stognienko et al. (1995), and Voschinnikov
et al. (2006).

For most combinations of materials, EMTs can be even further approximated by
simple volume averages such that the refractive indices of the particle as a
whole can be written (for a simple two-component particle with component 2
being vacuum) as $n_{i}=fn_{\mathit{i1}}+\left(1-f\right)n_{\mathit{i2}}$ and
$\left(n_{r}-1\right)=f\left(n_{\mathit{r1}}-1\right)+\left(1-f\right)\left(n_{\mathit{r2}}-1\right)$,
where nr and ni are the effective real and imaginary indices of the aggregate
as a whole, and component 1 has volume fraction f. If a material which has
large refractive indices, such as iron, is involved, the full expressions are
needed (see Wright, 1987; Bohren & Huffman, 1983; or other basic references).
We will assume the porous aggregates are roughly equidimensional, not needle-
like structures, but even that added complication can often be tractable in
semi-analytical ways for most materials (Wright 1987; Bohren & Huffman 1983).

In the $r/\lambda{\ll}1$ regime, porosity plays little role, both because the
opacity is simply proportional to the total mass regardless of how it is
distributed (see above), and because ever-smaller particles are unlikely to be
aggregates of ever-smaller monomer constituents, but are more likely to be
monomers themselves. Ferguson et al. (2007) present numerical calculations
which show that, for the tiny particles they modeled (0.35 nm - 0.17 μm)
indeed this expectation is fulfilled to first order, also showing that mixing
of materials within these tiny aggregates has little effect (see also Allard
et al. 2001, Helling et al. 2008, Witte et al. 2009 and references therein).
Particles this small are likely to be well-mixed at all levels of typical
exoplanet atmospheres (see below). Some targets of interest may have high haze
layers where this regime is of interest.

However, in the $r/\lambda{\gg}1$ regime, which covers typical planetary
clouds observed in the optical or near-infrared, porosity does affect opacity.
Specifically, for a particle of mass M and porosity $\varphi$, the opacity
$\kappa=C_{e}/M=3/4\rho r=3/4\rho_{s}\left(1-\varphi\right)r$ where
$r=\left(3M/4\rho_{s}\left(1-\varphi\right)\right)^{1/3}$ is the actual
radius. Comparing the opacity of this particle to that of a “solid” particle
with the same mass having radius rs , gives
$\kappa/\kappa_{s}=\left(1-\varphi\right)^{-2/3}$. The porous particle thus
has a larger opacity for the same mass, and indeed reaches the geometrical
optics (λ independent) limit at lower mass than a nonporous particle. In this
regime, effective medium theory should be a convenient and valid way of
modeling porous and/or aggregate grain properties (section 3.2; see also
Helling et al. 2008 and references therein).

Variable composition introduces further complexity to computing single-
scattering properties, requiring a mixing rule to compute the refractive index
for the composite particle. Different mixing rules can introduce uncertainties
of order 1 and 10% respectively in the real and imaginary refractive indices
for black carbon inclusions in liquid water droplets (Lesins et al., 2002).

##  4 Cloud Models

The cloud properties required to compute radiative fluxes are rather
extensive. Thermal emission requires the emissivity and temperature of the
cloud particles. Vertical fluxes can be computed from knowledge of the
vertical distribution of particle cross-section (for each species of particle)
together with the wavelength dependent scattering phase function and the
single-scattering albedo. For a basic two-stream approach these quantities can
be boiled down to a vertical profile of extinction coefficient, asymmetry
parameter, and single-scattering albedo.

Complexities can arise if particle temperature differs from the local
atmospheric temperature by virtue of latent heat exchange and radiative
heating of the particles, but this should be rare in exoplanet applications
(Woitke & Helling 2003) and typically the particles are assumed to be at
ambient atmospheric air temperature. Clouds that are not horizontally uniform
are another possible complication that can be considered.

The required scattering and absorption total cross sections and overall
asymmetry parameter needed for input to a radiative transfer model are found
by integration of the single-scattering properties and emissivities over the
particle size distributions. The size distributions in turn can be computed by
a so-called bin model which tracks the number of particles in multiple
different bin sizes as the particles interact with the atmosphere. Such an
approach is computationally expensive, and a more efficient treatment is to
assume a particular functional form with a small number of free parameters.

For terrestrial applications log-normal distributions, with three parameters
(total number, geometric mean radius, and geometric standard deviation) and
gamma distributions, also with three parameters (total number, slope
parameter, and shape parameter) are commonly used. Separate size distributions
are used for particles of different phases and bulk densities (e.g., raindrops
and snowflakes) and for each mode of a multimodal size distribution. For
example, the parameterized size distribution of cloud droplets, which grow
principally through condensation in the terrestrial atmosphere, are treated
distinctly from raindrops, which grow principally through collisions. Although
a poor approximation for cloud droplets, it is often assumed that the shape
parameter is zero for other cloud species, which allows the gamma distribution
to collapse to an exponential distribution, thereby reducing the free
parameters from three to two.

For the computation of heating rates from the divergence of radiative fluxes,
the vertical distribution of any absorbing and emitting species is obviously
critical. Also, any vertical redistribution by scatterers in bands with
emission or absorption requires that their vertical distribution is accurate.
If horizontal photon transport is unimportant, it is feasible to represent the
global atmosphere–or the atmosphere within any model column of finite
horizontal extent, for that matter–with two columns, one clear and one cloudy
(for example see the discussion in the context of brown dwarfs in Marley et
al. (2010)). Assumptions about the vertical overlap of clouds are critical and
can induce large errors in top-of-atmosphere fluxes as well as heating rates
(e.g., Barker et al., 1999).

In the context of extrasolar planet atmosphere modeling we must connect a
simple, usually 1D model of the atmosphere with what is potentially a complex
brew of cloud properties which we would ideally need to know. However it is
clear that the number of variables can quickly grow to unmanageable extent. In
this chapter we discuss various approaches that have been taken to address
this problem.

###  4.1 Conceptual Framework for Cloud Modeling

Given a profile of atmospheric temperature as a function of elevation or
pressure we can ask where clouds form and what are their radiative properties.
Here we give a simplified discussion of the problem following Sanchez-Lavega
et al. (2004).

The abundance of a given atmospheric species a in the vapor can be given by
its mass mixing ratio $m_{a}=\rho_{a}/\rho=\epsilon P_{a}/P$ where $\rho$ and
P are the density and pressure of the “background” atmosphere (that is, not
including species a) and $\rho_{a}$ and Pa are the density and partial
pressure of vapor species a. Here $\epsilon=\mu_{a}/\mu$ or the ratio of the
molecular weight of species a,  $\mu_{a}$, to the background gas mean
molecular weight $\mu$. A not unreasonable assumption is that any vapor in
excess of the saturation vapor pressure of a,  $P_{v,a}\left(T\right)$,
condenses out. We can define the saturation ratio of a as
$f_{a}=P_{a}/P_{v,a}\left(T\right)$.

Deep in the atmosphere, below cloud base, $P_{a}{\ll}P_{v,a}$ so the species
is found in the gas phase. In a Lagrangian framework one can imagine a rising
parcel of gas that cools adiabatically as it expands, and at some point it may
reach $P_{a}=P_{v,a}$ and a saturation ratio $f_{a}=1$. In an Eulerian
framework this condition requires that the thermal profile intersects with the
vapor pressure curve for a given constituent, for instance as the 550 K model
intersects the H2O vapor pressure curve in Figure 3. However the thermal
profile for the 1300 K model does not intersect the H2O vapor pressure curve.
Thus H2O would be expected to condense in the cooler, but not the warmer,
atmosphere. To this point the problem is relatively straightforward for those
species that condense directly from the gas phase, such as water. For other
species, however, condensation instead proceeds by a net chemical reaction
when the condensed phase has a lower Gibb’s free energy than the vapor phase.
One such example is H2S + 2Na $\rightleftharpoons$ Na2S(s) + H2.

Above the condensation level a number of issues arise. First, the assumption
that all vapor in excess of saturation condenses may be faulty. Condensation
may require a supersaturation $\left(f_{a}>1\right)$ before it proceeds, owing
to the thermodynamic energy barrier of forming new particles. If so, what
degree of supersaturation is required? Above cloud base, which is the lowest
level at which condensation occurs, abundance of the condensed phase will
depend on a balance between the convective mixing of particles from below and
the downward sedimentation of the condensate particles. If the sedimentation
rate of some portion of the size distribution of condensate is greater than
the vertical mixing velocity scale, the scale height for the condensed phase,
Ha, will be smaller than the atmospheric scale height H evaluated at cloud
base. In the solar system typical values of Ha/H range from 0.05 to 0.20
(Sanchez-Lavega et al. 2004).

