# Non-precipitating shallow cumulus convection is intrinsically unstable to
length-scale growth ††thanks: This preprint is intended for publication in a
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Martin Janssens12*, Jordi Vilà-Guerau de Arellano1,
Chiel C. van Heerwaarden1, Stephan R. de Roode2,
A. Pier Siebesma23, Franziska Glassmeier2

(1 Meteorology & Air Quality Department, Wageningen University & Research
2 Geoscience & Remote Sensing Department, Delft University of Technology
3 Royal Netherlands Meteorological Institute
)

###### Abstract

Condensation in cumulus clouds plays a key role in structuring the mean, non-
precipitating trade-wind boundary layer. Here, we summarise how this role also
explains the spontaneous growth of mesoscale ($>O(10)$ km) fluctuations in
clouds and moisture around the mean state in a minimal-physics, large-eddy
simulation of the undisturbed period during BOMEX on a large ($O(100)$ km)
domain. Small, spatial anomalies in latent heating in cumulus clouds, which
form on top of small moisture fluctuations, give rise to circulations that
transport moisture, but not heat, from dry to moist regions, and thus
reinforce the latent heating anomaly. We frame this positive feedback as a
linear instability in mesoscale moisture fluctuations, whose time-scale
depends only on i) a vertical velocity scale and ii) the mean environment’s
vertical structure. In our minimal-physics setting, we show both ingredients
are provided by the shallow cumulus convection itself: It is intrinsically
unstable to length scale growth. The upshot is that energy released by clouds
at kilometre scales may play a more profound and direct role in shaping the
mesoscale trade-wind environment than is generally appreciated, motivating
further research into the mechanism’s relevance.

##  1 Introduction

Shallow clouds organised into mesoscale patterns by convective instabilities
have been recognised as a ubiquitous feature of the subtropical marine
boundary layer since satellite imagery in the 1960s first revealed them (Agee
et al.,, 1973). While their discovery sparked much research on the role of
convective instabilities in patterning boundary layer clouds, much of that
research was long focused towards open and closed convective cells (e.g.
Fiedler,, 1985; Müller and Chlond,, 1996). Yet, a rich spectrum of cloud
patterns can be found outside the paradigm of such mesoscale cellular
convection (Wood and Hartmann,, 2006), including for shallow cumulus clouds
that top the trade-wind marine boundary layer (Stevens et al.,, 2020; Denby,,
2020; Janssens et al.,, 2021).

The interest in the self-organisation of trade-wind cumulus has risen in
recent years, in response to cloud-resolving simulations of deep convection
(Muller and Held,, 2012), which spontaneously develop mesoscale fluctuations
in their cloud structures. Since deep convective organisation plays an
important role in regulating radiative heat loss from the atmosphere (Tobin et
al.,, 2012), it seemed natural to ask whether the observed shallow convective
organisation plays a similarly important role. Bony et al., (2020) suggest
that the answer to this question is yes; in observations, different trade-wind
cumulus patterns, forming under different larger-scale conditions, have
different cloud feedbacks. Given the disparity between observations and
climate model simulations of the trade-cumulus feedback (Myers et al.,, 2021;
Cesana and Del Genio,, 2021), this provides ample motivation for better
understanding the processes that pattern shallow cumulus-topped marine
boundary layers.

Many mesoscale cumulus patterns may simply be either passive responses to
mesoscale heterogeneities in cloud-controlling conditions driven by larger-
scale dynamics, or are remnants of extratropical disturbances advected into
the trades (Schulz et al.,, 2021). However, several others appear to result
from the shallow convection itself. Sub-cloud layer rain evaporation can
trigger density currents that force new convection upon collision (Seifert and
Heus,, 2013; Zuidema et al.,, 2017), while heterogeneous radiative cooling can
drive circulations that lead to cloud clustering (Klinger et al.,, 2017;
Naumann et al.,, 2019). Even simulations of clear convective boundary layers
(Jonker et al.,, 1999) and stratocumulus-topped layers (de Roode et al.,,
2004) spontaneously develop appreciable mesoscale fluctuations in their
moisture fields. Building on these studies, Bretherton and Blossey, (2017)
(BB17 hereafter) in a remarkably thorough piece of work noted that even non-
precipitating shallow cumulus convection - stripped of all interactive
precipitation and radiation feedbacks - self-organises into clusters in large-
eddy simulations (LESs) on domains larger than 100 km, over similar time
scales as observed in nature (Narenpitak et al.,, 2021).

This finding is rather striking, since studies of the slab-averaged structure
of such layers have a rich history, dating back to the observational budget
surveys of the Northeast Pacific (Riehl et al.,, 1951), the Atlantic Tradewind
Experiment (ATEX, Augstein et al.,, 1973) and the Barbados Oceanographic and
Meteorological Experiment (BOMEX, Nitta and Esbensen,, 1974). These campaigns
established the classical picture of the trade-wind boundary layer: It is
maintained by transport of liquid water into the trade-inversion, where its
evaporation moistens and cools the layer sufficiently to balance the drying
and heating from the subsiding environment. Using data from the BOMEX
campaign, Betts, (1973, 1975) then traced the energy supporting the fluxes of
liquid water into the inversion to the slab-averaged net condensation in the
conditionally unstable cloud layer below.

Here, we will revisit minimal-physics LES of the “undisturbed” period during
phase 3 of BOMEX, detailed in section 2, with two objectives. First, we will
review BB17’s instability in this classical view of the the trade-wind layer,
to show that the instability can be understood as a natural extension of the
role played by net condensation in the slab mean (section 3) to mesoscale
fluctuations around that mean (section 4). Our hope is that making this
connection will aid the interpretative side of future studies of the
mechanism, e.g. in attempts to understand the relative importance of processes
that can pattern trade-wind clouds.

Our second objective is to study the origins of the instability more
quantitatively than BB17. To do so, we extend their theory to a linear
stability model for bulk mesoscale moisture fluctuations, and examine its
conditions for instability (section 5). We will show that these are satisfied
by the cumulus convection itself, and do not require anything from the large-
scale environment, other than that it supports a cumulus layer. Put
differently, we will conclude that shallow cumulus convection is intrinsically
unstable to length scale growth. We end the paper by discussing the relevance
of these findings to several ongoing studies of the self-organising cumulus
layer, and suggest a few directions that such future research could take
(section 6). A summmary is given in section 7.

##  2 Large-eddy simulation of the undisturbed period during BOMEX

###  2.1 Case study

We consider a situation based on observations performed on 22 and 23 June
1969, during phase 3 of BOMEX. There are many reasons for this. First, during
this so-called “undisturbed” period, the vertical slab-mean moisture and heat
profiles were observed to be in a nearly steady state, capped by a well-
defined inversion. In fact, the steadiness of these days was an important
reason to select them for the budget studies that diagnosed the main features
of cumulus convection in an undisturbed environment (Holland and Rasmusson,,
1973; Nitta and Esbensen,, 1974). Later, this also helped popularise the case
as a testbed for validating LES models (Siebesma and Cuijpers,, 1995; Siebesma
et al.,, 2003), as it allowed comparing statistics averaged over long time-
periods. As a result, the undisturbed period is perhaps the single most most
studied realisation of the trade-wind boundary layer. All these features make
the situation attractive for our study, since it is our objective to use LES
to study the development of fluctuations around a steady mean state, departing
from the well-established theory from the early observational work.

It is worth pausing here to note that Nitta and Esbensen, (1974) already show
that the trade-wind layer is usually not in steady state, but is highly
variable. Furthermore, recent observations of the subtropical Atlantic reveal
that the trades usually feature stronger winds, weaker subsidence and stronger
temperature inversions than observed during the undisturbed period, often
associated with larger-scale, precipitating cloud structures (Schulz et al.,,
2021). Therefore, the situation we study should be considered illustrative,
rather than representative.

The second reason we concentrate on BOMEX is that BB17 also report LES results
of the case. Their simulations produce significant mesoscale moisture and
cloud fluctuations, if run for several days on domains whose horizontal
dimensions exceed 100 km $\times$ 100 km. Hence, we will be able to translate
rather directly between their results and ours.

Finally, the BOMEX setup we consider excludes and simplifies a number of
processes. Of particular interest here are that the case i) ignores spatial
and temporal variability in the large-scale subsidence, horizontal wind and
surface fluxes of heat and moisture, instead imposing constant forcings for
all three, ii) does not locally calculate radiative heating rates,
approximating them with a slab-averaged cooling, and iii) explicitly ignores
the formation and impact of precipitation. This will suppress length scale
growth encouraged by large-scale vertical ascent (Narenpitak et al.,, 2021),
radiation (Klinger et al.,, 2017) and cold pools (Seifert and Heus,, 2013),
respectively, all of which appear to be important pathways to develop the
mesoscale cumulus patterns observed in nature.

We do not suggest that variable larger-scale forcing, radiation and
precipitation do not influence the length scale growth in shallow cumulus
fields. We merely note that BB17 find that they are not necessary ingredients;
they merely act to modulate an internal, dynmical growth mechanism that also
occurs without them. The mechanism in question is thus fundamentally rooted in
moist, shallow convection, and its understanding is clarified by only studying
this aspect.

###  2.2 Model setup

We simulate BOMEX using the Dutch Atmospheric Large Eddy Simulaton (DALES,
Heus et al.,, 2010; Ouwersloot et al.,, 2017). We run the case precisely as
reported by Siebesma et al., (2003), save for its computational grid,
integration time and advection scheme. To allow the formation of mesoscale
fluctuations with little influence from the finite domain size, the cases are
run on horizontally square domains spanning 102.4 km, with a height of 10 km,
for 36 hours. The horizontal grid spacing $\Delta x=\Delta y=200$ m, while the
vertical grid spacing $\Delta z=40$ m up to 6 km; it is stretched by 1.7% per
level above this height. The case is run with a variance-preserving, second-
order central difference scheme to represent advective transfer. We will
concentrate our analysis on the early phase of the simulation, since it
develops strong moisture fluctuations that approach the scale of the domain
length after around 18 hours. Subsequently, deep convective clouds develop.
Such situations are deemed unrealistic in our non-precipitating simulations
over domains with doubly-periodic boundary conditions.

##  3 The classical theory

In the anelastic approximation adopted by our LES code, the local budget of a
generic scalar $\chi$, which here will denote water or a measure of heat, can
be written as

$\frac{\partial\chi}{\partial t}=-\frac{\partial}{\partial
x_{jh}}\left(u_{jh}\chi\right)-\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}w\chi\right)+S_{\chi},$ (1)

where $u_{jh}$ contains the horizontal velocity vector, the subscript $jh$
indicates summation over the horizontal coordinate, $w$ is the vertical
velocity, $\rho_{0}(z)$ is a profile of reference density and $S_{\chi}$ is a
local source. Since we are interested in fluctuations in $\chi$, we introduce
the definition

$\chi=\overline{\chi}+\chi^{\prime},$ (2)

where $\overline{\chi}$ and $\chi^{\prime}$ respectively refer to the slab-
average and resulting fluctuation of any model variable. It is instructive to
use this partitioning to rewrite eq. 1 on the following form (see appendix A
for a step-by-step procedure):

$\frac{\partial\chi}{\partial
t}=-\overline{u_{jh}}\frac{\partial\chi}{\partial
x_{jh}}-u_{jh}^{\prime}\frac{\partial\overline{\chi}}{\partial
x_{jh}}-\frac{\partial}{\partial
x_{jh}}\left(u_{jh}^{\prime}\chi^{\prime}\right)-\overline{w}\frac{\partial\chi}{\partial
z}-w^{\prime}\frac{\partial\overline{\chi}}{\partial
z}-\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}w^{\prime}\chi^{\prime}\right)+S_{\chi},$ (3)

where we have used the anelastic conservation of mass. Eq. 3 will serve as our
point of departure for the rest of the study. In it, the first three terms
describe horizontal transport i) with the mean wind, ii) with fluctuations in
the horizontal velocity, and iii) through turbulent fluxes, respectively. The
fourth term represents transport with the mean vertical velocity, often
associated with the prevailing subsidence of the trades, the fifth term
denotes transport with vertical velocity fluctuations against the mean
gradient, the sixth term describes vertical turbulent transport, and the final
term is again reserved for sources.

###  3.1 Slab-averaged heat and moisture budgets

We will first briefly summarise the dynamics that govern the slab-averaged
thermodynamic structure, since these dynamics turn out to also mostly explain
the development of mesoscale fluctuations on top of it. The mean scalar
budgets can be derived from eq. 3 by i) slab-averaging it over a sufficiently
large region to represent an ensemble average and ii) assuming the horizontal
flux divergence out of the region over which we average is small:

$\underbrace{\frac{\partial\overline{\chi}}{\partial
t}}_{\text{Tendency}}+\underbrace{\overline{u_{jh}}\frac{\partial\overline{\chi}}{\partial
x_{jh}}}_{\text{Horizontal
advection}}+\underbrace{\overline{w}\frac{\partial\overline{\chi}}{\partial
z}}_{\text{Subsidence}}=-\underbrace{\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}\overline{w^{\prime}\chi^{\prime}}\right)}_{\text{Vertical
flux
convergence}}+\underbrace{\overline{S_{\chi}}}_{\text{Sources}}\equiv\underbrace{Q}_{\text{Apparent
source}},$ (4)

$Q$ defines Yanai et al., (1973)’s apparent heat source and moisture sink, if
the equations are posed for appropriate heat and moisture variables,
respectively. We will present these budgets for the two prognostic variables
in our LES model, which are conserved under non-precipitating cumulus
convection: The total water specific humidity $q_{t}$ and liquid-water
potential temperature $\theta_{l}$, approximated as

$\theta_{l}\approx\theta-\frac{L_{v}}{c_{p}\Pi}q_{l}.$ (5)

Here, $\theta=T/\Pi$ is (dry) potential temperature, $T$ is temperature,
$L_{v}$ is the latent heat of vaporisation, $c_{p}$ is the specific heat of
dry air at constant pressure, $q_{l}$ is the liquid water specific humidity
and

$\Pi=\left(\frac{p}{p_{0}}\right)^{\frac{R_{d}}{c_{p}}}$ (6)

is the Exner function, with $R_{d}$ the gas constant of dry air, $p$ the
reference pressure profile, and $p_{0}=10^{5}$ Pa.