Nucleation is the starting process for the formation of a condensed phase
(either liquid or solid) from a gaseous state, or the formation of solid from
a liquid state. It creates an initial distribution of nuclei (embryos) which,
if large enough, are stable with respect to the higher-entropy phase and tend
to grow into larger particles by condensation or freezing. Such a phase
transition can only occur spontaneously under thermodynamically favorable
conditions (for the following discussion we first focus on the process of
condensate nucleation from the vapor phase). Such conditions require the
saturation ratio $f_{a}$ to exceed unity ($f_{a}=1$ characterizes phase
equilibrium).

The simplest nucleation mechanism is homogeneous nucleation, where the initial
nuclei are formed by random spontaneous collisions of monomers in the vapor
phase (e.g. single H2O molecules) into a molecular cluster. This process is,
however, connected to an energy barrier that can prevent the formation of a
stable new phase. A supersaturated gas phase possesses a high Gibb’s free
energy, whereas at the same time molecules in the condensed phase would be at
a lower potential energy. Thus, removing molecules from the vapour phase and
adding them into a condensed phase would in principle lower the total free
energy of the entire thermodynamical system. The corresponding net change in
free energy depends mainly on the volume of an embryo (as a function of the
particle radius $r$) and the supersaturation of the gas.

However, the creation and growth of nuclei also implies that a new surface is
generated which additionally contributes to the free energy owing to surface
tension. It is only thermodynamically favourable to form a condensed phase
when the net reduction of free energy in the system $\left({\sim}r^{3}\right)$
is greater than (or equal to) the free energy required from the surface
tension $\left({\sim}r^{2}\right)$. Therefore, there exists a critical radius
for which these two contributions balance each other. Embryos smaller than the
critical size are unstable and will tend to evaporate again quickly whereas
those with sizes larger than the critical radius will tend to grow freely. The
critical radius is, in particular, a function of the saturation ratio. Low
supersaturations yield very large critical radii while, on the other hand, for
increasing values of $f_{a}$ the size of the critical embryos decreases.

Homogeneous nucleation usually requires a very high supersaturation and is
therefore unlikely to occur in atmospheres of terrestrial planets on a large
scale (observed supersaturations for water vapor in the atmosphere of Earth
are normally in the range of a few percent, for example), because
heterogeneous nucleation occurs at much lower supersaturations and thus
quenches the supersaturation well before homogeneous nucleation occurs.

The predominant nucleation process from the vapor for terrestrial planetary
atmospheres is thought to be heterogeneous nucleation. Here, the initial
distribution of nuclei is formed on pre-existing surfaces. The presence of
such surfaces substantially lowers the supersaturation required to form the
critical clusters. Depending on the properties of these surfaces (e.g. whether
they are soluble with respect to the condensed phase) nucleation is already
possible for saturation ratios close to unity. Such surfaces can be provided
by dust, sea salt, pollen, or even bacteria. In case of terrestrial planets
these particles (condensation nuclei) are usually largely available because of
mechanical process associated with the planetary surface–such as wave
breaking, bubble bursting, and dust saltation–and from the formation of haze
particles (such as sulfuric acid or sulfate droplets) that result from
photochemical processes. Therefore, heterogeneous nucleation can reasonably be
expected as the dominant nucleation mechanism for terrestrial extrasolar
planets. This assumption, however, complicates the treatment of cloud
formation because details on such condensation nuclei (composition, number
density, size distribution) are not available in case of exoplanets.

Additionally, the formation of a solid phase (e.g. ice crystals) can in
principle occur by different pathways. It can either form directly from the
gas phase by homogeneous or heterogeneous nucleation. On the other hand, it is
also possible to form the liquid phase as an intermediate step, followed by
freezing of the supercooled liquid into the solid phase afterwards. Whether
this indirect or the direct pathway occurs depends on the properties of the
condensing species and on the local atmospheric conditions. Water ice cloud
crystals in the Earth’s atmosphere form by homogeneous and heterogeneous
freezing of liquid in mixed-phase clouds (such as cumulonimbus and Arctic
stratiform clouds) and heterogeneous nucleation in cirrus clouds. In the
Martian atmosphere CO2 clouds form directly by heterogeneous nucleation from
the gas phase.

Given a composition of the cloud the task for any cloud model then becomes one
of computing cloud particle sizes and their number distribution through the
atmosphere above cloud base. A very simple solution would be to simply assume
a mean particle size and a cloud scale height and this is effectively the
approach many investigations have take to explore the effect of exoplanet
clouds. For the remainder of this section we consider efforts to more
rigorously derive expected particle sizes and the vertical distribution of
cloud particles. We begin by considering the most thoroughly modeled clouds,
terrestrial clouds of liquid water and water ice. We then move on to cloud
models that have been constructed for terrestrial and giant extrasolar planets
and conclude by considering the lessons learned from efforts to model clouds
expected in brown dwarf atmospheres.

###  4.2 Perspective from Earth Science

Cloud modeling of terrestrial clouds comes in a wide assortment of classes.
For detailed cloud studies of limited spatial extent, dynamical frameworks
range from 0-D parcel models, to 1-D column models, to 2-D eddy-resolving
models, to 3-D large-eddy simulations and cloud-resolving models. The
difference between such models is the number of spatial dimensions represented
in the governing equations. Another varying aspect of cloud models is the
degree of detail in describing cloud microphysics. The simplest models simply
assume that all vapor in excess of saturation condenses and assume a fixed
size for the cloud particles. Others assume a functional shape for the cloud
particle size distributions and parameterize the microphysical processes,
prognosing one, two, or three moments of the size distribution for each
condensate species (such as cloud water, rain, cloud ice, snow, graupel, and
hail). The most detailed approach is to resolve the cloud particle size
distributions without making any assumptions about the functional shape of the
size distributions.

Global climate model frameworks similarly range from one-dimensional radiative
convective models (e.g., Manabe & Wetherald, 1967), to two-dimensional,
zonally averaged models (e.g., Schneider, 1972), to modern three-dimensional
general circulation models (GCMs). (See Schneider & Dickinson, 1974 for an
early, comprehensive review of approaches to global climate modeling.) With
respect to clouds in global models, one-dimensional radiative convective
models suffer from a major deficiency, namely predicting plausible estimates
of horizontal cloud coverage (see Ramanathan & Coakley, 1978). A two-
dimensional framework is an intermediate step, though modern climate studies
rely principally on three-dimensional GCMs. The rest of this section will
focus on clouds in modern GCMs.

The representation of clouds in GCMs remains a major challenge, as cloud
feedbacks constitute a leading source of uncertainty in current model-derived
estimates of overall climate sensitivity, which are typically cast in terms of
the sensitivity of globally averaged surface temperature to changes in
radiative forcings (see Hansen et al., 1984). The response of tropical cirrus
clouds to increasing sea surface temperatures has been a topic of great debate
in the last two decades. At one extreme is the thermostat hypothesis of
Ramanathan & Collins (1991), which suggests that increased water vapor leads
to more extensive, thicker anvils that will on net cool the planet through
increased albedo, while at the other extreme is the argument of Lindzen et al.
(2001) that greater condensate loading in a warmer climate precipitate more
efficiently and lead to a dryer upper atmosphere that traps less infrared
energy, driving a negative water vapor feedback. Both hypotheses produce a
negative climate feedback for tropical clouds, one employing increased solar
reflection, the other relying on increased infrared emission. While these are
provocative ideas that have spawned countless evaluations of the hypothesis
(with neither withstanding scrutiny), the predominant concept currently is
that the tropical climate feedback for modern GCMs is determined primarily by
the response of clouds in the marine boundary layer (Bony & Dufresne, 2005),
of which the transition from overcast stratocumulus to broken cloud fields of
trade cumulus is a leading primary candidate responsible for that response.
With respect to cloud feedbacks generally, attention on the climate feedback
of cirrus clouds formed from the detrainment of deep tropical convection has
been supplanted to a large degree by more recent focus on the climate feedback
of shallow clouds.