In our LES model, which features doubly periodic boundary conditions, slab
averages taken of horizontal gradients and vertical velocity are zero by
definition. Therefore, we impose the horizontal transport and the vertical
velocity in the subsidence term on the left-hand side of eq. 4, in addition to
a slab-averaged radiative cooling sink in the budget for $\theta_{l}$. The
resulting contributions to eq. 4 are plotted in figs. 1 a) and b); they mirror
those simulated by Siebesma and Cuijpers, (1995), which in turn reasonably
match the apparent heat and moisture sources measured by Nitta and Esbensen,
(1974).

Figure 1: Slab-averaged contributions to the slab-averaged tendencies of
$q_{t}$ (a), $\theta_{l}$ (b) and $q_{l}$ (c) (eq. 4 rewritten for
$\partial\overline{\chi}/\partial t$ on the left-hand side), additionally
averaged over hours 2-16 of the simulation. The source of
$\overline{\theta_{l}}$ is radiative, negative and imposed, the source of
$\overline{q_{l}}$ is the net condensation rate.

These budgets quantify the effects of shallow cumulus convection on the slab-
average thermodynamic structure. Up to the trade-inversion’s base, convergence
of moisture and heat fluxes moisten and heat the layer. Conversely, these
fluxes moisten and cool the inversion layer, countering the drying and warming
from the mean environment’s subsidence. The imposed radiative source offers
additional cooling throughout the layer. In spite of our intentions, and
contrary to the models participating in Siebesma et al., (2003), these
processes do not quite balance, resulting in a negative
$\overline{\theta_{l}}$ tendency and positive $\overline{q_{t}}$ tendency in
the inversion layer: Our simulation is not a steady mean state. We will
elaborate on this in section 6.

###  3.2 The role of net condensation

The appropriate thermodynamic quantity for analysing the capacity of the
boundary layer to work against the subsiding environment is the buoyancy. It
also bridges the apparent gap between the budgets of conserved variables
discussed above, which do not refer to the presence of liquid water, and the
role of net condensation, which is key to our analysis.

We will interpret buoyancy through fluctuations in virtual potential
temperature $\theta_{v}$:

$\theta_{v}=\theta\left[1+\left(\frac{R_{v}}{R_{d}}-1\right)q_{t}-\frac{R_{v}}{R_{d}}q_{l}\right].$
(7)

Borrowing from Stevens, (2007), its fluctuations can to good approximation can
be written as

$\theta_{v}^{\prime}\approx
a_{1}\theta_{l}^{\prime}+a_{2}\overline{\theta_{l}}q_{t}^{\prime}+a_{3}\overline{\theta_{l}}q_{l}^{\prime}$
(8)

with constants $a_{1}-a_{3}$ set to

$a_{1}=\frac{\theta_{v_{c}}}{\theta_{l_{c}}}\frac{1}{1+\frac{q_{l_{c}}L_{v}}{c_{p}T_{c}}}\approx
1,$ (9a) $a_{2}=1-\frac{R_{v}}{R_{d}}\approx 0.608,$ (9b)
$a_{3}=a_{1}\frac{\theta_{l_{c}}L_{v}}{\theta_{c}c_{p}T_{c}}-\frac{R_{v}}{R_{d}}\approx
7.$ (9c)

In these relations, co-fluctuations among the thermodynamic variables are
neglected, and $\theta_{v_{c}}$, $\theta_{l_{c}}$, $\theta_{c}$, $q_{l_{c}}$
and $T_{c}$ are taken to be representative cloud layer constants. $R_{v}$ is
the ideal gas constants for dry moist air. If we reinterpret the spatial
fluctuations in eq. 8 as changes occurring in time, we may also to good
approximation write

$\frac{\partial\theta_{v}}{\partial
t}\approx\frac{\partial\theta_{l}}{\partial
t}+a_{2}\overline{\theta_{l}}\frac{\partial q_{t}}{\partial
t}+a_{3}\overline{\theta_{l}}\frac{\partial q_{l}}{\partial t}.$ (10)

We will make two notes on eq. 10 that prepare us for our subsequent
discussion. First, we note that $a_{3}\overline{\theta_{l}}\frac{\partial
q_{l}}{\partial t}$ is small in the slab-average (Betts,, 1973), giving a
powerful constraint that we return to in section 4 4.4. Consider eq. 4 with
$\chi=q_{l}$, plotted in fig. 1 c). In our simulations, we do not impose
large-scale horizontal transport of liquid water, while the effects of the
prescribed subsidence on $\overline{q_{l}}$ are negligible. This allows us to
simplify eq. 4 to

$\frac{\partial\overline{q_{l}}}{\partial
t}=-\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}\overline{w^{\prime}q_{l}^{\prime}}\right)+\overline{\mathcal{C}}\approx
0,$ (11)

where $\mathcal{C}$, the net condensation, is the only source in the absence
of precipitation. Eqs. 10 and 11 together explain how, in the conditionally
unstable cloud layer, the positive net condensation is not stored as local
potential energy ($\theta_{v}$), but is instead almost immediately exchanged
into upwards transport; this is the balance shown in fig. 1 c). In the stable
inversion layer, the transported liquid water entirely re-evaporates,
resulting in a local convergence of $\overline{w^{\prime}q_{l}^{\prime}}$ that
integrates to zero over the column.

Applying the constraint eq. 11 to eq. 10 brings us to our second note: Eq. 10
reduces to an equation for another quantity that is conserved in non-
precipitating convection, which Grenier and Bretherton, (2001) call the
liquid-water virtual potential temperature:

$\theta_{lv}=\theta_{l}+a_{2}\overline{\theta_{l}}q_{t}\equiv\theta_{v}-a_{3}\overline{\theta_{l}}q_{l}$
(12)

Inserting eq. 4 with $\chi\in\left[\theta_{l},q_{t}\right]$ in the slab-
averaged eq. 10 gives

$\frac{\partial\overline{\theta_{v}}}{\partial
t}\approx\frac{\partial\overline{\theta_{lv}}}{\partial
t}\propto-\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}\left(\overline{w^{\prime}\theta_{l}^{\prime}}+a_{2}\overline{\theta_{l}}\overline{w^{\prime}q_{t}^{\prime}}\right)\right),$
(13)

from which we may observe that the tendency of $\overline{\theta_{v}}$ is
proportional not to the slab-averaged buoyancy flux
$\overline{w^{\prime}\theta_{v}^{\prime}}$, but to fluxes of $\theta_{lv}$,
which may be derived from eq. 8 by multiplication with $w^{\prime}$:

$w^{\prime}\theta_{lv}^{\prime}=w^{\prime}\theta_{l}^{\prime}+a_{2}\overline{\theta_{l}}w^{\prime}q_{t}^{\prime}\equiv
w^{\prime}\theta_{v}^{\prime}-a_{3}\overline{\theta_{l}}w^{\prime}q_{l}^{\prime}.$
(14)

Figure 2: Contributions of the fluxes in eqs. 14 and 15 to
$\overline{w^{\prime}\theta_{lv}^{\prime}}$, averaged over hours 6-16 of the
simulation.

The slab-average of this flux, and its contributions, are plotted in fig. 2;
this figure largely explains the effect of $\mathcal{C}$ on the layer. In one
view of eq. 14, the structure of $\overline{w^{\prime}\theta_{lv}^{\prime}}$
(maroon line) closely follows that of
$\overline{w^{\prime}\theta_{l}^{\prime}}$ (yellow line). However, one may use
eq. 5 to write

$w^{\prime}\theta_{l}^{\prime}=w^{\prime}\theta^{\prime}-\frac{L_{v}}{c_{p}\Pi}w^{\prime}q_{l}^{\prime},$
(15)

which led Betts, (1975) to recognise that it is mainly
$L_{v}/(c_{p}\Pi)\overline{w^{\prime}q_{l}^{\prime}}$ (blue line) that is
responsible for maintaining the large, downward
$\overline{w^{\prime}\theta_{l}^{\prime}}$ in the cloud layer. The other view
of eq. 14 follows Stevens, (2007): $w^{\prime}\theta_{lv}^{\prime}$ is the
buoyancy flux, minus its contributions from liquid water fluxes. By
definition, $w^{\prime}\theta_{v}^{\prime}=w^{\prime}\theta_{lv}^{\prime}$ in
the subcloud layer. But in the cloud layer, fig. 2 shows how the former is
substantially outweighed by the latter: In energetic terms, latent heating and
cooling translate more efficiently into vertical kinetic energy ($w^{\prime
2}$) than into storage of local potential energy ($\theta_{v}^{\prime}$). In
both views, the structure of $\overline{w^{\prime}\theta_{lv}^{\prime}}$ is
mainly supported by that of $\overline{w^{\prime}q_{l}^{\prime}}$. Since eq.
11 shows how $\mathcal{C}$ governs the net divergence of
$\overline{w^{\prime}q_{l}^{\prime}}$ in the cloud layer and its convergence
in the inversion layer, this discussion buttresses the classical picture of
the cumulus-topped boundary layer that we drew in the introduction: Net
condensation and the associated latent heating of the cloud layer, and the
subsequent transport to, re-evaporation in and cooling of the inversion layer,
balance the heating of the subsiding environment, and thus maintain the
structure of $\overline{\theta_{v}}$ across the layer in general and across
the trade inversion in particular.

##  4 Summary of BB17’s model for mesoscale fluctuations

We are now ready to summarise BB17’s model for the development of mesoscale
fluctuations. We will do so using only the classical theory outlined above,
and a single assumption on the horizontal buoyancy field that also turns out
to have similar consequences as we have already discussed. In section 5, we
will then move beyond BB17’s theory, to a closed-form model of the instability
and an analysis of its conditions.

###  4.1 Definitions

Following BB17, we will frame length scale growth in our fields as an increase
in magnitude of mesoscale fluctuations that develop over the slab-average. One
can identify such mesoscale fluctuations in $\chi$ by partitioning
$\chi^{\prime}$, defined in eq. 2, into a mesoscale component
$\chi^{\prime}_{m}$ and sub-mesoscale component $\chi^{\prime}_{s}$, which
gives:

$\chi=\overline{\chi}+\chi^{\prime}_{m}+\chi^{\prime}_{s}.$ (16)

BB17 scale-partition their variables by extracting horizontal averages over
blocks of 16 km $\times$ 16 km. Here, we conduct the decomposition with a
spectral low-pass filter at the horizontal wavenumber that corresponds to
scales of 12.5 km. As an example, consider fig. 3; our spectral filter
extracts the field shown in the right panel from that shown in the left panel.
Of course, any choice of method and scale for this separation is somewhat
arbitrary. Yet, since its primary objective is to distinguish mesoscale
fluctuations from fluctuations that occur on the scale of a typical cumulus
cloud, we expect that any consistently performed scale separation at a scale
that is larger than this typical cumulus scale (around 1 km), but sufficiently
smaller than our finite domain size (100 km), would suffice to illustrate what
we intend to show.

In the following, special attention will be paid to “moist, mesoscale
regions”. To define such regions, we use the density-weighted vertical average

$\langle\chi\rangle=\frac{\int_{0}^{z_{\infty}}\rho_{0}\chi
dz}{\int_{0}^{z_{\infty}}\rho_{0}dz},$ (17)

where $z_{\infty}$ refers to the domain top at 10 km. From this follows a
definition of the column-averaged, or bulk moisture $\langle q_{t}\rangle$. We
will take moist mesoscale regions to be horizontal coordinates where $\langle
q_{t_{m}}^{\prime}\rangle>0$, and dry mesoscale regions where $\langle
q_{t_{m}}^{\prime}\rangle<0$. The black contour line in the right panel of
fig. 3 gives a visual impression of this delineation.

Figure 3: Fluctuations of column-averaged total specific humidity $\langle
q_{t}\rangle$ (left), and its mesoscale-filtered component $\langle
q_{t_{m}}^{\prime}\rangle$, overlaid with a contour separating mesoscale
regions that are moister and drier than the spatial mean (right) after 24
hours.

With these definitions, we formulate a budget for $\chi_{m}^{\prime}$, by
subtracting eq. 4 from eq. 3, making several assumptions that make the
equation consistent with our LES, mesoscale-filtering the result, and
rearranging the terms (see appendix A for the derivation’s details):

$\frac{\partial\chi_{m}^{\prime}}{\partial
t}=-w_{m}^{\prime}\Gamma_{\chi}-\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}F_{\chi^{\prime}_{m}}\right)-\frac{\partial}{\partial
x_{jh}}\left(u_{jh}\chi^{\prime}\right)_{m}-\overline{w}\frac{\partial\chi_{m}^{\prime}}{\partial
z}+S_{\chi_{m}}^{\prime}.$ (18)

In this relation, the mean vertical gradient $\partial\overline{\chi}/\partial
z=\Gamma_{\chi}$ and $F_{\chi_{m}^{\prime}}$ is the anomalous mesoscale
vertical flux of $\chi^{\prime}$ away from the slab average

$F_{\chi_{m}^{\prime}}=\left(w^{\prime}\chi^{\prime}\right)_{m}-\overline{w^{\prime}\chi^{\prime}}.$
(19)

In spite of the number of steps taken to derive it, we draw attention to eq.
18’s similarity to the slab-averaged budget, eq. 4. It features the horizontal
and vertical convergence of $\chi_{m}^{\prime}$ (second and third terms on the
right-hand side of eq. 18), and the mesoscale anomalous effect of subsidence
(fourth term) and sources (fifth term). The most important difference with eq.
4 is the first term, which describes transport along the mean gradient of
$\chi$ with mesoscale vertical velocity fluctuations. This term will prove to
be central.