A fundamental problem in representing clouds in the GCMs is that the native
GCM grid cells are very coarse, of order 100 km horizontally and 1 km
vertically. Although model resolution steadily improves as computing power
advances, the problem of convection and clouds being unresolved persists in
models designed to span the globe and simulate climate over time scales of
decades to millennia. The basic assumption in a conventional treatment of
convection and clouds in a GCM is that cloud properties and precipitation
rates in a model grid cell, which is much larger horizontally than any cloud,
can be computed based on the mean properties of the grid cell. The purpose of
a cloud parameterization is to compute cloud properties and precipitation
rates from those mean properties.

A somewhat recent version of the NASA GISS (Goddard Institute for Space
Studies) GCM (Schmidt et al., 2006) includes a conventional treatment of
clouds and convection. The atmospheric model grid spacing is roughly 2 × 2.5°
with 20 or 23 vertical layers, and thus all convection and cloud physics are
necessarily highly parameterized. Deep convection is parameterized based on
the convective instability of model columns using idealized updrafts and
downdrafts, which detrain air into stratiform cloud layers. The stratiform
cloud cover is computed as a diagnostic function of grid-scale relative
humidity, and the relative humidity thresholds used to compute cloud cover in
that diagnostic function serve as the principle tuning knobs for the GCM, with
the dual tuning targets of top-of-atmosphere radiative balance and an overall
albedo reasonably close to satellite-based estimates (note that such tuning
can easily result in exaggerated cloudiness in some regions that make up for
insufficient cloudiness in others). In the GISS GCM the only prognostic cloud
variable is the mass mixing ratio of cloud condensate, which is a fundamental
component of the cloud parameterization. Any precipitation is assumed to
evaporate or fall out in one time step (30 minutes) and the phase of the
condensate is probabilistically determined from temperature to allow for a
modest amount of supercooling (liquid colder than the melting point) on
average. The standard version of the GCM assumes different cloud droplet
concentrations over ocean and land and also assumes a fixed number
concentration for ice particles, and a fixed effective variance of the
condensate size distributions is assumed for computing cloud optical
thickness.

A more complex approach for stratiform cloud microphysics is used in the most
recent version of the NCAR (National Center for Atmospheric Research) GCM.
That scheme uses two moments (mass and number) for two prognostic (cloud water
and cloud ice) and two diagnostic (rain and snow) hydrometeor species
(Morrison & Gettelman, 2009). The rapidly sedimenting species are treated
diagnostically with a tridiagonal solver to allow for long time steps (20
minutes with 2 microphysics substeps) in a manner that avoids numerical
instability associated with falling through more than one layer during a time
step. A novel aspect of this microphysics scheme is that by assuming a
particular subgrid-scale distribution of cloud water, microphysical processes
that involve cloud water take into account the problem of grid-averaging over
nonlinear process rates (Pincus & Klein, 2000). Taking into account joint
subgrid-scale distributions of just two moments for two species complicates
the math considerably (Larson & Griffin, 2012).

An alternative to parameterizing convection within grid scales O(100 km) is
the multiscale modeling framework (Randall et al., 2003), in which two-
dimensional cloud-resolving models are embedded within each GCM column. While
some convection-related aspects of the global circulation are treated well by
such an approach, like the more traditional approach to GCM cloud
parameterization, the pervasive and climatologically important regime of
shallow marine convection is poorly represented in such models (e.g., Marchand
& Ackerman, 2010). (Explicit resolution of such clouds requires horizontal
resolution O(100 m) and vertical resolution O(10 m).) Avoiding issues related
to embedding 2-D slices within GCM columns (problems including how to orient
the slices and shortcomings including the use of periodic lateral boundary
conditions for each embedded 2-D sub-model), the most expensive, yet perhaps
straightforward approach to climate modeling is the Earth Simulator, a global
cloud-resolving model with simulations run on a 3.5-km horizontal grid (Satoh
et al., 2008). The computational demands of such an approach are vast, with
order 1019 grid cells in such a model. Even such a brute-force approach falls
far short of the grid resolution required to explicitly simulate shallow
marine convection, however. It is safe to say that even on the most powerful
computing platforms that global simulations of Earth will be saddled with
cloud and convective parameterizations for the foreseeable future.

###  4.3 Exoplanet Clouds

In extreme contrast to the situation for Earth outlined above, there are
currently no observational constraints for atmospheres of terrestrial
exoplanets that would provide information about what kind of clouds may have
to be considered for a particular exoplanet. Available observables are
confined to basic planetary parameters, like radius and orbital inclination
(if a transit event can be observed), planetary mass (from the radial velocity
method), orbital eccentricity and distance, and additionally the type of the
central star. Consequently, we are faced with the difficult problem of
modeling clouds in an environment without actually knowing any further
details.

Cloud formation can only be treated theoretically in compliance with the
chemical composition of the atmosphere because it determines the condensing
species forming a cloud. Considering how diverse atmospheres of terrestrial
exoplanets can be expected to be, the self-consistent modelling of cloud
formation in such atmospheres without observational constraints or theoretical
predictions is somewhat ambiguous. The composition of a terrestrial planet
cannot be easily deduced from simple theoretical arguments. In the case of a
planet which has lost its primordial atmosphere the atmospheric composition is
determined by the combination of the outgassed chemical species from the
planet’s interior and the volatiles delivered by impacts of asteroids and
comets and will therefore depend on the planet’s mantle composition, physical
processes in the planetary interior, and the composition and sizes of
impactors. Another factor with a huge impact on the atmospheric composition is
also the possible existence of a biosphere that interacts chemically with the
atmosphere.

The long-term evolution of the atmospheric composition, such as that resulting
from the carbonate-silicate cycle as on Earth, depends also on the occurrence
of plate tectonics. To date, the only known planet with plate tectonics is
Earth. It is currently not quite well understood under which conditions a
planetary crust will start plate tectonics and how this process is maintained
over an extended period of time. This is especially true for more massive
terrestrial planets like super-Earths, where there is much controversy
regarding plate tectonics (see e.g. Valencia et al. 2007, O’Neill et al.
2007).

Other important processes determining the atmospheric composition are the
escape mechanisms of atmospheric gas to space. While thermal escape is a
function of the atmospheric temperature, planetary mass, and the molecular
weight, the non-thermal escape processes (erosion of the atmosphere by a
stellar wind, for example) are much more complicated (e.g. Lammer et al. 2008
or Tian, this volume). They not only depend on the activity of the central
star–which itself is a function of the stellar type and its particular stellar
evolution–but also on the possible existence of a planetary magnetic field
(linked to the rotation rate of the planet), which can protect the planetary
atmosphere against loss processes.

In contrast to a low-mass planet like Earth, a more massive terrestrial super-
Earth might retain a part of its primordial hydrogen dominated atmosphere,
partly enriched by volcanic outgassing or additional external delivery. Such
atmospheres may be vastly different from those known within our solar system.
A discussion of possible atmospheric compositions can be found in Seager &
Deming (2010).

The secondary atmosphere of an Earth-like planet (Earth-like with respect to
the chemical composition of the planetary mantle) would most likely be rich in
H2O and CO2 (Schaefer et al. 2012). Therefore, clouds composed of these
species are of prime interest for habitable Earth-like planets. Cooler
atmospheres can also contain significant amounts of CH4 and CH3, or SO2 in
case of high temperatures (Schaefer et al. 2012).

In this environment effective cloud models must match the problem at hand. For
example highly sophisticated terrestrial cloud microphysics and dynamics
models are not required in order to ascertain the range of plausible albedos
for a hypothetical Earth-like terrestrial planet. However more sophisticated
approaches than simple ad-hoc models may be needed to interpret the colors of
a directly imaged planet.

###  4.4 Cloud Models for Terrestrial Exoplanets

In principle the basic mathematical description of cloud microphysics in
atmospheres of terrestrial exoplanets do not deviate from their solar system
counterparts. The temporal and spatial evolution of the cloud particle size
distribution can be described by means of a master equation (“general dynamic
equation”) incorporating all relevant gain and loss processes. This includes
nucleation, evaporation, sedimentation, coagulation/coalescence, diffusion, or
hydrodynamical transport (see Pruppacher & Klett, 1996 or Lamb & Verlinde
2011). While the numerical solution of the master equation is still quite
challenging it can be efficiently performed by the methods summarised in
Williams & Loyalka (1991) or by applying more advanced methods (e.g.
continuous and discontinuous Galerkin methods by Sandu & Borden, 2003).

Self-consistent modeling of the formation and temporal and spatial evolution
of clouds is an unsolved problem. An ideal treatment would require a thorough
knowledge of the state of the atmosphere, including its composition, the
spatial distribution of chemical species, atmospheric temperature and
dynamics, and the size distribution and density of potential cloud
condensation nuclei and heterogeneous freezing nuclei. However, even in the
terrestrial atmosphere the formation of ice at temperatures warmer than the
homogeneous freezing temperature for water (about 233 K for typical drop
sizes) is not well understood (e.g., Fridlind et al., 2007, 2012); far less,
if anything, is known about heterogeneous freezing and condensation nuclei in
other atmospheres and would hardly be detectable by remote observations of
terrestrial extrasolar planets. Lack of laboratory data to derive the
necessary microphysical rates under atmospheric conditions more “exotic” than
found in the solar system further complicates the problem of describing cloud
formation in atmospheres of exoplanets.

Additionally, many atmospheric models for exoplanets are restricted to one
spatial dimension and are often considered to be stationary, which makes a
detailed description of cloud microphysics very difficult. While the climatic
effects of clouds can be approximately treated in a one-dimensional model
atmosphere, a consistent modelling of cloud formation would, in principle,
require a three-dimensional dynamical atmospheric model as described in
Section 4.2. In comparison to one-dimensional models, however, three-
dimensional general circulation models contain many more free parameters and
are very computationally intensive. Properties such as the surface orography,
distribution of surface types (fractions of oceans and land mass), and the
local distribution of chemical species (affected by volcanism for example)
play a major role for the dynamics and chemistry of the atmosphere (Joshi,
2003), and affect directly the formation and evolution of clouds. Since none
of these detailed properties are known for a terrestrial exoplanet many
additional assumptions about the planet and its atmosphere have to enter the
calculations. On the other hand, three-dimensional models are the only
opportunity to obtain information about the possible temporal variation and
fractional coverage of clouds and their distribution throughout the
atmosphere. Such results might be required to analyze transmission spectra of
terrestrial exoplanetary atmospheres containing patchy clouds.

Given the aforementioned challenges due to the lack of observational
constraints, clouds in atmospheres of terrestrial exoplanets are usually
treated in a simplified way. The simplest method to account for the effects of
clouds in an atmospheric model is a modified surface albedo. This approach has
been widely used in the past (e.g., Kasting et al. (1993), Segura et al.
(2003, 2005), or Grenfell et al. (2007)). The surface albedo of these kind of
models is modified to yield a specified surface temperature for a given
reference scenario. For example, a common reference scenario is an Earth-like
planet around the Sun at a known orbital distance of 1 AU. The surface albedo
is then adjusted to obtain the mean surface temperature of Earth (288 K),
thereby mimicking the climatic effects of clouds. This adjusted surface albedo
value is then used in all subsequent model calculations, assuming that the net
effect of the clouds is invariant from changes of the atmospheric conditions,
or type of central star. This approach makes no assumptions about the physical
nature of the clouds, their composition, size, or optical properties but
instead estimates their effects based on the original tuning to surface
temperature. While a modified surface albedo can crudely describe the climatic
effect of clouds, it cannot simulate their impact on the planetary spectra.

A more detailed approach is to consider model scenarios where the properties
of clouds can be assumed to be approximately known. For a completely Earth-
like planet, one could expect that the properties of clouds in such an
atmosphere would closely resemble those found on Earth. This approach was used
by Kitzmann et al. (2010) to study the impact of mean Earth water clouds in
the atmospheres of Earth-like extrasolar planets. In contrast to a modified
surface albedo, the wavelength-dependent optical properties of the cloud
particles are explicitly taken into account to study their effects for
different incident stellar spectra and other parameters. Thus, in addition to
the influence on the atmospheric and surface temperatures, their impact on the
planetary spectra can also be studied in detail using this modelling approach
(e.g., Kaltenegger et al., 2007). If the properties of the clouds are not
known (e.g., for a CO2 cloud in a thick CO2 dominated atmosphere of a super-
Earth) parameter studies can be performed, varying the cloud properties over a
wide range to estimate the possible effect of clouds. This approach has been
used by Forget & Pierrehumbert (1997) for CO2 ice clouds in an early Martian
atmosphere, for example.

More detailed treatments of cloud formation in exoplanetary atmospheres
include simplified air parcel and vertically resolved one-dimensional cloud
models based on those originally developed for the Earth atmosphere. Such
models can be used to determine mean cloud properties under different
atmospheric conditions (see Neubauer et al. (2011, 2012) for several cloud
species (e.g. H2O and H2SO4) or Zsom et al. (2012) for water (droplet and ice)
clouds, for example). However, these more detailed descriptions of cloud
formation already need to include many additional assumptions, such as the
distribution of cloud condensation nuclei, which strongly influence the
resulting cloud properties.

While cloud models for terrestrial exoplanets so far lack many of the
sophisticated and detailed cloud microphysics needed to reproduce the
complicated cloud structures known from Earth observations they nonetheless
offer an important first-order estimate of cloud effects in exoplanetary
atmospheres. One of the largest uncertainties for one-dimensional models is
the treatment of fractional clouds. Unless the atmosphere is globally
supersaturated, thus resulting in a completely cloud-covered planet, the
fraction of the atmosphere where clouds are present has to be introduced as a
free parameter. Thus, one-dimensional models are incapable of organically
describing planets with fractional cloud-cover.

Although one-dimensional models are commonly employed for many terrestrial
exoplanet applications, three-dimensional models have also been used. For the
terrestrial super-Earth Gliese 581d, Wordsworth et al. (2011) adapted a Mars
global circulation model that included a simplified treatment of CO2 ice cloud
formation. This microphysical model assumes a certain size and number density
of cloud condensation nuclei and equally distributes the condensable material
among them, accounting also for the sedimentation and hydrodynamical transport
of the cloud particles within the atmosphere. The corresponding formation of
CO2 clouds lead to an increase of the surface temperature of Gliese 581d in
their model calculations. The same approach was also used by Wordsworth et al.
(2010) in a one-dimensional atmospheric model for the same planet.

###  4.5 Giant Planet Cloud Models

Unlike the case for Earthlike planets where water clouds are the greatest
concern, a wide variety of species may condense in the hydrogen-helium
dominated atmospheres of giant planets. Sánchez-Lavega et al. (2004) review
the standard framework for cloud formation in giant planets. Homogeneous
condensation occurs when the partial pressure of a species in the gas phase
exceeds its saturation vapor pressure at a given temperature in the
atmosphere. Sánchez-Lavega et al. tabulate the vapor pressures for many
relevant species. Other expressions for additional species can be found in
Ackerman & Marley (2001) and Morley et al. (2012). Curves tracing the set of
pressure and temperature conditions at which a given species condenses
assuming equilibrium chemistry (“condensation equilibrium curves”) are shown
for many species in Figure 3 and a schematic illustration of the resulting
cloud decks is shown in Figure 4.

Once a cloud layer forms the condensate is removed from the overlying
atmosphere and thus is no longer available to react at lower temperatures
higher in the atmosphere. Thus the calculation of the chemical equilibrium
state for the atmosphere must account for the presence of the cloud. Such a
“condensation chemistry” is distinct from equilibrium chemistry calculations
in which the condensate remains in communication with the gas phase and is
available for reaction at lower temperatures. Condensation chemistry is
discussed in detail by Fegley & Lodders (1996), Lodders & Fegley (2002), and
Visscher et al. (2006) as well as by Burrows & Sharp (1999) in the context of
brown dwarf models and Sudarsky et al. (2003) for extrasolar giant planets.
The schematic Figure 4 accounts for condensation chemistry. If iron were not
sequestered into a deep cloud layer in Jupiter’s atmosphere the Fe would react
with gaseous H2S to form FeS, thus removing H2S from the observable
atmosphere, in contradiction to observations (Fegley & Lodders, 1996).