Many of our results will show averages of eq. 18 over moist and dry mesoscale
regions, which, because such regions are not entirely stationary, introduces
two nuances (see appendix A c). First, it allows us to remove the effects of
transport of the mesoscale fluctuations with $\overline{u_{jh}}$, and second,
it introduces a divergence term that describes the net expansion of the
regions with velocity $u_{jh}^{e}$. Denoting averages over moist or dry
regions by $\widetilde{\cdot}$, this finally gives:

$\underbrace{\frac{\partial\widetilde{\chi_{m}^{\prime}}}{\partial
t}}_{\text{Tendency}}=\underbrace{\widetilde{-w_{m}^{\prime}}\Gamma_{\chi}}_{\text{Gradient
production}}-\underbrace{\frac{\partial}{\partial
x_{j_{h}}}\widetilde{\left(u_{jh}^{\prime}\chi^{\prime}\right)_{m}}}_{\text{Horizontal
transport}}+\underbrace{\frac{\partial}{\partial
x_{jh}}\widetilde{\left(u_{jh}^{e}\chi^{\prime}\right)_{m}}}_{\text{Expansion}}\\\
-\underbrace{\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}\widetilde{F_{\chi^{\prime}_{m}}}\right)}_{\text{Vertical
transport}}-\underbrace{\overline{w}\frac{\partial\widetilde{\chi_{m}^{\prime}}}{\partial
z}}_{\text{Subsidence}}+\underbrace{\widetilde{S_{\chi_{m}}^{\prime}}}_{\text{Source}}$
(20)

To keep the text uncluttered, we will only discuss explicitly regions where
$\langle q_{t_{m}}^{\prime}\rangle>0$, since observations pertaining to such
regions are generally the opposite in dry, mesoscale regions, albeit of
different magnitude. To give the reader an impression of this asymmetry, the
figures will generally present both moist and dry profiles.

###  4.2 A sketch of the instability

The top row of fig. 4 shows how small disturbances in $\langle
q_{t}^{\prime}\rangle$ grow into significant mesoscale fluctuations over an
eight hour time window. The figure’s bottom row identifies growing clusters of
shallow cumulus clouds that develop on top of these mesoscale regions,
becoming more vigorous and reaching deeper into the inversion as they grow.
Fig. 6 a) and b) quantifies this in the form of vertical profiles. Since the
fluctuations in the temperature variables discussed in section 3 remain small
with respect to their root-mean-square (see fig. 6 d-f), this suggests that to
understand the length scale growth of our clouds, we must understand what
drives the formation of $\langle q_{t_{m}}^{\prime}\rangle$.

Figure 4: Time evolution (left to right) of $\langle
q_{t_{m}}^{\prime}\rangle$ (top row) and cloud top height (bottom row),
overlaid by contours that separate moist and dry mesoscale regions at 12 and
15 hours, and annotated with cloud fraction.

Fig. 5 offers a sketch of the explanation. Over the vertical dimension, clouds
(black contour lines) develop favourably on top of a patch of
$q_{t_{m}}^{\prime}>0$ (black, dashed contour) in the upper cloud layer. The
$q_{t_{m}}^{\prime}$ structure is produced by a mesoscale circulation of
approximately 1 cm/s (overlaid streamlines), which converges in the sub-cloud
layer beneath the structure, transports moisture upwards along the negative,
slab-mean vertical moisture gradient, and detrains it laterally near the
inversion base around 1500m, where the mesoscale vertical velocity
$w_{m}^{\prime}$ becomes negative. Fig. 6, which shows the temporal
development of mesoscale flucutations in $q_{t}$, $q_{l}$ and
$w_{m}^{\prime}$, averaged over moist and dry mesoscale regions, quantifies
these statements. The reader will recognise that we have here merely described
the action of the first term in eq. 18 and eq. 20; we will make this
connection explicit in section 44.3.

Figure 5: Cross-section over an example y-z plane of our simulation at 16
hours, coloured by filled contours of $q_{t}$ (red to blue) and overlaid by
contour lines of i) $q_{t_{m}}^{\prime}>0$ (black, dashed), ii) clouds (black,
unbroken) and iii) $\left(w^{\prime}q_{l}^{\prime}\right)_{m}$ (white to blue;
it does not coincide with the clouds because it is filtered at slab level).
Also overlaid are streamlines of the mesoscale-filtered, in-plane velocity
fluctuations (defined by $\left[u_{m}^{\prime},w_{m}^{\prime}\right]$), whose
line thickness is weighted by this velocity’s local magnitude. Horizontal,
dashed lines represent cloud base and inversion base.

The mesoscale circulations themselves arise from corresponding mesoscale
variations in the classical theory of the slab-averaged layer that we have
discussed in section 3, supplemented by a single, well-known assumption from
mesoscale tropical meteorology, namely that horizontal fluctuations in density
remain small. We observe the resulting “weak temperature gradients” in the
profiles of mesoscale buoyancy fluctuations $\theta_{v_{m}}^{\prime}$, plotted
in fig. 6 e), which do not differ appreciably between moist and dry mesoscale
regions. This allows the circulations to develop directly from mesoscale
fluctuations in condensational heating in the cloud layer, and evaporative
cooling in the inversion layer; we show this in section 44.4. With reference
to our discussion in section 3, such heating fluctuations are anticipated by
the mesoscale-filtered vertical flux of liquid water in fig. 5 (white-to-blue
contours).

The mesoscale condensation anomalies again favour regions with positive
mesoscale moisture fluctuations, which control the mesoscale relative humidity
fluctuations when the (potential) temperature fluctuations, shown in fig. 6
d)-f) are small. In all, BB17 then identify a self-reinforcing feedback:
Mesoscale fluctuations in condensation and evaporation in cumulus clouds give
rise to mesoscale circulations, which in turn enhance mesoscale moisture
fluctuations, on top of which stronger mesoscale fluctuations in condensation
and evaporation develop.

Figure 6: Time evolution of mesoscale fluctuations, averaged over moist and
dry mesoscale regions, for total specific humidity $q_{t}$ (a), liquid-water
specific humidity $q_{l}$ (b), vertical velocity $w$ (c), liquid-water
potential temperature $\theta_{l}$ (d), virtual potential temperature
$\theta_{v}$ (e) and liquid-water virtual potential temperature $\theta_{lv}$
(f). Upper axes indicate the maximum of these fluctuations relative to the
maximum root-mean-square fluctuation in each quantity at the last, plotted
time.

###  4.3 Mesoscale moisture fluctuations develop from mesoscale circulations

Fig. 7 shows the terms in eq. 20 with $\chi=q_{t}$. It identifies the main
reason for the rise of $q_{t_{m}}^{\prime}>0$ in the moist cloud layer to be
the production of $q_{t_{m}}^{\prime}$ by vertical, mesoscale transport along
the mean, negative moisture gradient $\Gamma_{q_{t}}$. We will call this
“gradient production”, in the spirit of variance-budget studies (e.g. de Roode
et al.,, 2004; Heinze et al.,, 2015; Anurose et al.,, 2020), which show that
this term, when scaled with the moisture flucutation itself, is the main
driver of moisture variance in cloud-topped boundary layers. Our gradient
production essentially quantifies a similar process as variance production,
with the exception of tracing it in the mesoscales specifically, and with the
advantage of being more easily visualised. The term follows directly from the
$w_{m}^{\prime}$ profiles plotted in fig. 6 c). These are increasingly
positive in the moist cloud layer, and increasingly negative in the moist
inversion layer, and thus accelerate the gradient production in time.

Figure 7: Vertical profiles of the terms in the $q_{t_{m}}^{\prime}$ budget
averaged over moist and dry mesoscale regions (eq. 20), and over 10-16 hr.

Fig. 7 also shows that gradient production of $q_{t_{m}}^{\prime}$ is, to a
large extent, balanced by the convergence of vertical moisture fluxes. These
fluxes transport the positive moisture fluctuation that is produced in the
moist cloud layer into the overlying inversion. Since the term’s vertical
integral is zero, it does not add or remove $\langle
q_{t_{m}}^{\prime}\rangle$ from a column; it just translates the vertical
structure of $-w_{m}^{\prime}\Gamma_{q_{t}}$ into profiles of its tendency
(black line in fig. 7). It is therefore also mainly responsible for situating
the peak of the mesoscale fluctuations in cloudiness ($q_{l_{m}}^{\prime}$,
see fig. 6 b) in the inversion layer.

Horizontal transport enters the budget through i) the mesoscale horizontal
moisture fluxes from moist to dry mesoscale regions and ii) the net region
expansion with $u_{jh}^{e}$. In the cloud layer, the region expansion (yellow
line) is negative, i.e. the mesoscale circulation’s positive cloud-layer
moisture convergence acts to concentrate the mesoscale moisture fluctuation.
In the inversion, the work done in broadening the inversion-layer moisture
fluctuation by the mesoscale, horizontal, outflows shown in fig. 5 makes the
term positive. This effect generally outweighs the drying (olive line) from
transport across the region boundary with the mesoscale flow and through
turbulent mixing of $q_{t_{m}}^{\prime}$ down the horizontal moisture
gradient, which draws $q_{t_{m}}^{\prime}$ from the mesoscales.

The subsidence term is only a small direct contributor to the budget; as we
have seen, its primary role is in setting the slab-mean environment in which
the moisture fluctuations can develop. The budget has no further sources, i.e.
in the absence of precipitation, $S_{{q_{t}}_{m}}^{\prime}=0$.

The relative importance of the gradient production and horizontal advection of
$q_{t_{m}}^{\prime}$ to the development of moist and dry mesoscale regions is
adequately captured by vertically averaging eq. 20 with $\chi=q_{t}$ using eq.
17, which gives a budget for $\widetilde{\langle q_{t_{m}}^{\prime}\rangle}$;
the time-evolution of this budget is plotted in fig. 8. It shows that the
column-averaged mesoscale moistening rate increases roughly exponentially in
moist areas, and that it is initially well-approximated by the gradient
production. Only once significant mesoscale moist patches have formed (see
fig. 4 at 12 and 15 hours) do horizontal fluxes begin substantially opposing
it. This suggests that $q_{t_{m}}^{\prime}$ development is initially
production-driven, while the horizontal structure and mature development of
the fluctuations also depend on the efficiency with which horizontal transport
can redistribute them.

The net expansion’s column average is small, but slightly negative in moist
areas, i.e. not only are large, moist areas becoming moister, they are also
becoming slightly smaller. We will briefly discuss this clustering tendency
and its significance in section 6.

Figure 8: Time-evolution of the $\langle q_{t_{m}}^{\prime}\rangle$ budget
(eq. 17 applied to eq. 20), averaged over moist and dry mesoscale regions,
between 6-16 hr, in the spirit of fig.13 of Bretherton and Blossey, (2017).
The dashed line plots the linear instability eq. 30b.

###  4.4 Mesoscale circulations develop from anomalous latent heating in
clouds

####  4.4.1 Weak temperature gradients

To understand why moist mesoscale regions grow, we must deduce the source of
$w_{m}^{\prime}$ in the gradient production of $q_{t_{m}}^{\prime}$. In other
words: Why does the mesoscale circulation shown in fig. 5 develop? BB17 argue
that this is best understood through a Weak Temperature Gradient (WTG)
framework (e.g. Held and Hoskins,, 1985; Sobel et al.,, 2001), which has
proven useful in explaining self-organised, circulation-driven scale growth in
the moisture fields of tropical atmospheres in radiative-convective
equilibrium (e.g. Emanuel et al.,, 2014; Chikira,, 2014; Beucler et al.,,
2018; Ahmed and Neelin,, 2019).

While deep convective clouds are the most spectacular example, any sort of
convection in a stably stratified fluid generates density fluctuations which
gravity waves continually redistribute horizontally, also the shallow cumuli
under consideration here. Since these waves travel at a characteristic speed
much higher than that with which advection can transport mixed scalars such as
moisture, they may prevent buoyancy fluctuations from accumulating over the
time scale with which $q_{t_{m}}^{\prime}$ grows, i.e. the mesoscale buoyancy
fluctuations remain small. A WTG interpretation of the governing equations
then allows using the mesoscale buoyancy budget to diagnose $w_{m}^{\prime}$.

To understand this, we first mesoscale filter our definition for buoyancy (eq.
8):

$\theta_{v_{m}}^{\prime}=\theta_{l_{m}}^{\prime}+a_{2}\overline{\theta_{l}}q_{t_{m}}^{\prime}+a_{3}\overline{\theta_{l}}q_{l_{m}}^{\prime}$
(21)

From eq. 21, it is not immediately obvious that $\theta_{v_{m}}^{\prime}$
should be small. For instance, if $\theta_{l_{m}}^{\prime}\approx 0$,
$\theta_{v_{m}}^{\prime}\approx\overline{\theta_{l}}\left(a_{2}q_{t_{m}}^{\prime}+a_{3}q_{l_{m}}^{\prime}\right)$.
So, upon inspecting fig. 6 a) and b), one may expect to find mesoscale
buoyancy fluctuations that correlate to the moisture and liquid water
fluctuations. This turns out to be a very good approximation in layers of
relatively continuous cloud cover, such as closed cell convection (de Roode et
al.,, 2004; de Roode and Los,, 2008), in which thermals hardly penetrate the
stable layer above the mixed layer and thus give gravity waves much less of a
chance to redistribute their buoyancy. In such situations, significant
$\theta_{v_{m}}^{\prime}$ are observed to develop in the cloud layer, which
may contribute directly to the development of $w_{m}^{\prime}$ through the
mesoscale-filtered vertical momentum equation.

Figure 9: Vertical profiles of $\theta_{v_{m}}^{\prime}$ (eq. 8) and
$\theta_{lv_{m}}^{\prime}$ (eq. 12), and their contributions from
$\theta_{l_{m}}^{\prime}$, $q_{t_{m}}^{\prime}$ and $q_{l_{m}}^{\prime}$,
averaged over moist and dry mesoscale regions (left, right), over 10-16 hr
(dark lines) and over every hour between 10-16 hr (transparent lines, in order
of increasing opacity).