Once the cloud base pressure is found the challenge is to describe the
variation in cloud particle sizes and number densities above this level. Early
attempts to develop cloud models for use in giant solar system atmospheres
included the work of Lewis (1969), Rossow (1978), and Carlson et al. (1988).
These and other early works are reviewed by Ackerman & Marley (2001). In this
subsection we focus on more recent modeling approaches that are in use today,
in particular the cloud models of Ackerman & Marley and of Helling and
collaborators.

Figure 4: Schematic illustration (modified from Lodders 2004) of cloud layers
expected in extrasolar planet atmospheres based on consideration of
equilibrium chemistry in the presence of precipitation. The three panels
correspond roughly to effective temperatures Teff of approximately 120 K
(Jupiter-like, left), to 600 K (middle) to 1300 K (right). Note that with
falling atmospheric temperature the more refractory clouds form at
progressively greater depth in the atmosphere and new clouds composed of more
volatile species form near the top of the atmosphere.

####  4.5.1 Ackerman & Marley

Iron and silicates condense in the atmospheres of warm brown dwarfs (Figure 3)
and these clouds must be accounted for in models of brown dwarf emergent
spectra. Early modeling attempts for such clouds simply computed the mass of
dust that would be found in chemical equilibrium for a given assumed initial
abundance of gas (as if the gas were isolated at a given pressure and
temperature from the rest of the atmosphere. The lower the atmospheric
temperature the more dust that would be present. Atmospheric models following
such a prescription (such as the “DUSTY” models of Allard et al. 2001)
adequately reproduced the near-infrared colors of the warmest brown dwarfs but
predicted far too great of a dust load in cooler objects. Thus it was apparent
that an accounting for sedimentation of grain particles was required. One
approach used in the literature was to set a variable “critical” temperature
for a given cloud such that cloud particles would only be found between cloud
base and the specified temperature (Tsuji, 2002). Another approach was to
limit the cloud to be confined within an arbitrary distance, usually one
scale-height, of cloud base. Both such approaches required the choice of an
arbitrary particle size for the grains. The advantage of such approaches is
that they are computationally very tractable for modeling and thus allow the
exploration of a large parameter space. One disadvantage is that it is
difficult to consider particle size effects and other complexities.

In order to allow for vertically-varying particle number densities and sizes a
second approach was suggested by Ackerman & Marley (2001). In their
formulation downward transport of particles by sedimentation is balanced by
upwards mixing of vapor and condensate (either solid grains or liquid drops),

$-K_{\mathrm{zz}}\frac{{\partial}q_{t}}{{\partial}z}-f_{\mathrm{sed}}w^{*}q_{c}=0$
(1)

where Kzz is the vertical eddy diffusion coefficient, qt is the mixing ratio
of condensate and vapor, qc is the mixing ratio of condensate, w* is the
convective velocity scale, and fsed is a dimensionless parameter that
describes the efficiency of sedimentation. The solution of this equation
allows computation of a self-consistent variation in condensate number density
and particle size with altitude above an arbitrary cloud base.

In their model the cloud base is found by determining at which point in the
atmosphere the local gas abundance exceeds the local condensate saturation
vapor pressure $P_{v,a}$ at which point the atmosphere becomes saturated. In
cases where the formation of condensates does not proceed by homogeneous
condensation an equivalent vapor pressure curve is computed, as described by
Morley et al. (2012).

The Ackerman & Marley cloud model has the advantage of not requiring knowledge
of microphysical processes to compute particle sizes. For a given
sedimentation efficiency clouds are simply assumed to have grown large enough
to provide the required downward mass flux that balances Equation (1). Since
the solution is numerically rapid a large number of models can be computed and
compared with data within a tractable amount of time. Sample model
temperature-pressure profiles along with equilibrium condensation curves are
illustrated in Figure 3.

Considering the simplicity of this approach, the model has fared fairly well
in comparisons with data. Stephens et al. (2009) for example compared model
spectra for L and T dwarfs to a large database of near- to mid-infrared
spectra. They found that cloudy L dwarfs can generally be well fit by clouds
computed with fsed = 1 to 2 while early T dwarfs, which exhibit thinner
clouds, are better fit with fsed = 3 to 4. The model thus provides a framework
for describing mean global clouds in a 1-dimensional sense, but the model
lacks the ability to explain why the sedimentation efficiency might change at
effective temperatures around 1200 K, where the near-infrared spectra of L
dwarfs evolves over a small temperature range.

Marley et al. (2010) considered the effect of partial cloudiness on L dwarf
spectra computed with the Ackerman & Marley (2001) cloud model. Their method
assumed that clear and cloudy columns of atmosphere had the same temperature
profile and together emitted the flux corresponding to a specified effective
temperature. They found that partially cloudy L dwarfs would have emergent
spectra comparable to standard models with homogeneous cloud cover but with
larger values of fsed. Thus a dwarf with 50% clear skies and 50% cloudy skies
with fsed = 2 ends up with a model spectrum comparable to that of a
homogeneous cloud cover with fsed = 4.

####  4.5.2 Helling and Collaborators

The most extensive body of work on cloud formation in giant exoplanet and
brown dwarf atmospheres has been undertaken by Helling and her collaborators
(Helling & Woitke, 2006; Helling et al., 2008; Witte et al., 2009, 2011; de
Kok et al., 2011) who follow the trajectory of seed particles from the top of
their model atmospheres as they sink downwards. The seeds grow and accrete
condensate material as they fall, resulting in “dirty” or compositionally
layered grains. This work extends the dust moment method from Gail et al.
(1984) and Gail & Sedlmayr (1988). It accounts for the microphysics of grain
growth given these conditions and available relevant laboratory data. Particle
nucleation is explicitly computed, taking into account barriers to grain
formation.

Because condensation is envisioned in this framework to proceed downwards from
the top of the atmosphere rather than upwards from the deep atmosphere and
because grains are allowed to interact with the surrounding gas, the cloud
composition predicted by the Helling approach differs substantially from that
employed in the Ackerman & Marley framework. An example is shown in Figure 5
for a model brown dwarf (log g = 5 in CGS units, Teff = 1600 K). Here Helling
et al. (2008) predict that in addition to the usual Fe, MgSiO3 and Mg2SiO4
cloud layers, additional condensates including SiO2 and MgO will form. These
latter species are not predicted by equilibrium condensation for a cooling gas
mixed upwards from higher temperature and pressure conditions The TiO2 cloud
seeds arising at the model at the top of the atmosphere in this approach is
also evident.

In fact the presence of these initial TiO2 seed particles at the top of the
model atmosphere in the Helling framework deserves some discussion. In the
equilibrium chemistry condensation framework, Ti-bearing condensates (e.g.,
CaTiO3) would form much deeper in the atmosphere as a gas parcel rises
vertically and cools. Precipitation of such particles would remove the
condensate from the gas phase and notable amounts of refractory TiO2 seed
particles would not be expected to arrive at the top of the atmosphere.
Conversely in the Helling conceptual framework the formation of CaTiO3 at the
expected equilibrium cloud base is thought to be kinetically inhibited (since
multiple collisions of molecules would be required) resulting in the
refractory TiO2 clusters being mixed further upwards to ultimately seed
condensation in the cooler upper reaches of the atmosphere. Thus the Helling
approach fundamentally assumes that vertical mixing timescales, at least in
localized columns, are faster than condensation timescales. As the seeds
eventually fall from the top of the atmosphere they then encounter other
condensable molecules which are likewise assumed not to have not been cold
trapped below and the seeds then accrete these species and grow. The model
iterates to find a self-consistent solution. This top-down approach thus
differs from most of the other cloud modeling approaches discussed in the
literature which generally conceive of a condensation sequence operating from
the depths of the atmosphere upwards with species sequentially condensing, as
conceptually shown in Figure 4. Atmospheric mixing by breaking gravity waves
(Freytag et al. 2010) might provide a mechanism to stir the atmosphere
sufficiently to deliver the seed particles.

Because of the complexity of the computational approach required to compute
cloud properties in this framework there have been fewer comparisons between
model spectra computed with the Helling clouds and data than has been the case
with the Ackerman & Marley cloud model. Some direct comparisons between
different cloud models are shown by Helling et al. (2008). Ultimately only a
thorough comparison of the predictions of all cloud modeling schools and data
will be required to establish which conceptual framework is a better
approximation over which ranges of conditions. An application of the Helling
framework to the clouds of Jupiter would be enlightening.

Figure 5: Composition of atmospheric cloud layers for a Teff = 1600 K,
log(g)=5 brown dwarf as computed by the dust model of Helling and
collaborators (Helling et al. 2008). The vertical axis is atmospheric
temperature with the top of the atmosphere to the top of the figure. The
horizontal axis gives the relative volumes Vs of each dust species indicated
by line labels as a ratio to the total dust volume Vtot. Unlike the Ackerman &
Marley condensation equilibrium approach, this model predicts that MgO and
SiO2 are important condensates along with the TiO2 seed particles that are
formed at the top of the model.

####  4.5.3 Other Approaches

More simplified approaches have also been taken for cloud models, such as
specifying particle sizes and cloud heights. Sudarsky et al. (2000; 2003)
computed model exoplanet albedo spectra given various particle size and cloud
height assumptions. In particular they utilized the Deirmendjian (1964) size
distribution and explored the effects of changing mean cloud particle sizes
and widths of the size distribution. Sudarsky et al. (2000) also considered
the effects of various plausible photochemical hazes on giant planet albedo
spectra.

Cooper et al. (2003) employed the timescale comparison framework pioneered by
Rossow (1978) to compute cloud models for brown dwarfs and extrasolar giant
planets. In this approach various timescales for particle nucleation, growth,
and sedimentation are compared to derive expected condensate particle sizes.
As discussed by Ackerman & Marley (2001), a difficulty with the Rossow method
is that some of the critical timescales depend upon unknown factors,
particularly the assumed supersaturation. Nevertheless the Cooper et al. model
provides a useful survey of likely particle sizes for species of interest
expected under various combinations of gravity and effective temperature. For
example, in agreement with most of the other cloud models Cooper et al.
predict typical silicate grain sizes in the range of 10 to 200 μm.

Tsuji and collaborators (Tsuji 2002, 2005; Tsuji et al. 2004) have computed
brown dwarf models by specifying cloud-top and cloud-base temperatures. For
the directly imaged planets Currie et al. (2011), Madhusudhan et al. (2011),
and Bowler et al. (2010) employ a variety of approaches to specify cloud
properties and explore parameter space. Approaches such as these offer the
advantage of quickly exploring the phase space of possible models and
establishing the effect of plausible cloud models on spectra. The lack of
physical complexity in such models is offset by their useful ability to offer
qualitative understanding of the effect of various condensate properties.

###  4.6 Lessons Learned from Cloudy Brown Dwarfs

Brown dwarfs–hydrogen-helium rich objects with masses between about 12 and 80
times that of Jupiter (MJ)–have been a proving ground for understanding the
role of clouds in exotic atmospheres. This is because the class of brown
dwarfs known as the L dwarfs have atmospheric temperatures in the regime in
which iron and silicate grains condense from the gas phase. It is apparent
from the available data that these refractory condensates do not form a pall
of particles mixed through the entire atmosphere, but rather form discrete
cloud layers. As such L dwarfs were the first objects outside the solar system
for which a detailed description of clouds was required in order to interpret
their emitted spectra.

There have been several comparisons of cloudy brown dwarf atmosphere models to
observations. The most extensive to date are presented by Cushing et al.
(2008) and Stephens et al. (2009). These authors compared atmosphere models of
Saumon & Marley (2008) computed using the Ackerman & Marley (2001) cloud model
to a variety of L- and T-type brown dwarf spectra from 0.8 to 15 μm. For the L
dwarfs and early T-dwarfs the cloudy models clearly did a better job
reproducing the data than cloudless models. The tunable fsed parameter, with
typical values between 1 and 3, allowed sufficient dynamic range to generally
reproduce most of the observed spectra. Witte et al. (2011) meanwhile compare
model spectra computed with the Helling et al. (2008) cloud formulation to
spectra of L-type brown dwarfs. Likewise their cloud model shows much better
agreement with data than either cloudless or very cloudy models with no dust
sedimentation. In all of these studies the matches between models and data are
very good in many cases, but nevertheless there remain notable spectral
mismatches and it is clear that a more sophisticated cloud model would be
required to fit all objects.

The most important lesson learned from the campaign to model brown dwarfs may
be that large grids of atmosphere models–including a variety of cloud
descriptions–are required. Models should not be so complex that the creation
of such grids are a challenge. In the next section we review the available
exoplanet data relevant to clouds. As we will see the exoplanet data do not
yet require large systematic model grids, but such models will undoubtedly be
required as more data become available. Ideally more models will be available
than has been the case for brown dwarfs, thereby permitting more systematic
comparisons of various cloud modeling frameworks to large exoplanet datasets.

##  5 Observations of exoplanet clouds

###  5.1 Transiting Planets

####  5.1.1 Transmission Spectra

Perhaps the most convincing evidence of high altitude clouds or hazes on any
extrasolar planet to date is found in the case of the transiting planet HD
189733b. This 1.1-Jupiter mass planet orbits a bright nearby K star and is
thus an excellent target for detailed studies of its atmosphere. This planet
is notable because its transit radius–the apparent size of the planet as a
function of wavelength–follows a smooth power law as first measured by Pont et
al. (2008). Signatures of molecular or atomic absorption expected from a
clear, solar composition atmosphere are almost absent, although Na is detected
at high spectral resolution (Huitson et al., 2012). Figure 6 illustrates the
smooth variation in atmospheric transmission as measured from 0.3 through 1 μm
(Pont et al., 2008; Sing et al., 2009; Gibson et al., 2011). The red curve
presents a pure Rayleigh scattering model while a gas opacity only model is
also shown. With the possible exception of a spectral feature at 1.5 μm the
smooth variation of planet radius with wavelength extends to at least 2.5 μm
(Gibson et al., 2011).

Figure 6: Observed transmission spectrum (points) for transiting hot Jupiter
planet HD 189733b (Sing et al. 2011). The width of the wavelength bin for each
measurement is indicated by the x-axis error bars. The y-axis error bars
denote the 1-ς error. The smooth curve denotes the prediction of a simple haze
Rayleigh scattering-only model while the lower curve is a model for gaseous
absorption only from Fortney et al. (2010). Further details in Sing et al.
(2011). Figure courtesy J. Fortney.

The most natural explanation for the HD 189733b transmission spectrum is that
a population of small, high albedo particles is present in the atmosphere at
low pressures (Lecavelier des Etangs et al., 2008). In the terminology of Mie
scattering this requires that the scattering efficiency is large compared to
the absorption efficiency or $Q_{\mathit{abs}}{\ll}Q_{\mathit{scat}}$.
Lecavelier des Etangs et al. demonstrate that this in turn requires a material
with an imaginary index of refraction that is small compared to the real index
and suggest MgSiO3 as a possible candidate. In the condensation chemistry
framework (Section 3.4) however, silicates are expected to condense much
deeper in the atmosphere and the mechanism by which small grains could be
transported to the upper atmosphere has yet to be fully explored, although
vigorous mixing is a likely explanation.

Another transiting planet for which particulate opacity may be important is GJ
1214b. This 6.5-Earth mass planet orbits an M star. Planetary structure models
that fit the observed mass and radius of the planet can be found with either a
thick hydrogen atmosphere comprising less than 3% of the mass of the planet or
a water rich planet surrounded by a steam atmosphere. Transmission spectra of
a large scale height, hydrogen-dominated atmosphere (Figure 7) are predicted
to exhibit well defined absorption bands while a water dominated atmosphere
would have a much smaller scale height and consequently a very smooth
transmission spectrum (Miller-Ricci & Fortney, 2010).

The observed flat transmission spectrum from the optical to perhaps 5 μm (Bean
et al., 2010, 2011; Berta et al., 2012; Desert et al., 2011) is consistent
with a small scale-height atmosphere, thus apparently favoring the water-rich
alternative. However high altitude clouds or hazes, as with HD 189733 b, could
also be concealing atmospheric absorption bands if they lie at altitudes above
200 mbar (Bean et al., 2010 and Figure 7). Since the nominal model pressure-
temperature profile does not cross the condensation curve of any major species
at solar abundance (although ZnS and KCl do condense around 1 bar), Bean et
al. proposed photochemical products as the likely source of the haze rather
than clouds. However as shown in Figure 7, Morley et al. (2013) find that very
extended (fsed=0.1) equilibrium sulphide clouds, including Na2S, in an
atmosphere enriched in heavy elements can also attenuate the flux sufficiently
to reproduce the data. Such small values of fsed are seen in some terrestrial
regimes (Ackerman & Marley 2001) but whether or not this would be plausible in
the atmosphere of GJ 1214b remains to be investigated.