However, our broken cumulus layer demands a different view: Fig. 9, which
plots the contributions to eq. 21 (opaque lines) and their time evolution
(increasingly dark, transparent lines), indicates
$\widetilde{\theta_{l_{m}}^{\prime}}\neq 0$ and
$a_{2}\overline{\theta_{l}}\widetilde{q_{t_{m}}^{\prime}}\gg
a_{3}\overline{\theta_{l}}\widetilde{q_{l_{m}}^{\prime}}$. Instead of
remaining small, $\theta_{l_{m}}^{\prime}$ becomes increasingly negative in
moist mesoscale regions, while
$a_{3}\overline{\theta_{l}}\widetilde{q_{l_{m}}^{\prime}}$ remains almost
negligible. The result is an approximate balance between the
$\theta_{l_{m}}^{\prime}$ and $q_{t_{m}}^{\prime}$ contributions in eq. 21,
and a comparatively small $\widetilde{\theta_{v_{m}}^{\prime}}$ (black line in
fig. 9) with little temporal development. Hence, if we differentiate eq. 21 to
time, we may write the mesoscale-fluctuation equivalent of eq. 10, and
recognise that it is stationary, akin to its slab-averaged counterpart being
steady to maintain a stable trade-inversion:

$\frac{\partial\theta_{v_{m}}^{\prime}}{\partial
t}\approx\frac{\partial\theta_{l_{m}}^{\prime}}{\partial
t}+a_{2}\overline{\theta_{l}}\frac{\partial q_{t_{m}}^{\prime}}{\partial
t}+a_{3}\overline{\theta_{l}}\frac{\partial q_{l_{m}}^{\prime}}{\partial
t}\approx 0.$ (22)

Eq. 22 is our statement of the WTG approximation. Before using it, we pause to
employ the observation that the temporal development of
$a_{3}\overline{\theta_{l}}\widetilde{q_{l_{m}}^{\prime}}$ does not seem to
appreciably influence $\widetilde{\theta_{v_{m}}^{\prime}}$ (fig. 9). Indeed,
we may approximate eq. 18 for $q_{l}$ as a slightly more stringent version of
eq. 11:

$\frac{\partial q_{l_{m}}^{\prime}}{\partial
t}\approx-\frac{\partial}{\partial
z}\left(F_{q_{l_{m}}^{\prime}}\right)+\mathcal{C}_{m}^{\prime}\approx 0.$ (23)

Hence, in another parallel of the slab-mean theory, mesoscale anomalies in the
rate of net condensation $\mathcal{C}$ are approximately balanced by anomalies
in their vertical transport, represented by the divergence of
$F_{q_{l_{m}}^{\prime}}$, making storage of $q_{l_{m}}^{\prime}$ passive.
These assumptions are confirmed by fig. 10.

The upshot of this discussion is that even though we in section 3 posed
budgets for $\theta_{v}$ to analyse the stability of the trade-wind layer’s
slab-averaged structure to vertical growth, while we here pose it to analyse
the growth of horizontal fluctuations, the consequences of applying eq. 23 are
similar: Eq. 21 again reduces to a budget for $\theta_{lv}$, only here for its
mesoscale fluctuations. Because eq. 23 holds, $\theta_{lv_{m}}^{\prime}$
satisfies the WTG approximation as well as $\theta_{v_{m}}^{\prime}$. In
adopting it, we again follow BB17, who perform their analysis in terms of
liquid virtual static energy, the energy-equivalent to $\theta_{lv}$.

Figure 10: Vertical profiles of the terms in the $q_{l_{m}}^{\prime}$ budget
averaged over moist and dry mesoscale regions (eq. 20, which approximately
reduces to eq. 23), and over 10-16 hr.

####  4.4.2 Modelling $w^{\prime}_{m}$

The budget for $\theta_{lv_{m}}^{\prime}$ follows from inserting the (scaled)
tendencies of $\theta_{l_{m}}^{\prime}$ and $q_{t_{m}}^{\prime}$, as written
in eq. 22, into eq. 18. Subsuming transport and (diabatic) sources of
$\theta_{lv_{m}}^{\prime}$ under $S_{\theta_{lv_{m}}}^{\prime}$ then gives the
WTG formulation on a commonly written form:

$\frac{\partial\theta_{lv_{m}}^{\prime}}{\partial
t}=-w_{m}^{\prime}\frac{\partial\overline{\theta_{lv}}}{\partial
z}+S_{\theta_{{lv}_{m}}}^{\prime}\approx 0.$ (24)

Figure 11: Vertical profiles of the terms in the $\theta_{lv_{m}}^{\prime}$
budget (eq. 20, which approximately reduces to eq. 24), averaged over moist
and dry mesoscale regions, and over 10-16 hr.

Fig. 11 plots all terms in this budget. It reveals that the gradient
production is primarily balanced by anomalous, vertical mesoscale convergence
of $\theta_{lv}^{\prime}$ fluxes ($F_{{\theta_{lv}^{\prime}}_{m}}$); both
terms are at least an order of magnitude larger than
$\theta_{lv_{m}}^{\prime}$’s tendency, horizontal advection and subsidence
heating. Hence, the figure invites us to rewrite eq. 24 as a diagnostic
equation for $w_{m}^{\prime}$, which can be understood as the vertical
velocity needed to move air parcels heated by $S_{\theta_{lv_{m}}}^{\prime}$
quasi-statically to their level of neutral buoyancy under a stable
stratification (Klein,, 2010). In our non-precipitating simulations with
homogenoud, imposed radiation, the anomalous heat source
$S_{\theta_{lv_{m}}^{\prime}}$ is reduced to the vertical convergence of
$F_{{\theta_{lv}^{\prime}}_{m}}$. Hence, the model reads:

$w_{m}^{\prime}\approx-\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}F_{{\theta_{lv}^{\prime}}_{m}}\right)\frac{1}{\Gamma_{\theta_{lv}}}.$
(25)

Fig. 12 a) and b) confirm the accuracy of this approximation, except in the
sub-cloud layer where the gradients we divide through are too small for our
simulations to resolve.

Figure 12: $w_{m}^{\prime}$ derived directly from the LES model (black line)
and modelled by eq. 25 (maroon line), averaged over moist and dry regions and
over 10-16 hr. Note also the comparatively small error made if
$F_{{\theta_{l}^{\prime}}_{m}}$ or
$a_{3}\overline{\theta_{l}}\mathcal{C}_{m}^{\prime}$ is used instead of
$F_{{\theta_{lv}^{\prime}}_{m}}$ in eq. 25 (yellow and sea green lines).

####  4.4.3 The role of condensation

What governs the vertical convergence of $F_{{\theta_{lv}^{\prime}}_{m}}$?
Fig. 11 shows that it is positive in the moist region’s cloud layer, and
negative in the corresponding inversion layer. To explain this, we will for a
final time return to our discussion from section 3.

Fig. 13 plots the mesoscale-filtered and slab-averaged contributions to
$w^{\prime}\theta_{lv}^{\prime}$, obtained from mesoscale filtering eq. 14,
and compares it to the slab-averaged fluxes previously presented in fig. 2.
The vertical structure of the fluxes in the cloud layer is qualitatively
similar when averaging them over either the entire slab, the moist mesoscale
regions, or the dry mesoscale regions; they only differ in their magnitude.
Therefore, just like we found $\overline{w^{\prime}\theta_{lv}^{\prime}}$ to
be supported by $\overline{w^{\prime}q_{l}^{\prime}}$, we recognise that
$\left(w^{\prime}\theta_{lv}\right)_{m}$ is supported by
$\left(w^{\prime}q_{l}^{\prime}\right)_{m}$, when averaged over moist and dry
regions. This is possible in spite of
$\theta_{lv_{m}}^{\prime}\approx\theta_{v_{m}}^{\prime}$ throughout the
boundary layer (fig. 9), because eq. 23 shows that mesoscale condensation
anomalies do not accumulate as $q_{l}^{\prime}$, but instead give rise to an
anomalous divergence of liquid water flux in the moist cloud layer. The
contribution from $q_{l_{m}}^{\prime}$ to $\theta_{lv_{m}}^{\prime}$ (eq. 21)
is small; the contribution from $\left(w^{\prime}q_{l}^{\prime}\right)_{m}$ to
$\left(w^{\prime}\theta_{lv}^{\prime}\right)_{m}$ (filtered eq. 14) is not.

Figure 13: Grid-resolved, vertical mesoscale-filtered fluxes
$\left(w^{\prime}\theta_{v}^{\prime}\right)_{m}$ and
$\left(w^{\prime}\theta_{lv}^{\prime}\right)_{m}$ and their contributions from
$\left(w^{\prime}\theta_{l}^{\prime}\right)_{m}$,
$\left(w^{\prime}q_{t}^{\prime}\right)_{m}$ and
$\left(w^{\prime}q_{l}^{\prime}\right)_{m}$ as defined in eq. 14, averaged
over moist and dry mesoscale regions (unbroken lines) and over the entire
horizontal slab (dashed lines, copied from fig. 2), over 10-16 hr. The region-
averaged flux anomaly $\widetilde{F_{\chi_{m}^{\prime}}}$ (eq. 19) is the
difference between the unbroken and dashed lines for each variable.

In moist regions, fig. 13 shows that $F_{{q_{l}^{\prime}}_{m}}>0$. Using eq.
23, we recognise this to result directly from $\mathcal{C}_{m}^{\prime}>0$,
which consequently explains most of the moist cloud layer’s anomalous heating,
i.e. why $-\partial F_{{\theta_{lv}^{\prime}}_{m}}/\partial z>0$ in the left
panel of fig. 11. Eq. 25 then shows that this heating is immediately
compensated by mesoscale ascent along the mean stratification, as mandated by
the WTG model. As a result of the slab mean-exceeding liquid water transport
from the layer below, the moist inversion layer must also evaporate more
liquid water than the slab average, i.e. $\mathcal{C}_{m}^{\prime}<0$.
Consequently, the moist inversion layer experiences an anomalous convergence
of $\left(w^{\prime}q_{l}^{\prime}\right)_{m}$ and an associated anomalous
evaporative cooling, i.e. $-\partial F_{{\theta_{lv}^{\prime}}_{m}}/\partial
z<0$ in fig. 11. Just like in the cloud layer below, this cooling is quickly
balanced by a negative $w_{m}^{\prime}$.

Hence, we have arrived at the heart of the mechanism: Horizontal, mesoscale
anomalies in the same vertical structure of the net condensation $\mathcal{C}$
that governs the slab-mean layer’s evolution have, under WTG, as a consequence
that they develop mesoscale vertical motion of a few centimetres per second.
To demonstrate this explicitly, $w_{m}^{\prime}$ remains accurately predicted
even when substituting $a_{3}\overline{\theta_{l}}\mathcal{C}_{m}^{\prime}$
for $F_{{\theta_{lv}^{\prime}}_{m}}$ in eq. 25 (green lines in fig. 12).

Note that if we proceed along similar lines as above using Betts, (1973)’s
original view of the slab-averaged problem, i.e. using mesoscale anomalies in
$w^{\prime}\theta_{l}^{\prime}$ instead of $w^{\prime}\theta_{lv}^{\prime}$ to
explain $w_{m}^{\prime}$, the analysis remains largely unchanged, because,
following the discussion in section 3, $w^{\prime}\theta_{l}^{\prime}$ too is
well-known to be chiefly governed by $\mathcal{C}$. Since this view ignores
virtual effects on the evolution of $\theta_{v_{m}}^{\prime}$ (see eq. 21), it
is slightly less accurate, but remains adequate for predicting
$w_{m}^{\prime}$ (yellow lines in fig. 12). We emphasise that also in this
view, $w_{m}^{\prime}$ remains rooted in latent heating.

####  4.4.4 BB17’s model

To complete the argument, one may multiply eq. 25 with the negative, mean
moisture gradient to finally arrive at a model for the onset of
$q_{t_{m}}^{\prime}$, formulated in terms of anomalous heat fluxes and the
ratio of mean flow gradients

$\frac{\partial q_{t_{m}}^{\prime}}{\partial t}\sim-
w_{m}^{\prime}\Gamma_{q_{t}}\approx\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}F_{{\theta_{lv}^{\prime}}_{m}}\right)\frac{\Gamma_{q_{t}}}{\Gamma_{\theta_{lv}}},$
(26)

which fig. 14 show is also accurate, and mostly captured even when replacing
$F_{{\theta_{lv}^{\prime}}_{m}}$ by
$a_{3}\overline{a_{3}}\mathcal{C}_{m}^{\prime}$ or
$F_{{\theta_{l}^{\prime}}_{m}}$. Eq. 26 is a succinct summary of the model
presented by BB17, combining the significant terms in their eq.20 and eq.32.

Figure 14: $w_{m}^{\prime}\Gamma_{q_{t}}$ derived directly from the LES model
(ground truth) following eq. 26 (maroon line), averaged over moist and dry
regions and over 10-16 hr. Also shown are results if
$F_{{\theta_{l}^{\prime}}_{m}}$ or
$a_{3}\overline{\theta_{l}}\mathcal{C}_{m}^{\prime}$ is used instead of
$F_{{\theta_{lv}^{\prime}}_{m}}$ eq. 26 (yellow and sea green lines).

##  5 Bulk model for the instability

As BB17 note, eq. 30a can in the language of tropical meteorology be
understood as negative gross moist stability of moisture fluctuations (Neelin
and Held,, 1987; Raymond et al.,, 2009). Because of the absence of horizontal
heterogeneity in radiation and precipitation in our simulations, we can here
simplify the instability a little further than studies of deep convection
typically do. In particular, we will close a simple, linear bulk instability
model for the development of the moisture fluctuations, and examine the
conditions of this model in some detail.

###  5.1 Linear instability model

To close a positive feedback loop driving the development of
$q_{t_{m}}^{\prime}$, such fluctuations must lead directly to
$F_{{\theta_{lv}^{\prime}}_{m}}$, because vertical gradients in
$F_{{\theta_{lv}^{\prime}}_{m}}$ lead to mesoscale moistening anomalies
following eq. 26. Since $\mathcal{C}_{m}^{\prime}$ supports
$F_{{\theta_{lv}^{\prime}}_{m}}$, an intuitive basis for this closure is to
assume that a moister cloud layer is all that is needed for cumuli growing
into it to condense more water. Fig. 5 confirms that cloud-layer
$q_{t_{m}}^{\prime}$ and $\mathcal{C}_{m}^{\prime}$ are well collocated. BB17
sketch a similar picture. However, a mathematical, theoretically founded
description of the closure is still missing. Therefore, we suggest one here.

Our model will describe the time-evolution of bulk mesoscale moisture
fluctuations, $\langle q_{t_{m}}^{\prime}\rangle$. Vertical, partial
integration of eq. 26 using eq. 17 gives:

$\frac{\partial\langle q_{t_{m}}^{\prime}\rangle}{\partial
t}\approx-\left.\frac{\rho_{0}}{\int_{0}^{z_{\infty}}\rho_{0}dz}F_{{\theta_{lv}^{\prime}}_{m}}\frac{\Gamma_{q_{t}}}{\Gamma_{\theta_{lv}}}\right|^{z_{\infty}}_{0}-\left\langle
F_{{\theta_{lv}^{\prime}}_{m}}\frac{\partial}{\partial
z}\left(\frac{\Gamma_{q_{t}}}{\Gamma_{\theta_{lv}}}\right)\right\rangle.$ (27)

$F_{{\theta_{lv}^{\prime}}_{m}}=0$ above the cloud layer and at the surface,
courtesy of our lower boundary condition, setting the first term to zero. If
we additionally assume that $\partial/\partial
z\left(\Gamma_{q_{t}}/\Gamma_{\theta_{lv}}\right)$ is approximately constant
with height, we may move it outside the integral:

$\frac{\partial\langle q_{t_{m}}^{\prime}\rangle}{\partial
t}\approx-\frac{\partial}{\partial
z}\left(\frac{\Gamma_{q_{t}}}{\Gamma_{\theta_{lv}}}\right)\left\langle
F_{{\theta_{lv}^{\prime}}_{m}}\right\rangle.$ (28)

This assumption is not entirely accurate, but sufficiently reasonable through
the cloud layer (not shown) that it is worth making in the interest of showing
that bulk moistening of moist, mesoscale regions is governed by the integrated
heat flux anomaly. Any parameterisation that relates such heat fluxes to
$q_{t_{m}}^{\prime}$ suffices to close the model, and many such
parameterisations can be imagined. In the spirit of Betts, (1975), we will
merely use a simplified mass-flux approximation:

$\displaystyle F_{{\theta_{lv}^{\prime}}_{m}}$ $\displaystyle\approx-
a_{3}\overline{\theta_{l}}F_{q_{l_{m}}^{\prime}}$ (29a) $\displaystyle\approx-
C_{1}a_{3}\overline{\theta_{l}}w^{*}q_{l_{m}}^{\prime}$ (29b)
$\displaystyle\approx-C\overline{\theta_{l}}w^{*}q_{t_{m}}^{\prime}.$ (29c)

To write eq. 29b, we take $w^{*}$ to be a single vertical velocity averaged
over all cloudy cells in both space and time, which ignores entrainment and
detrainment effects, and which is less accurate than using cloud core
variables (Siebesma and Cuijpers,, 1995). However, it allows us to relate
mesoscale fluctuations in in-cloud liquid water directly to
$q_{l_{m}}^{\prime}$ only. Since these assumptions yield errors only in the
flux anomaly’s magnitude, but not in the shape of its vertical profile
(Siebesma and Cuijpers,, 1995), we correct them with a second constant
$C_{1}$. To write eq. 29c, we also assume $q_{l_{m}}^{\prime}\propto
q_{t_{m}}^{\prime}$, and that equality can be restored by a single model
constant $C$ which subsumes $C_{1}$ and $a_{3}$. This amounts to assuming that
mesoscale fluctuations in the saturation specific humidity are small.

In spite of all these assumptions, we consider eq. 29c with $C=0.3$ adequate
for the present discussion, c.f. fig. 15 a). Inserting this relation in eq. 28
allows framing the growth of column-averaged mesoscale moisture fluctuations
as a linear instability problem, whose time scale is
$\tau_{q_{t_{m}}^{\prime}}$:

$\frac{\partial\langle q_{t_{m}}^{\prime}\rangle}{\partial
t}\approx\frac{\left\langle
q_{t_{m}^{\prime}}\right\rangle}{\tau_{q_{t_{m}}^{\prime}}},$ (30a)
$\tau_{q_{t_{m}}^{\prime}}=\frac{1}{C\overline{\theta_{l}}w^{*}\frac{\partial}{\partial
z}\left(\frac{\Gamma_{q_{t}}}{\Gamma_{\theta_{lv}}}\right)}.$ (30b)

Eq. 30b remains rather accurate (fig. 15 b), diagnosing a time scale for the
instability of almost 4 hours in our simulation. The model is also plotted in
fig. 8.

Figure 15: Scatterplots of moist and dry region averaged a)
$q_{t_{m}}^{\prime}$ against $-F_{{\theta_{lv}^{\prime}}_{m}}$ and b) $\langle
q_{t_{m}}^{\prime}\rangle$ against $\partial\langle
q_{t_{m}}^{\prime}\rangle/\partial t$, for all fields between 6-16 hours
(dots). Lines follow eq. 29, with constants $w^{*}=0.52$ m/s and $C=0.3$
(panel a), and eq. 30a, with $\partial/\partial
z\left(\Gamma_{q_{t}}/\Gamma_{\theta_{lv}}\right)=1.5\cdot 10^{-3}$ g/kg/K/m
(panel b). The constants are derived from the LES as described in the text.
The time scale implied by the line in panel b (eq. 30b) is annotated.

While illustrative, it is prudent to ask if eq. 29, upon which this time scale
estimate rests, is reliable. Since it depends heavily on $w^{*}$, which is not
well-constrained by any argument we have made, but is energetically supported
by in-cloud turbulence at the very smallest scales our numerical model
resolves, this is in fact quite questionable. BB17, who estimate the time
scale without reference to a model for it, obtain a significantly larger
number (15 hours) than we do, suggesting that the mechanism may exhibit a
strong numerical dependence, as is observed for models of self-aggregating
deep convection (e.g. Muller and Held,, 2012; Wing et al.,, 2020). We devote a
separate manuscript to this issue.

###  5.2 Condition for instability

If the assumptions made in deriving eq. 30b are generally valid, its only
condition for unabated growth of $q_{t_{m}}^{\prime}$ is that
$\partial/\partial_{z}\left(\Gamma_{q_{t}}/\Gamma_{\theta_{lv}}\right)>0$.
BB17 arrive at the same condition, but formulate it as a demand that a convex
relation must exist between $\overline{q_{t}}(z)$ and
$\overline{\theta_{lv}}(z)$ (or another pair of conserved thermodynamic
variables). This follows from rewriting

$\frac{\partial}{\partial
z}\left(\frac{\Gamma_{q_{t}}}{\Gamma_{\theta_{lv}}}\right)=\Gamma_{\theta_{lv}}\frac{\partial^{2}\overline{q_{t}}}{\partial\overline{\theta_{lv}}^{2}}$
(31)

with the quotient rule of calculus. Hence, a stable stratification
($\Gamma_{\theta_{lv}}>0$) and
$\partial^{2}\overline{q_{t}}/\partial\overline{\theta_{lv}}^{2}>0$ are needed
to translate $F_{{\theta_{lv}^{\prime}}_{m}}$ into a positive, column-averaged
gradient production of $q_{t_{m}}^{\prime}$. This requirement arises because
the divergence of $F_{{\theta_{lv}^{\prime}}_{m}}$ itself integrates to zero.
Were there a linear, mixing-line relation between $\overline{\theta_{lv}}$ and
$\overline{q_{t}}$, the gradient production of $q_{t_{m}}^{\prime}$ would
integrate to zero also. In contrast, a concave relation between
$\overline{\theta_{lv}}$ and $\overline{q_{t}}$ would result in a self-
stabilising feedback, where heating fluctuations would lead to column-drying.

Our simulations’ initial profiles of $\overline{q_{t}}$ and
$\overline{\theta_{lv}}$ are straight lines, i.e. initially
$\partial^{2}\overline{q_{t}}/\partial\overline{\theta_{lv}}^{2}=0$. Any
convexity in this relation must consequently develop in time. BB17 suggest
that the convexity may be brought about by large-scale cooling and advection.
However, by deriving an evolution equation for
$\partial^{2}\overline{q_{t}}/\partial\overline{\theta_{lv}}^{2}$, one can
show that in LES simulations where the thermodynamic state, subsidence and
large-scale forcing profiles are initially at most linear functions of height,
these processes cannot induce
$\partial^{2}\overline{q_{t}}/\partial\overline{\theta_{lv}}^{2}>0$ (we
present the full derivation in appendix B). Instead, it must develop from
slab-averaged fluxes of $\theta_{lv}$ and $q_{t}$, whose vertical structure
must satisfy

$\frac{1}{\rho_{0}}\frac{\partial^{3}}{\partial
z^{3}}\left(\rho_{0}\overline{w^{\prime}q_{t}^{\prime}}\right)<0$ (32a)
$\frac{1}{\rho_{0}}\frac{\partial^{3}}{\partial
z^{3}}\left(\rho_{0}\overline{w^{\prime}\theta_{lv}^{\prime}}\right)<0$ (32b)

when $\Gamma_{\theta_{lv}}>0$ and $\Gamma_{q_{t}}<0$.

Once $\overline{\theta_{lv}}$ and $\overline{q_{t}}$ are no longer linear
functions of height, these constraints can be relaxed to include effects from
linear profiles of subsidence and large-scale forcing, and curved profiles of
the heat and moisture fluxes. However, since BOMEX features large-scale
forcings in the cloud layer that are constant in height, these processes are
prohibited from having any effect on the development of convexity in our cloud
layer’s mean state. Hence, in contrast to what BB17 suggest, we find that the
inception of
$\partial^{2}\overline{q_{t}}/\partial\overline{\theta_{lv}}^{2}>0$ is not
imposed by large-scale forcing, but is an internal feature of our simulated,
slab-averaged cumulus convection’s vertical structure.

In our simulations, the initially linear profiles of $\overline{q_{t}}$ and
$\overline{\theta_{lv}}$ acquire a convex relation in the lower cloud layer
and in the inversion (fig. 16, hatched regions in fig. 17 a and b). This
structure is generated by slab-mean heating and drying in the lower cloud
layer, and moistening and cooling in the inversion layer (dashed lines in fig.
16); these tendencies can in turn largely be attributed to slab-mean flux
convergence of heat and moisture (fig. 17 c and d, see also figs. 1 a and b).
The hatched regions in fig. 17 c and d show that their vertical structure
satisfies eq. 32b precisely where
$\partial^{2}\overline{q_{t}}/\partial\overline{\theta_{lv}}^{2}>0$, as
expected.

Figure 16: Development of the vertical relation between
$\overline{\theta_{lv}}$ and $\overline{q_{t}}$ over four time samples,
superimposed by their tendency at the last point in time, at several
altitudes. Circles indicate the inversion base, squares the cloud base, and
triangles cloup top, as they vary with time. Figure 17: Time-evolution of
vertical profiles of $\overline{\theta_{lv}}$ (a) and $\overline{q_{t}}$ (b),
$\frac{\partial}{\partial
z}\left(\overline{w^{\prime}\theta_{lv}^{\prime}}\right)$ (c) and
$\frac{\partial}{\partial z}\left(\overline{w^{\prime}q_{t}^{\prime}}\right)$
(d), and vertical distribution of cloud-top height throughout the simulation
(e). Hatches indicate areas where
$\partial^{2}\overline{q_{t}}/\partial\overline{\theta_{lv}}^{2}>0$ (a and b),
$\partial^{3}\overline{w^{\prime}\theta_{lv}^{\prime}}/\partial z^{3}<0$ (c)
and $\partial^{3}\overline{w^{\prime}q_{t}^{\prime}}/\partial z^{3}<0$ (d) at
$t=16$ hr. Arrows denote the action of condensation and evaporation in very
shallow clouds over the transition layer.

A rather elegant explanation for this fact is suggested by Albright et al.,
2022b . In observations, the cloud-top height distribution features two modes
surrounded by a rather wide spread: One due to very shallow cumuli between
500m-1000m and one due to deeper clouds, which reach the inversion layer
between 1600m and 2000m. Our simulations exhibit the same bimodality (fig. 17
e)), with the modes coinciding precisely with the layers where
$\partial^{2}\overline{q_{t}}/\partial{\overline{\theta_{lv}}}>0$.

The first mode spans the so-called transition layer, which we loosely
interpret as the layer where $\overline{q_{t}}$ is no-longer well-mixed, while
$\overline{\theta_{lv}}$ is. Observations both old (Augstein et al.,, 1973)
and very recent (Albright et al., 2022a, ) indicate that thick, curved
transition layers are omnipresent in the trades. Albright et al., 2022b
suggest that such a structure may be brought about by the population of very
shallow clouds that inhabit this layer (see arrows in fig. 17): At cloud base,
they warm and moisten in accordance with deeper clouds. Yet, they quickly
evaporate (around 700m), yielding a cooling and moistening that is analogous
to how deeper clouds cool and moisten the inversion, and that is visible in
the flux convergence profiles. Once the very shallow clouds dissipate, the
positive net condensation in the remaining, deeper clouds returns the heating
and moistening the layer. This is what brings about the structure in the heat
and moisture fluxes that ensure
$\partial^{2}\overline{q_{t}}/\partial{\overline{\theta_{lv}}}>0$. Similarly,
the thickness and curvature of the inversion layer is brought about by
variability in how deep into that layer individual, deeper cumuli reach, and
thus where they evaporate, cool and moisten the layer.

Demanding curvature of the flux convergence profiles takes us two orders
beyond quasi-steady assumptions that sometimes feature in simple models of
cumulus layers (Stevens,, 2007); these require the slab-mean flux divergence
to be constant with height. Yet, above our simulation’s well-mixed layer,
gradients and curvature in the flux divergence are the norm, rather than the
exception, especially near cloud base and the inversion. In fact, the
resulting, subtle deviations from linearity in the vertical structure of
$\overline{\theta_{l}}$ and $\overline{q_{t}}$ are a ubiquitous feature across
the canon of LESs that exhibit length scale growth in their trade-wind cumulus
layers, e.g. the original Rain in Cumulus over Ocean (RICO) ensemble (van
Zanten et al.,, 2011) and its derivatives (e.g. Seifert et al.,, 2015; Anurose
et al.,, 2020), the idealised framework developed by Bellon and Stevens,
(2012) and later used for studies of cloud organisation by Vogel et al., (e.g.
2016), the simulations of length scale growth presented by Narenpitak et al.,
(2021) and even the simulations (Blossey et al.,, 2013) that BB17 develop
their theory upon, in spite of all these cases including much more detailed
physics than we do here. The condition eq. 32a is satisfied even for the moist
static energy fluxes diagnosed over the undisturbed BOMEX period by Nitta and
Esbensen, (1974); their satellite images also indicate that large cloud
structures developed even during the undisturbed period so long associated
only with small, stable cumuli.

In all, we find rather convincing evidence that the conditions required to
destabilise mesoscale moisture fluctuations are usually satisfied, and that it
is not necessary to resort to explanations involving the larger-scale forcing
on the system. In making this statement, we are essentially arguing for the
more general validity of BB17’s results: Shallow cumulus convection is
intrinsically unstable to length scale growth.