Miller-Ricci Kempton et al. (2012) explored possible photochemical pathways in
the atmosphere of this planet to explore possible mechanisms for haze
production and found that second order hydrocarbons, including C2H2, C2H4, and
C2H6 are efficiently produced by UV photolysis of methane in the atmosphere.
While their model did not explore the chemistry to hydrocarbons of order
higher than C2H6, they argue that polymerization of the initial photochemical
products are likely and that complex hydrocarbons including tholins and soots
are likely to form. Morley et al. (2013) demonstrate that such a hydrocarbon
haze could well explain the flat transmission spectrum observed for this
planet if the atmosphere is indeed hydrogen rich with efficient methane
photolysis (with results very similar to the cloudy case shown in Figure 7).
In an unpublished manuscript Zahnle et al. (2009) argue that for solar
composition atmospheres in general there is a range of atmospheric
temperatures near 1000 K that would be expected to favor methane photolysis
and the formation of higher order hydrocarbon soots or hazes. A definitive
exploration of this point would require new generations of computer codes that
can follow the fate of hydrocarbon species produced by photochemistry.

Figure 7: Observed (points) and model (lines) transit radius of GJ 1214b.
Datapoints are from multiple observations as described in Morley et al.
(2013). Models are for a cloudless H2-He rich atmosphere with 50 times the
solar abundance of heavy elements (grey) and the same atmosphere with the
equilibrium abundance of clouds computed with fsed = 0.1 (red). Figure
modified from Morley et al. (2013).

While it is still too soon to definitely characterize the atmospheres of
either HD 189733b or GJ 1214b, it is apparent that cloud or haze opacity is
likely important in at least some transiting planet atmospheres. The James
Webb Space Telescope will obtain transit spectra of dozens of planets and
characterize atmospheric absorbers as a function of planet mass and
composition and the degree of stellar insolation. Such comprehensive studies
will help to map out the conditions under which clouds and hazes are found.

####  5.1.2 Transiting Planets in Reflected Light

In addition to the light transmitted through the atmosphere of transiting
planets, measurements have also been made of the the apparent brightness of
the day side of transiting planets. By comparing the brightness of a
transiting system immediately before and after a planet is occulted by its
primary star the combination of light reflected and emitted by the planet can
be measured. In the limit in which the planet does not emit but rather shines
only by reflected light within the passband such a measurement provides the
mean geometric albedo of the planet. However since such a measurement is most
tractable for large planets orbiting close to their stars, in other words the
hot Jupiter class of planets, thermal emission cannot generally be
disregarded.

Upper limits on the geometric albedo have been placed on many planets (see the
summary in Demory et al. (2011)), generally finding the passband averaged
geometric albedo to be less than about 30% and in some cases much less
(Kipping & Spiegel, 2011). Such a finding is not surprising for hot giant
planet atmospheres (Marley et al., 1999; Sudarsky et al., 2000, 2003) since in
the absence of a cloud layer most of the incident flux is absorbed rather than
scattered. Despite these predictions as constraints became available from
studies of transiting planets the preponderance of low albedos was often
treated as surprising when compared to the bright albedo of Jupiter.

Persuasive evidence has been found, however, indicating that at least one hot
Jupiter has a large geometric albedo that is likely attributable to a haze or
cloud layer. Kepler-7b (Latham et al., 2010) has a geometric albedo averaged
over the Kepler bandpass (423 to 897 nm, Koch et al., 2010) of $0.32\pm 0.03$
(Demory et al., 2011). Such a high albedo is more typical of the cloudy solar
system giants (e.g., Karkoschka, 1994) than a deep scattering atmosphere.
Demory et al. thus argue that the most likely explanation is that this
particular planet has a bright photochemical haze or cloud layer, perhaps
similar to that seen in the transit spectra of HD 189733b. More detailed
modeling to constrain the properties of the scattering layer has not yet been
done and to date transit spectra have not been obtained for Kepler 7b.

As albedos are measured for more planets with appropriate corrections for
thermal emission, it may become apparent which particular combinations of
planet mass, composition, and stellar incident flux (particularly including UV
flux) conspire to produce high albedo planets.

###  5.2 Directly Imaged Planets

####  5.2.1 Young Jupiters

Giant planets start their lives in a warm, extended state with luminosities
much greater than at later times after they have cooled. For this reason giant
planets are easier to detect at young ages of a few hundred million years or
less and several have already been detected, including 2M 1207b and the
planets orbiting the A star HR 8799 (Marois et al., 2008; 2010). The near-
infrared thermal emission of all of these objects (which have effective
temperatures near 1000 K) points to the presence of substantial refractory
cloud decks, most likely comprised of iron and silicate grains (e.g., Marois
et al., 2008; Barman et al., 2011). Much of the literature on these objects
has focused on constraining the properties of these clouds because in field
brown dwarfs clouds have largely dissipated by 1000 K. Currie et al. (2011),
Madhusudhan et al. (2011), and Bowler et al. (2010) also all explored models
for the directly imaged planets and agreed that clouds played a critical role
in shaping their thermal emission at a lower effective temperature than is
typical for more massive field brown dwarfs.

Marley et al. (2012) also constructed model atmospheres for these planets and
concurred that clouds are present in the atmospheres of HR 8799 b, c, and d.
They applied the cloud model of Ackerman & Marley (2001) and found that they
could reproduce much of the available data by using cloud parameters typically
seen in warmer L dwarfs. They argue from mass balance considerations that mean
cloud particle size likely varies inversely with $\sqrt{g}$ where g is the
gravitational acceleration and that all else being equal the column optical
depth of a cloud varies proportionately to $\sqrt{g}$. The net result of these
scaling relationships is that clouds in lower gravity objects tend to be
similar to clouds found at warmer effective temperatures in higher gravity,
more massive brown dwarfs. They find that as objects cool their spectra can be
modeled by increasing the sedimentation efficiency, fsed in Equation (1). The
effective temperature at which fsed begins to increase varies with gravity
such that lower gravity objects begin to lose their refractory clouds at
warmer effective temperatures than low mass objects.

At cooler effective temperatures near 600 K another set of condensates become
important in field dwarfs. Morley et al. (2012) have demonstrated that Na2S
and other clouds (Figure 3) moderately redden the near-IR colors of late T
type brown dwarfs. Once young giant planets in this effective temperature
range are discovered it will be possible to test if the scaling relationships
seen at higher effective temperature persist.

Ultimately understanding the nature of clouds in the warm, young giant planets
hinges on understanding why the cloud clearing effective temperature varies as
it does. In the next few years many more young giant planets are expected to
be discovered by the upcoming GPI and SPHERE exoplanet surveys.

####  5.2.2 Giants in Reflected Light

No giant planet has yet been imaged in reflected light although such an
observation is less technically difficult than imaging a terrestrial planet in
reflected starlight and a number of space telescope missions have been
proposed. As with terrestrial planets, clouds play a large role in affecting
the reflection spectra of a giant planet (Marley et al., 1999; Sudarsksy et
al., 2002; Cahoy et al., 2010).

One way of visualizing the effect of clouds in a giant planet atmosphere is to
imagine a Jupiter twin lying at progressively closer distances to the sun
(Figure 4). The optical geometric albedo spectrum of Jupiter generally
consists of a bright continuum punctuated by methane absorption bands. The
bright continuum is formed by scattering from gas and stratospheric hazes in
the blue and the bright ammonia clouds at longer wavelengths. The methane
bands are formed from the column of gas overlying the clouds. Below the
ammonia cloud-tops lie cloud decks of NH4SH and H2O.

A Jupiter twin (with the same internal heat flux) lying closer to its star at
2AU would have a warmer atmosphere in which the ammonia and ammonium
hydrosulphide clouds would not condense. Instead the planet would be covered
by a bright global layer of water clouds which would give a very high albedo
(Figure 1). At even closer distances to its star the atmosphere would be too
warm for water clouds to form. Gas absorption thus overwhelms gaseous Rayleigh
scattering and the planet becomes much darker (Figure 1) than the cloudless
case.