##  6 Discussion and outlook

Before summarising, let us review three consequences of this rather striking
conclusion.

###  6.1 Relevance of circulation-driven scale growth

To what extent does condensation-driven scale growth matter in nature? To lay
bare the essence of the mechanism, we have here made a number of
simplifications that are probably too restrictive for us to speak
authoritatively on this matter. In particular, our assumptions that i) the
surface fluxes and large-scale advection of heat and moisture are fixed in
space and time, ii) precipitation and interactive radiation do not play a role
and iii) the mean environment is rather stationary, imply that we ignore
several important processes that in nature will modulate the instability we
discuss. Since the relative effects of such processes in patterning the trades
is a topic of active research, we briefly discuss some anticipated
consequences of these assumptions here.

First, if $q_{t_{m}}^{\prime}$ reach the surface, eq. 27 shows that surface
flux anomalies could potentially oppose further moistening. At constant sea
surface temperatures and with mild precipitation, BB17 show such effects to be
of second order importance, indicating that the well-mixed layer may remain
quasi-stationary and tied to the local surface conditions at large spatial
scales, as is often assumed in models of deep convective self-organisation
(e.g. Emanuel et al.,, 2014). This matters, since sea-surface temperature
influences the vigour of the convection (measured here through $w^{*}$) and
thus, through eq. 30b, can explain e.g. the more rapid production of mesoscale
circulations and moist patches over a warmer ocean simulated by Vogel et al.,
(2016). Extensions to also study sea-surface temperature heterogeneities,
which amplify circulations (Park et al.,, 2006), are easily imagined.

Heterogeneous radiation can support shallow circulations in detailed (Klinger
et al.,, 2017) and conceptual (Naumann et al.,, 2019) simulations of cumulus-
topped boundary layers. In particular, when the circulations are sufficiently
strong to begin detraining significant amounts of inversion cloud atop the
moist region’s boundary layer, they reinforce the anomalous heating that here
drives the circulations (; ); these effects would accelerate the mechanism
beyond the time-scale eq. 30b.

If the shallow cumulus layer deepens sufficiently for precipitation to form,
we must further amend our estimates. Slab-averaged precipitation will on one
hand reinforce $\Gamma_{q_{t}}<0$ in the cloud layer, but on the other reduce
the inversion-layer evaporation upon which $F_{{\theta_{lv}}_{m}}$ relies
(Albrecht,, 1993); the relative effects of these factors seem to enhance
length scale growth in the simulations conducted by BB17. However,
precipitation that is sufficiently vigorous to produce cold pools will locally
discourage the formation of convection upon which our simulated circulations
rely, and may thus relieve gradient production driven by cloudy updrafts as
the leading-order dynamics that govern the spatial distribution of moisture
and clouds; such transitions seem to take place in mesoscale LES of RICO
(Seifert and Heus,, 2013; Seifert et al.,, 2015; Anurose et al.,, 2020; Thomas
et al.,, 2021). When cold pools dominate, their length scale replaces that of
the clouds as the descriptor of the mesoscales (Matheou and Lamaakel,, 2021),
and the length scales of cloud-free areas emerge as the natural complement to
those of the clouds themselves as measures of the resulting cloud organisation
(Janssens et al.,, 2021; Schulz et al.,, 2021). However, a broad regime of
conditions can be imagined where both circulations and cold pools act in
concert to pattern the convection, e.g. under strong inversions and high
surface fluxes, as for the patterns labelled Flowers and Fish in Schulz et
al., (2021), or through an imposed large-scale ascent which may vary from
being upwards and later downwards (Narenpitak et al.,, 2021). Such situations
deserve more study.

Finally, we note that recent observations (George et al., 2021a, ; George et
al., 2021b, ) suggest that mesoscale circulations with a similar magnitude and
vertical structure as we find in our simulations pervade the trades. Also, we
note that any process which gives rise to mesoscale circulations will be
amplified by the mechanism discussed here. Hence, while more research is
needed to explicitly root these observations in the dynamics described here
and by BB17, the mechanism should be a central hypothesis in any study
attempting to explain mesoscale variability in clouds and moisture.

###  6.2 Connection to trade-inversion growth

Application of the WTG approximation implies that the level of the trade
inversion is rather constant over our domain, in spite of large moisture
fluctuations accumulating over the layer. The level of this trade-inversion is
governed by $w^{\prime}\theta_{lv}^{\prime}$ (section 3; Stevens,, 2007),
fluctuations in which also govern scale growth. Combining these observations
gives two insights.

First, it highlights a very practical way in which mesoscale cloud
fluctuations affect the slab-averaged layer: Since
$\overline{w^{\prime}\theta_{lv}^{\prime}}$ skews towards the profiles set by
the deeper clouds in moist regions (see fig. 13), the scale growth raises the
inversion. This explains why, in contrast to Siebesma and Cuijpers, (1995)’s
small-domain simulations, our domain-averaged thermodynamics are not steady
(fig. 1). Unless balanced by an increased subsidence with height or
precipitation, scale growth and deepening of the inversion go hand in hand.

The upshot is that the self-aggregation of moisture discussed here may
influence transitions to deep convection. In fact, after around 20 hours, the
moist patches in our simulation develop deep, organised clouds, aided by the
lack of subsidence above 2 km in our simulation setup. This feedback is
similar to that described by Jeevanjee and Romps, (2013); the fundamental
instability to scale-growth of shallow convection may thus also initiate the
self-organisation of deep convection.

###  6.3 Connection to cloud feedback estimates

How does the scale growth mechanism affect cloud fraction, which to first
order govern the trades’ contribution to the equilibrium climate sensitivity,
and which remains poorly constrained in general circulation models (Zelinka et
al.,, 2020)? Fig. 4 shows the cloud fraction is remarkably robust over our 16
hours of simulation, as cloudiness increases in moist regions compensate
reductions in dry regions. The small, observed reduction can be attributed to
the developing circulation’s tendency to contract the moist regions at the
expense of the dry regions (fig. 8), an observation which is consistent with
e.g. Vogel et al., (2016).

The weak dependence of the cloud fraction on the cloud length scale is
consistent with observations (Janssens et al.,, 2021), which indicate a rather
weak relation between measures of a cloud field’s characteristic length or
aggregtion, and cloud fraction. Even if the mechanism would strengthen above
warmer sea surfaces or in more weakly subsiding environments, it would thus
likely support the emerging picture that trade-wind cloudiness is rather
insensitive to changes in the overall climate (Myers et al.,, 2021; Cesana and
Del Genio,, 2021).

However, two notes on this statement motivate further research. First, the
cloud fraction will be sensitive to the developing inversion-layer outflows’
ability to sustain extensive sheets of inversion cloud, which it does not do
in our simulations, but which is observed in other studies (Bretherton and
Blossey,, 2017; Vogel et al., 2020b, ; Narenpitak et al.,, 2021; Bony et al.,,
2020). Many situations can be imagined to feature higher inversion cloud
fractions than BOMEX, whose inversion is rather dry and warm. More systematic
study of the mechanism over different environmental conditions using more
realistic physics than we do here is warranted.

This is particularly pertinent because approaches such as those taken by Myers
et al., (2021); Cesana and Del Genio, (2021) essentially assume large-scale
cloud-controlling variables set the cloud fraction. Recent observations lend
credence to this approach, suggesting that it is (presumably externally
induced) variability in vertical velocity at larger-scales which raises the
subcloud layer and thus gives rise to stronger cloud-base mass fluxes and
cloud fractions (Bony and Stevens,, 2019; Vogel et al., 2020a, ; George et
al., 2021a, ).

Our results, however, suggest the opposite view: Here, spatial variability in
the convective mass flux, filtered and averaged over mesoscale moist and dry
regions, give rise to variability in larger-scale vertical velocity. The role
of the resulting circulation is to set the right cloud-layer thermodynamic
environment for clouds to preferentially form in, which results in the
stronger mass fluxes in regions of positive, large-scale vertical velocity. We
also do not observe the subcloud layer height to differ appreciably between
moist and dry spatial regions (see fig. 5). If the view suggested by our
simulations turns out to matter in nature, questions arise regarding the
validity of approaches such as those taken by Myers et al., (2021); Cesana and
Del Genio, (2021), because they ignore that shallow convective clouds may
simply control their own cloud-controlling variables. Reconciling the views
put forward on the basis of recent observations with ours is thus a
recommendation with substantial ramifications (Bony et al.,, 2015).
Fortunately, the data from the recent EUREC4A field campaign (Bony et al.,,
2017; Stevens et al.,, 2021) may be sufficiently detailed to begin answering
such questions, boding well of our understanding of the significance of self-
organising shallow cloud patterns to climate.

##  7 Summary and concluding remarks

Building on Bretherton and Blossey, (2017), we have formulated an idealised
model for a linear instability that leads to uninhibited length scale growth
of moisture fluctuations in layers of non-precipitating trade-wind cumulus
(eq. 30b). Using only well-established theory and a classical large-eddy
simulation setup (BOMEX) with no heterogeneous surface forcing, radiation or
precipitation, the model explains how small spatial differences in the amount
of latent heating in shallow cumulus clouds produce a mesoscale circulation
under the assumption of weak, horizontal mesoscale temperature gradients. The
circulation converges moisture into regions that consequently support more
cumulus clouds, diabatic heating and a stronger circulation; these regions
grow exponentially in intensity and scale (fig. 8) until they are modulated by
an outer length scale (here the finite size of our LES domains) or translate
the problem to a regime of different leading-order dynamics, e.g. driven by
precipitation or radiation, which we do not simulate.

Contrary to what BB17 suggest, in our idealised LES simulations the imposed
environment of a scale larger than the domain does not directly influence the
convective instability, as long as it supports a mean cumulus layer; the
instability is free to develop on top of the mean state as a function only of
turbulent fluxes of heat and moisture (eq. 32b), because they naturally give
rise to curvatures in the mean state.

In all, we conclude that shallow convection is intrinsically unstable to scale
growth, a result which is implied even by the results reported by Nitta and
Esbensen, (1974) for the “undisturbed” BOMEX period, upon which many ideas
that assume horizontal homogeneity in trade-wind cloudiness rely. Our results
should aid to disassemble the last vestiges of such ideas.

As a final remark, we note how striking it is that we have only required well-
established, classical theory for our discussion. As noted at the outset, the
structure of the mean trades was elucidated sixty years ago. WTG’s utility has
been known for forty years in the community of tropical meteorology (Held and
Hoskins,, 1985). Even the interpretation of the instability we have discussed
as negative values in moist gross stability (BB17), relates to classical,
influential concepts from tropical meteorology (Neelin and Held,, 1987). This
motivates us to conclude simply by asking: What else might we learn from the
insights of the giants of tropical meteorology when exploring the rather
uncharted territory of the mesoscale trades?

## Acknowledgments

MJ warmly acknowledges illuminating conversations with Anna Lea Albright on
the role of very shallow cumulus in shaping the transition layer. CvH
acknowledges funding from the Dutch Research Council (NWO) (grant:
VI.Vidi.192.068). A. Pier Siebesma acknowledges funding by the European
Union’s Horizon 2020 research and innovation program under grant agreement no.
820829 (CONSTRAIN project). FG acknowledges support from The Branco Weiss
Fellowship - Society in Science, administered by ETH Zürich, and from an NWO
Veni grant. Finally, we thank the Dutch Reserach Council NWO for use of its
computer facilities (project 2021/ENW/01081379).

## Open science

To ensure the reproducibility of the results, we make public the version of
DALES
(https://doi.org/10.5281/zenodo.6545655), the numerical settings
(https://doi.org/10.6084/m9.figshare.19762219.v1), and routines used to
generate the plots presented herein (https://doi.org/10.5281/zenodo.6545917).
Living repositories for DALES and the postprocessing scripts are available at
https://github.com/dalesteam/dales and
https://github.com/martinjanssens/ppagg.

##  Appendix A Budgets of scalars

###  A.1 Derivation of eq. 3

Eq. 3 in the main text may be derived from eq. 1 by making use of the
decomposition into slab-averaged $\overline{\cdot}$ and fluctuating
$\cdot^{\prime}$ quantities (eq. 2), yielding the following expansions for the
horizontal and vertical advection terms:

$\displaystyle\frac{\partial}{\partial x_{jh}}\left(u_{jh}\chi\right)=$
$\displaystyle\frac{\partial}{\partial
x_{jh}}\left(\overline{u_{jh}}\chi+u_{jh}^{\prime}\overline{\chi}+u_{jh}^{\prime}\chi^{\prime}\right)$
(33a) $\displaystyle=$
$\displaystyle\chi\frac{\partial\overline{u_{jh}}}{\partial
x_{jh}}+\overline{\chi}\frac{\partial u_{jh}^{\prime}}{\partial
x_{jh}}+\overline{u_{jh}}\frac{\partial\chi}{\partial
x_{jh}}+u_{jh}^{\prime}\frac{\partial\overline{\chi}}{\partial x_{jh}}$ (33b)
$\displaystyle\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}w\chi\right)=$
$\displaystyle\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}\left(\overline{w}\chi+w^{\prime}\overline{\chi}+w^{\prime}\chi^{\prime}\right)\right)$
(33c) $\displaystyle=$
$\displaystyle\chi\left(\frac{\partial\overline{w}}{\partial
z}+\frac{1}{\rho_{0}}\frac{\partial\rho_{0}}{\partial
z}\overline{w}\right)+\overline{\chi}\left(\frac{\partial w^{\prime}}{\partial
z}+\frac{1}{\rho_{0}}\frac{\partial\rho_{0}}{\partial
z}w^{\prime}\right)+\overline{w}\frac{\partial\chi}{\partial
z}+w^{\prime}\frac{\partial\overline{\chi}}{\partial
z}+\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}w^{\prime}\chi^{\prime}\right)$ (33d)

In the anelastic approximation, conservation of mass demands:

$\frac{\partial u_{jh}}{\partial x_{jh}}+\frac{\partial w}{\partial
z}+\frac{1}{\rho_{0}}\frac{\partial\rho_{0}}{\partial z}w=0$ (34)

with the last term required to conserve the reference mass (Lilly,, 1996); it
holds for both fluctuations and in the slab-average. Therefore, when adding
the expansions eq. 33b and eq. 33d, the respective sum of the first terms and
second terms in these equations (those scaled by $\chi$ and $\overline{\chi}$)
are zero, resulting in eq. 3 of the main text. Note that because we solve our
equations on a doubly periodic domain, eq. 34 requires $\overline{w}=0$, its
effects in eq. 3 are thus prescribed by setting $\overline{w}$ and scaling it
with the local vertical gradient of $\chi$.

###  A.2 Derivation of eq. 18 for mesoscale scalar fluctuations

The main text’s eq. 18 can be derived from eq. 3 and eq. 4 by subtracting the
latter from the former, which gives:

$\frac{\partial\chi^{\prime}}{\partial
t}=-\overline{u_{jh}}\frac{\partial\chi^{\prime}}{\partial
x_{j}}-u_{jh}^{\prime}\frac{\partial\overline{\chi}}{\partial
x_{jh}}-\frac{\partial}{\partial
x_{jh}}\left(u_{jh}^{\prime}\chi^{\prime}-\overline{u_{jh}^{\prime}\chi^{\prime}}\right)-\overline{w}\frac{\partial\chi^{\prime}}{\partial
z}-w^{\prime}\frac{\partial\overline{\chi}}{\partial
z}-\frac{1}{\rho_{0}}\frac{\partial}{\partial
z}\left(\rho_{0}\left(w^{\prime}\chi^{\prime}-\overline{w^{\prime}\chi^{\prime}}\right)\right)$
(35)

To derive eq. 18 from eq. 35, we have made a number of assumptions. In our LES
model, whose results we analyse, use of doubly periodic boundary conditions
enforces the following conditions on the horizontal advection of fluctuations:

$\overline{u_{jh}}\frac{\partial\chi^{\prime}}{\partial
x_{j}}=\frac{\partial}{\partial
x_{jh}}\left(\overline{u_{jh}}\chi^{\prime}\right)$ (36a)
$u_{jh}^{\prime}\frac{\partial\overline{\chi}}{\partial x_{jh}}=0$ (36b)
$\frac{\partial}{\partial
x_{jh}}\left(\overline{u_{jh}^{\prime}\chi^{\prime}}\right)=0$ (36c)

Thus, unless their effects would be prescribed or parameterised, our model
does not account for i) the advection of scalar fluctuations into the analysed
domain with the mean wind (eq. 36a), ii) interactions between horizontal
gradients in $\chi$ larger than the domain (eq. 36b) or iii) horizontal eddy-
fluxes into the domain (eq. 36c). For our idealised analysis of the onset of
scale-growth from local processes, these assumptions seem reasonable, but
probably become untenable for analyses of finite, real-world domains with open
boundaries.

Furthermore, we have in our analysis neglected the explicit effects of
unresolved-scales effects. These would enter the analysis through additional
diffusion terms on the right-hand side of eq. 1. We do not present them in our
equations, but we do compute them and include them in the appropriate flux
divergence terms in the budgets presented in the text. At the mesoscales,
which are far removed from their action on the smallest, resolved scales,
their direct effects are small. Nevertheless, their influence in setting the
fluxes which drive the model is non-trivial, as we will show in future work.

Making these assumptions and applying a mesoscale filter to the resulting
equation results in eq. 18, and ensures its consistency with our LES model.

###  A.3 Moist and dry region averaging

Eq. 18 is formulated in an Eularian manner. When averaging it over moist and
dry regions, as done in the main text, we risk that our local budgets become
dominated by mean-flow advection of the regions. Since we are more interested
in the evolution of the regions themselves, we note that we can make use of
Reynolds’ transport theorem (here manipulated with the divergence theorem)

$\widetilde{\frac{\partial\chi_{m}^{\prime}}{\partial
t}}=\frac{\partial\widetilde{\chi_{m}^{\prime}}}{\partial
t}-\frac{\partial}{\partial
x_{jh}}\widetilde{\left(\chi_{m}^{\prime}u^{b}_{jh}\right)}$ (37)

where $\widetilde{\cdot}$ represents the moist or dry region averaging
operator and $u^{b}_{j}$ is the horizontal velocity of the region’s boundary.
The first term on the right-hand side of eq. 37 captures the evolution of
$\chi_{m}^{\prime}$ averaged over the moist and dry regions, in which we are
primarily interested, and which is plotted in fig. 7 and 11. The second term
accounts for the advection and net expansion of the regions with $u^{b}_{jh}$.
By decomposing $u^{b}_{j_{h}}$ into its contributions from mean flow advection
($\overline{u}_{jh}$) and net expansion ($u^{e}_{jh}$)

$u^{b}_{jh}=\overline{u}_{jh}+u^{e}_{jh},$ (38)

and inserting this into the second term on the right-hand side of eq. 37, we
recognise that we may cancel the mean-flow advection term that results with
the mean-flow advection contribution to the second term on the right-hand side
of eq. 18, if it is expanded as follows:

$\frac{\partial}{\partial
x_{jh}}\left(u_{jh}\chi^{\prime}\right)_{m}=\frac{\partial}{\partial
x_{jh}}\left(\overline{u}_{jh}\chi_{m}^{\prime}\right)+\frac{\partial}{\partial
x_{jh}}\left(u_{jh}^{\prime}\chi^{\prime}\right)_{m}$ (39)

These operations leave a residual when comparing the region-averaged budgets
to the first term on the right-hand side of eq. 37 due to $u^{e}_{jh}$. This
is the term we dub “net region expansion” in the main. Since this residual
also includes errors from numerical integration of simulation output and
Reynolds averaging, it is the least well-constrained term in our budgets, but
we find it plausible to attribute its main vertical structure to net expansion
of the moist regions at the expense of dry regions.

##  Appendix B Derivation of evolution equation for $\partial/\partial
z\left(\Gamma_{q_{t}}/\Gamma_{\theta_{lv}}\right)$

To analyse the onset and evolution of mean-state convexity, i.e.
$\partial^{2}\overline{q_{t}}/\partial\overline{\theta_{lv}}^{2}>0$, we use
the equivalence indicated in eq. 31 and write an evolution equation for
$\partial/\partial z\left(\Gamma_{q_{t}}/\Gamma_{\theta_{lv}}\right)$ by
differentiating it to time and applying the quotient rule of calculus twice.
This results in the following relation, where we have attempted to retain some
brevity by writing (repeated) vertical derivatives as (repeated) subscripts
$z$:

$\frac{\partial}{\partial
t}\left(\frac{\overline{q_{t}}_{z}}{\overline{\theta_{lv}}_{z}}\right)_{z}=\frac{1}{\overline{\theta_{lv}}_{z}}\left(\frac{\partial\overline{q_{t}}}{\partial
t}\right)_{zz}-\frac{\overline{\theta_{lv}}_{zz}}{\overline{\theta_{lv}}_{z}^{2}}\left(\frac{\partial\overline{q_{t}}}{\partial
t}\right)_{z}-\frac{\overline{q_{t}}_{z}}{\overline{\theta_{lv}}^{2}_{z}}\left(\frac{\partial\overline{\theta_{lv}}}{\partial
t}\right)_{zz}+\left(2\frac{\overline{q_{t}}_{z}\overline{\theta_{lv}}_{zz}}{\overline{\theta_{lv}}_{z}^{3}}-\frac{\overline{q_{t}}_{zz}}{\overline{\theta_{lv}}_{z}^{2}}\right)\left(\frac{\partial\overline{\theta_{lv}}}{\partial
t}\right)_{z}$ (40)

To determine which processes influence the left-hand side of this equation, we
may substitute for the tendencies that appear on its right-hand side terms
from the budget for slab-averaged scalars eq. 4. Applying the vertical
derivatives term by term and inserting in eq. 40 results in eq. 41:

$\frac{\partial}{\partial
t}\left(\frac{\overline{q_{t}}_{z}}{\overline{\theta_{lv}}_{z}}\right)_{z}=\frac{1}{\overline{\theta_{lv}}_{z}}\left(-\left(\overline{w_{ls}}\overline{q_{t}}_{z}\right)_{zz}-\frac{1}{\rho_{0}}\left(\rho_{0}\overline{w^{\prime}q_{t}^{\prime}}\right)_{zzz}+\overline{S_{q_{t}}}_{zz}\right)\\\
-\frac{\overline{\theta_{lv}}_{zz}}{\overline{\theta_{lv}}_{z}^{2}}\left(-\left(\overline{w_{ls}}\overline{q_{t}}_{z}\right)_{z}-\frac{1}{\rho_{0}}\left(\rho_{0}\overline{w^{\prime}q_{t}^{\prime}}\right)_{zz}+\overline{S_{q_{t}}}_{z}\right)\\\
-\frac{\overline{q_{t}}_{z}}{\overline{\theta_{lv}}_{z}^{2}}\left(-\left(\overline{w_{ls}}\overline{\theta_{lv}}_{z}\right)_{zz}-\frac{1}{\rho_{0}}\left(\rho_{0}\overline{w^{\prime}\theta_{lv}^{\prime}}\right)_{zzz}+\overline{S_{\theta_{lv}}}_{zz}\right)\\\
+\left(2\frac{\overline{q_{t}}_{z}\overline{\theta_{lv}}_{zz}}{\overline{\theta_{lv}}_{z}^{3}}-\frac{\overline{q_{t}}_{zz}}{\overline{\theta_{lv}}_{z}^{2}}\right)\left(-\left(\overline{w_{ls}}\overline{\theta_{lv}}_{z}\right)_{z}-\frac{1}{\rho_{0}}\left(\rho_{0}\overline{w^{\prime}\theta_{lv}^{\prime}}\right)_{zz}+\overline{S_{\theta_{lv}}}_{z}\right)$
(41)

Eq. 41 highlights a few interesting requirements for the development of
convexity in the mean state. First, it shows that in the limit of linear mean
profiles ($\overline{\theta_{lv}}_{zz}=\overline{q_{t}}_{zz}=0$), the second
and fourth term in eq. 41 are zero, constraining the responsibility for the
onset of convexity development in our simulations to processes that have
curvature in their mean profiles (those in terms 1 and 3). However, the
subsidence and large-scale forcing profiles of BOMEX are initially at most
linear functions of height. Therefore, the only non-zero terms that remain in
eq. 41 are third derivatives of the slab-mean fluxes. In fact, for
$\Gamma_{\theta_{lv}}>0$ and $\Gamma_{q_{t}}<0$, these third derivatives must
be negative to initiate the development of the required convexity, i.e.
$\frac{\partial}{\partial
t}\left(\frac{\overline{q_{t}}_{z}}{\overline{\theta_{lv}}_{z}}\right)_{z}>0$.
The result is the condition eq. 32b in the main text.

If $\overline{\theta_{lv}}$ and $\overline{q_{t}}$ have curvature in their
profiles, the second and fourth terms are no longer necessarily zero, such
that linear variations in subsidence and large-scale forcing, as well as
curvature in the flux profiles, may have an effect.