##  6 Additional Topics

###  6.1 Clouds of Low Porosity Aggregates

Most of the discussion in this chapter has focused on cloud or haze particles
as fully dense spheres. However fluffy or porous aggregate particles may very
well form and such particles behave differently both as they interact with the
atmosphere and with radiation. In this section we simplify the Ackerman &
Marley (2001) approach to illustrate how low-porosity aggregates would behave.
In this simplified version, we neglect the fraction of the condensible species
present as vapor above the cloud base, so $q_{t}$= $q_{c}$. We stress that
this treatment is not a replacement for the complete model.

When the particles are too small to settle (see below) the condensate mixing
ratio $q=q_{c}=\rho_{c}/\rho_{g}$ is constant and so has an infinite scale
height $H_{q}$ (the condensate has the same scale height as the atmosphere).
However, when the cloud is significantly settled such that the condensate
mixing ratio scale height $H_{q}{\ll}H,$ and if it is assumed that the vapor
mixing ratio is negligible compared to the condensate ratio then Equation (1)
governing the vertical cloud distribution can be approximated as
$K_{\mathit{zz}}\mathit{dq}/\mathit{dz}-v_{f}q=0$, where Kzz = Lw* is the
vertical eddy diffusion coefficient, a property of macroscopic turbulence with
typical lengthscale L and large (energy-containing) eddy velocity w*; L is
usually taken to be the atmospheric scale height H, although this might not be
valid if the associated large eddy timescale L/w* is much longer than the
rotation period of the object (Schubert & Zhang, 2000) or the temperature
profile is stable (Ackerman & Marley use an ad hoc stability correction).
Then, for a settled cloud with uniform mass mixing ratio q and effective
thickness Hq, the above equation is approximated by
$K_{\mathit{zz}}q/H_{q}-v_{f}q=0$, thus the cloud thickness is roughly
$H_{q}=Hw^{*}/v_{f}$. The scale height in the simplified exponential solution
for condensate mass mixing ratio of Ackerman & Marley (2001) is essentially
the same. That is, when the convective eddy velocity is much larger than the
settling velocity, the cloud particles are not able to settle (
$f_{\mathit{sed}}=0$).

Particle settling velocities depend upon the dynamical regime. The settling
velocity vf is the product of the local gravity g and the gas drag stopping
time of the particle, ts. Ackerman & Marley (2001), equation B2, contains a
bridging expression β that covers both the so-called Stokes and Epstein drag
regimes, in which r is respectively greater than, or less than, the gas
molecule mean free path $l_{m}=m_{H_{2}}/\sigma_{H_{2}}\rho_{g}$, where
$m_{H_{2}}$ and $\sigma_{H_{2}}$ are the mass and cross section of a hydrogen
molecule and $\rho_{g}$ is the gas density. In equations B1 and B3 of Ackerman
& Marley (2001) the density contrast is essentially the particle density
because $\rho{\gg}\rho_{g}$ for all applications of interest. So, vf is
proportional to ρ, and for particles with porosity $\varphi$,
$\rho=\rho_{s}\left(1-\varphi\right)=\rho_{s}f$. In the Epstein regime (r/lm <
1) the expression for ts is very simple: $t_{s}=r\rho/c\rho_{g}$ where c is
the sound speed. Cuzzi & Weidenschilling (2006) show how the stopping time in
the Stokes regime (r/lm > 1) is essentially larger by a factor of r/lm.

We can thus show that for porous particles under local gravity g, having the
same mass as a particle with no porosity,
$v_{f}=\mathit{gr}\rho/c\rho_{g}=gr_{s}\rho_{s}\left(1-\varphi\right)^{2/3}/c\rho_{g}$,
and thus
$H_{q}=Hw^{*}/v_{f}=H_{q\left(\mathit{solid}\right)}\left(1-\varphi\right)^{-2/3}$.
Porous particles are lofted to greater heights than non-porous particles of
the same mass, because of their slower settling velocity. In this sense they
behave like smaller particles; however, their radiative behavior is not that
of smaller particles, as noted above; porous particles in the regime
$r/\lambda{\gg}1$ indeed have large and wavelength-independent opacity.

###  6.2 Polarization

Polarization may provide an additional avenue for characterizing cloudy
planets. Unlike the stellar radiation emitted by the central star, the light
scattered by clouds will in general be polarized. Thus, investigating the
polarized radiation scattered by clouds in contrast to the non-polarized
stellar radiation may be an opportunity for characterizing cloudy exoplanetary
atmospheres at short wavelengths in the future (see Stam 2008).

Giant planets can also be polarized in the thermal infrared. In this case the
required asymmetry in radiation emitted across the apparent planetary disk
must be provided either by rotational flattening or irregularities in the
global cloud cover. Marley & Sengupta (2011) investigated the former mechanism
and de Kok et al. (2011) the latter. Both sets of authors found that in the
most favorable circumstances polarization fractions of a few percent were
plausible and that in such cases polarization confirms the presence of a
scattering condensate layer.

While polarization undoubtedly provides additional constraints on cloud
particle sizes (e.g., Bréon & Colzy, 2000), condensate phase (e.g., van
Diedenhoven et al., 2012a), and even asymmetry parameter (van Diedenhoven et
al., 2012b) and atmospheric structure, it may in practice be of limited value
for studies of exoplanets. Within the solar system the most well known
discovery attributable to a polarization measurement is the particle size of
the clouds of Venus (Hansen & Hovenier, 1974). An imaging photopolarimeter
carried on the Pioneer 10 and 11 spacecraft also constrained the vertical
structure of Jupiter’s clouds (Gehrels et al., 1974). However in general other
techniques have proven more valuable. Especially given the low signal to noise
and difficulty inherent in any measurement of an extrasolar planet, the value
of further dividing the light into polarization channels must be weighed
against other potential measurements (e.g., obtaining higher resolution
spectroscopy).

##  7 Conclusions

As with the planetary atmospheres of solar system planets, clouds are expected
to play major roles in the vertical structure, chemistry, and reflected and
emitted spectra of extrasolar planets. That said, the most extensive
experience to date with clouds outside of the solar system has been with the
brown dwarfs. The presence of refractory clouds in L dwarfs and sulphide and
salt clouds in late T dwarfs has been well established and a number of methods
have been developed to model these clouds.

Compared to cloud modeling approaches within the solar system or particularly
on Earth, exoplanet cloud models are still in their infancy. In many cases
arbitrary clouds are employed that specify a range of plausible cloud
properties that are usually sufficient to explore parameter space. More
sophisticated efforts attempt to derive particle sizes, composition, and
number density as a function of height through the atmosphere from a given set
of assumptions will be needed once higher resolution, broad wavelength
spectral data become available.

As of the time of this chapter’s writing the best evidence for clouds in
extrasolar planet atmospheres lies in the spectra of the planets orbiting the
nearby A star HR 8799. Planets b, c, and d each have red near-infrared colors
that are best explained by global refractory cloud decks that have persisted
to lower effective temperatures than in higher mass field brown dwarfs. This
persistence of clouds to lower effective temperatures for lower gravity
objects continues a trend that has already been recognized among brown dwarfs.

Among the transiting planets there is convincing evidence for high altitude
clouds–or perhaps a photochemical haze–in the atmosphere of HD 189733b. The
transit spectra of this planet lacks the deep absorption bands expected for a
clear atmosphere of absorbing gas but rather exhibits a smoothly varying
absorption profile likely caused by small grain scattering. A second planet,
GJ 1214b also has a transit spectrum lacking absorption features. In this case
the spectrum may be attributable either to a small atmospheric scale height
resulting from a high mean molecular weight composition or from clouds.

These early detections are like distant clouds seen at sunset presaging a
coming storm. The GPI, SPHERE, and many other direct imaging planet searches
are expected to discover dozens of young, self-luminous extrasolar giant
planets over the coming decade (Oppenheimer & Hinkley, 2009; Traub &
Oppenheimer, 2010). Many of these planets will exist in the effective
temperature range in which clouds shape their emergent spectra. Meanwhile
continued studies of transiting planets, particularly by the upcoming James
Webb Space Telescope will probe the atmospheres of the transiting planets and
test the hypothesis that hazes of photochemical origin are important in some
regimes. Finally future space based coronagraphic telescopes may eventually
image extrasolar giant and terrestrial planets in reflected light. All such
efforts to discover and characterize extrasolar planets will hinge upon an
understanding of the role clouds play in shaping the climate and atmospheres
of planets.

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