## References

  * Agee et al., (1973) Agee, E. M., Chen, T., and Dowell, K. (1973).  A review of mesoscale cellular convection.  Bulletin of the American Meteorological Society, 54(10):1004–1012.
  * Ahmed and Neelin, (2019) Ahmed, F. and Neelin, J. D. (2019).  Explaining scales and statistics of tropical precipitation clusters with a stochastic model.  Journal of the Atmospheric Sciences, 76(10):3063–3087.
  * Albrecht, (1993) Albrecht, B. A. (1993).  Effects of precipitation on the thermodynamic structure of the trade wind boundary layer.  Journal of Geophysical Research: Atmospheres, 98(D4):7327–7337.
  * (4) Albright, A. L., Bony, S., Stevens, B., and Vogel, R. (2022a).  Observed subcloud layer moisture and heat budgets in the trades.  Journal of the Atmospheric Sciences, in review.
  * (5) Albright, A. L., Bony, S., Vogel, R., and Stevens, B. (2022b).  A new conceptual picture of the transition layer.  In EGU General Assembly 2022, Vienna.
  * Anurose et al., (2020) Anurose, T., Bašták Ďurán, I., Schmidli, J., and Seifert, A. (2020).  Understanding the moisture variance in precipitating shallow cumulus convection.  Journal of Geophysical Research: Atmospheres, 125(1):e2019JD031178.
  * Augstein et al., (1973) Augstein, E., Riehl, H., Ostapoff, F., and Wagner, V. (1973).  Mass and energy transports in an undisturbed atlantic trade-wind flow.  Monthly Weather Review, 101(2):101–111.
  * Bellon and Stevens, (2012) Bellon, G. and Stevens, B. (2012).  Using the sensitivity of large-eddy simulations to evaluate atmospheric boundary layer models.  Journal of the Atmospheric Sciences, 69(5):1582–1601.
  * Betts, (1973) Betts, A. (1973).  Non-precipitating cumulus convection and its parameterization.  Quarterly Journal of the Royal Meteorological Society, 99(419):178–196.
  * Betts, (1975) Betts, A. K. (1975).  Parametric interpretation of trade-wind cumulus budget studies.  Journal of Atmospheric Sciences, 32(10):1934–1945.
  * Beucler et al., (2018) Beucler, T., Cronin, T., and Emanuel, K. (2018).  A linear response framework for radiative-convective instability.  Journal of Advances in Modeling Earth Systems, 10(8):1924–1951.
  * Blossey et al., (2013) Blossey, P. N., Bretherton, C. S., Zhang, M., Cheng, A., Endo, S., Heus, T., Liu, Y., Lock, A. P., de Roode, S. R., and Xu, K.-M. (2013).  Marine low cloud sensitivity to an idealized climate change: The cgils les intercomparison.  Journal of Advances in Modeling Earth Systems, 5(2):234–258.
  * Bony et al., (2020) Bony, S., Schulz, H., Vial, J., and Stevens, B. (2020).  Sugar, gravel, fish, and flowers: Dependence of mesoscale patterns of trade-wind clouds on environmental conditions.  Geophysical research letters, 47(7):e2019GL085988.
  * Bony and Stevens, (2019) Bony, S. and Stevens, B. (2019).  Measuring area-averaged vertical motions with dropsondes.  Journal of the Atmospheric Sciences, 76(3):767–783.
  * Bony et al., (2017) Bony, S., Stevens, B., Ament, F., Bigorre, S., Chazette, P., Crewell, S., Delanoë, J., Emanuel, K., Farrell, D., Flamant, C., et al. (2017).  Eurec4a: A field campaign to elucidate the couplings between clouds, convection and circulation.  Surveys in Geophysics, 38(6):1529–1568.
  * Bony et al., (2015) Bony, S., Stevens, B., Frierson, D. M., Jakob, C., Kageyama, M., Pincus, R., Shepherd, T. G., Sherwood, S. C., Siebesma, A. P., Sobel, A. H., et al. (2015).  Clouds, circulation and climate sensitivity.  Nature Geoscience, 8(4):261–268.
  * Bretherton and Blossey, (2017) Bretherton, C. and Blossey, P. (2017).  Understanding mesoscale aggregation of shallow cumulus convection using large-eddy simulation.  Journal of Advances in Modeling Earth Systems, 9(8):2798–2821.
  * Cesana and Del Genio, (2021) Cesana, G. V. and Del Genio, A. D. (2021).  Observational constraint on cloud feedbacks suggests moderate climate sensitivity.  Nature Climate Change, 11(3):213–218.
  * Chikira, (2014) Chikira, M. (2014).  Eastward-propagating intraseasonal oscillation represented by chikira–sugiyama cumulus parameterization. part ii: Understanding moisture variation under weak temperature gradient balance.  Journal of the Atmospheric Sciences, 71(2):615–639.
  * de Roode et al., (2004) de Roode, S. R., Duynkerke, P. G., and Jonker, H. J. (2004).  Large-eddy simulation: How large is large enough?  Journal of the atmospheric sciences, 61(4):403–421.
  * de Roode and Los, (2008) de Roode, S. R. and Los, A. (2008).  The effect of temperature and humidity fluctuations on the liquid water path of non-precipitating closed-cell stratocumulus clouds.  Quarterly Journal of the Royal Meteorological Society: A journal of the atmospheric sciences, applied meteorology and physical oceanography, 134(631):403–416.
  * Denby, (2020) Denby, L. (2020).  Discovering the importance of mesoscale cloud organization through unsupervised classification.  Geophysical Research Letters, 47(1):e2019GL085190.
  * Emanuel et al., (2014) Emanuel, K., Wing, A. A., and Vincent, E. M. (2014).  Radiative-convective instability.  Journal of Advances in Modeling Earth Systems, 6(1):75–90.
  * Fiedler, (1985) Fiedler, B. H. (1985).  Mesoscale cellular convection: is it convection?  Tellus A, 37(2):163–175.
  * (25) George, G., Stevens, B., Bony, S., Klingebiel, M., and Vogel, R. (2021a).  Observed impact of mesoscale vertical motion on cloudiness.  Journal of the Atmospheric Sciences, 78(8):2413–2427.
  * (26) George, G., Stevens, B., Bony, S., Pincus, R., Fairall, C., Schulz, H., Kölling, T., Kalen, Q. T., Klingebiel, M., Konow, H., et al. (2021b).  Joanne: Joint dropsonde observations of the atmosphere in tropical north atlantic meso-scale environments.  Earth System Science Data, 13(11):5253–5272.
  * Grenier and Bretherton, (2001) Grenier, H. and Bretherton, C. S. (2001).  A moist pbl parameterization for large-scale models and its application to subtropical cloud-topped marine boundary layers.  Monthly weather review, 129(3):357–377.
  * Heinze et al., (2015) Heinze, R., Mironov, D., and Raasch, S. (2015).  Second-moment budgets in cloud topped boundary layers: A large-eddy simulation study.  Journal of Advances in Modeling Earth Systems, 7(2):510–536.
  * Held and Hoskins, (1985) Held, I. M. and Hoskins, B. J. (1985).  Large-scale eddies and the general circulation of the troposphere.  In Advances in geophysics, volume 28, pages 3–31. Elsevier.
  * Heus et al., (2010) Heus, T., van Heerwaarden, C. C., Jonker, H. J., Siebesma, A. P., Axelsen, S., van den Dries, K., Geoffroy, O., Moene, A. F., Pino, D., de Roode, S. S., et al. (2010).  Formulation of and numerical studies with the Dutch Atmospheric Large-Eddy Simulation (DALES).  Geosci. Model Dev, 3:415–444.
  * Holland and Rasmusson, (1973) Holland, J. Z. and Rasmusson, E. M. (1973).  Measurements of the atmospheric mass, energy, and momentum budgets over a 500-kilometer square of tropical ocean.  Monthly Weather Review, 101(1):44–55.
  * Janssens et al., (2021) Janssens, M., Vilà-Guerau de Arellano, J., Scheffer, M., Antonissen, C., Siebesma, A. P., and Glassmeier, F. (2021).  Cloud patterns in the trades have four interpretable dimensions.  Geophysical Research Letters, 48(5):e2020GL091001.
  * Jeevanjee and Romps, (2013) Jeevanjee, N. and Romps, D. M. (2013).  Convective self-aggregation, cold pools, and domain size.  Geophysical Research Letters, 40(5):994–998.
  * Jonker et al., (1999) Jonker, H. J., Duynkerke, P. G., and Cuijpers, J. W. (1999).  Mesoscale fluctuations in scalars generated by boundary layer convection.  Journal of the atmospheric sciences, 56(5):801–808.
  * Klein, (2010) Klein, R. (2010).  Scale-dependent models for atmospheric flows.  Annual review of fluid mechanics, 42:249–274.
  * Klinger et al., (2017) Klinger, C., Mayer, B., Jakub, F., Zinner, T., Park, S.-B., and Gentine, P. (2017).  Effects of 3-d thermal radiation on the development of a shallow cumulus cloud field.  Atmospheric Chemistry and Physics, 17(8):5477–5500.
  * Lilly, (1996) Lilly, D. (1996).  A comparison of incompressible, anelastic and boussinesq dynamics.  Atmospheric research, 40(2-4):143–151.
  * Matheou and Lamaakel, (2021) Matheou, G. and Lamaakel, O. (2021).  Growth rates of turbulence length scales in precipitating shallow cumulus convection.  In AGU Fall Meeting 2021. AGU.
  * Muller and Held, (2012) Muller, C. J. and Held, I. M. (2012).  Detailed investigation of the self-aggregation of convection in cloud-resolving simulations.  Journal of the Atmospheric Sciences, 69(8):2551–2565.
  * Müller and Chlond, (1996) Müller, G. and Chlond, A. (1996).  Three-dimensional numerical study of cell broadening during cold-air outbreaks.  Boundary-Layer Meteorology, 81(3):289–323.
  * Myers et al., (2021) Myers, T. A., Scott, R. C., Zelinka, M. D., Klein, S. A., Norris, J. R., and Caldwell, P. M. (2021).  Observational constraints on low cloud feedback reduce uncertainty of climate sensitivity.  Nature Climate Change, 11(6):501–507.
  * Narenpitak et al., (2021) Narenpitak, P., Kazil, J., Yamaguchi, T., Quinn, P. K., and Feingold, G. (2021).  From sugar to flowers: A transition of shallow cumulus organization during atomic.  Journal of Advances in Modeling Earth Systems, 13:e2021MS002619.
  * Naumann et al., (2019) Naumann, A. K., Stevens, B., and Hohenegger, C. (2019).  A moist conceptual model for the boundary layer structure and radiatively driven shallow circulations in the trades.  Journal of the Atmospheric Sciences, 76(5):1289–1306.
  * Neelin and Held, (1987) Neelin, J. D. and Held, I. M. (1987).  Modeling tropical convergence based on the moist static energy budget.  Monthly Weather Review, 115(1):3–12.
  * Nitta and Esbensen, (1974) Nitta, T. and Esbensen, S. (1974).  Heat and moisture budget analyses using bomex data.  Monthly Weather Review, 102(1):17–28.
  * Ouwersloot et al., (2017) Ouwersloot, H., Moene, A., Attema, J., and De Arellano, J. V.-G. (2017).  Large-eddy simulation comparison of neutral flow over a canopy: Sensitivities to physical and numerical conditions, and similarity to other representations.  Boundary-Layer Meteorology, 162(1):71–89.
  * Park et al., (2006) Park, K.-A., Cornillon, P., and Codiga, D. L. (2006).  Modification of surface winds near ocean fronts: Effects of gulf stream rings on scatterometer (quikscat, nscat) wind observations.  Journal of Geophysical Research: Oceans, 111(C3).
  * Raymond et al., (2009) Raymond, D. J., Sessions, S. L., Sobel, A. H., and Fuchs, Ž. (2009).  The mechanics of gross moist stability.  Journal of Advances in Modeling Earth Systems, 1(3).
  * Riehl et al., (1951) Riehl, H., Yeh, T., Malkus, J. S., and La Seur, N. E. (1951).  The north-east trade of the pacific ocean.  Quarterly Journal of the Royal Meteorological Society, 77(334):598–626.
  * Schulz et al., (2021) Schulz, H., Eastman, R., and Stevens, B. (2021).  Characterization and evolution of organized shallow convection in the downstream north atlantic trades.  Journal of Geophysical Research: Atmospheres, 126(17):e2021JD034575.
  * Seifert and Heus, (2013) Seifert, A. and Heus, T. (2013).  Large-eddy simulation of organized precipitating trade wind cumulus clouds.  Atmospheric Chemistry and Physics, 13(11):5631–5645.
  * Seifert et al., (2015) Seifert, A., Heus, T., Pincus, R., and Stevens, B. (2015).  Large-eddy simulation of the transient and near-equilibrium behavior of precipitating shallow convection.  Journal of Advances in Modeling Earth Systems, 7(4):1918–1937.
  * Siebesma and Cuijpers, (1995) Siebesma, A. and Cuijpers, J. (1995).  Evaluation of parametric assumptions for shallow cumulus convection.  Journal of Atmospheric Sciences, 52(6):650–666.
  * Siebesma et al., (2003) Siebesma, A. P., Bretherton, C. S., Brown, A., Chlond, A., Cuxart, J., Duynkerke, P. G., Jiang, H., Khairoutdinov, M., Lewellen, D., Moeng, C.-H., et al. (2003).  A large eddy simulation intercomparison study of shallow cumulus convection.  Journal of the Atmospheric Sciences, 60(10):1201–1219.
  * Sobel et al., (2001) Sobel, A. H., Nilsson, J., and Polvani, L. M. (2001).  The weak temperature gradient approximation and balanced tropical moisture waves.  Journal of the atmospheric sciences, 58(23):3650–3665.
  * Stevens, (2007) Stevens, B. (2007).  On the growth of layers of nonprecipitating cumulus convection.  Journal of the atmospheric sciences, 64(8):2916–2931.
  * Stevens et al., (2020) Stevens, B., Bony, S., Brogniez, H., Hentgen, L., Hohenegger, C., Kiemle, C., L’Ecuyer, T. S., Naumann, A. K., Schulz, H., Siebesma, P. A., et al. (2020).  Sugar, gravel, fish and flowers: Mesoscale cloud patterns in the trade winds.  Quarterly Journal of the Royal Meteorological Society, 146(726):141–152.
  * Stevens et al., (2021) Stevens, B., Bony, S., Farrell, D., Ament, F., Blyth, A., Fairall, C., Karstensen, J., Quinn, P. K., Speich, S., Acquistapace, C., et al. (2021).  Eurec 4 a.  Earth System Science Data, 13(8):4067–4119.
  * Thomas et al., (2021) Thomas, M. L., Bašták Ďurán, I., and Schmidli, J. (2021).  Toward parametrization of precipitating shallow cumulus cloud organization via moisture variance.  Journal of Geophysical Research: Atmospheres, 126(15):e2021JD034939.
  * Tobin et al., (2012) Tobin, I., Bony, S., and Roca, R. (2012).  Observational evidence for relationships between the degree of aggregation of deep convection, water vapor, surface fluxes, and radiation.  Journal of Climate, 25(20):6885–6904.
  * van Zanten et al., (2011) van Zanten, M. C., Stevens, B., Nuijens, L., Siebesma, A. P., Ackerman, A., Burnet, F., Cheng, A., Couvreux, F., Jiang, H., Khairoutdinov, M., et al. (2011).  Controls on precipitation and cloudiness in simulations of trade-wind cumulus as observed during rico.  Journal of Advances in Modeling Earth Systems, 3(2).
  * (62) Vogel, R., Bony, S., and Stevens, B. (2020a).  Estimating the shallow convective mass flux from the subcloud-layer mass budget.  Journal of the Atmospheric Sciences, 77(5):1559–1574.
  * Vogel et al., (2016) Vogel, R., Nuijens, L., and Stevens, B. (2016).  The role of precipitation and spatial organization in the response of trade-wind clouds to warming.  Journal of Advances in Modeling Earth Systems, 8(2):843–862.
  * (64) Vogel, R., Nuijens, L., and Stevens, B. (2020b).  Influence of deepening and mesoscale organization of shallow convection on stratiform cloudiness in the downstream trades.  Quarterly Journal of the Royal Meteorological Society, 146(726):174–185.
  * Wing et al., (2020) Wing, A. A., Stauffer, C. L., Becker, T., Reed, K. A., Ahn, M.-S., Arnold, N. P., Bony, S., Branson, M., Bryan, G. H., Chaboureau, J.-P., et al. (2020).  Clouds and convective self-aggregation in a multimodel ensemble of radiative-convective equilibrium simulations.  Journal of advances in modeling earth systems, 12(9):e2020MS002138.
  * Wood and Hartmann, (2006) Wood, R. and Hartmann, D. L. (2006).  Spatial variability of liquid water path in marine low cloud: The importance of mesoscale cellular convection.  Journal of Climate, 19(9):1748–1764.
  * Yanai et al., (1973) Yanai, M., Esbensen, S., and Chu, J.-H. (1973).  Determination of bulk properties of tropical cloud clusters from large-scale heat and moisture budgets.  Journal of Atmospheric Sciences, 30(4):611–627.
  * Zelinka et al., (2020) Zelinka, M. D., Myers, T. A., McCoy, D. T., Po-Chedley, S., Caldwell, P. M., Ceppi, P., Klein, S. A., and Taylor, K. E. (2020).  Causes of higher climate sensitivity in cmip6 models.  Geophysical Research Letters, 47(1):e2019GL085782.
  * Zuidema et al., (2017) Zuidema, P., Torri, G., Muller, C., and Chandra, A. (2017).  A survey of precipitation-induced atmospheric cold pools over oceans and their interactions with the larger-scale environment.  Surveys in Geophysics, 38(6):1283–1305.