Quarterly Journal of the Royal Meteorological Society Leif Denby, Institute
for Climate and Atmospheric Science, School of Earth and Environment,
University of Leeds, Leeds, LS2 9JT, United Kingdom<EMAIL_ADDRESS>Met
Office/NERC Grant NE/N013840/1 Research Article

# Characterising the shape, size and orientation of cloud-feeding coherent
boundary layer structures

Leif Denby  School of Earth and Environment, University of Leeds  Steven J.
Böing  School of Earth and Environment, University of Leeds  Douglas J. Parker
School of Earth and Environment, University of Leeds  Andrew N. Ross  School
of Earth and Environment, University of Leeds  Steven M. Tobias  School of
Mathematics, University of Leeds

###### Abstract

This paper presents two techniques for characterisation of cloud-feeding
coherent boundary layer structures through analysis of large-eddy simulations
of shallow cumulus clouds, contrasting conditions with and without ambient
shear. The first technique is a generalisation of the two-point correlation
function where the correlation length-scale as well as orientation can be
extracted. The second technique decomposes the vertical transport by coherent
structures by the shape, size and orientation of these structures. It is found
that the structures dominating the vertical flux are plume-like in character
(extending from the surface into cloud), show small width/thickness asymmetry
and rise near-vertically in the absence of ambient wind. The planar stretching
and tilting of boundary layer structures caused by the introduction of ambient
shear is also quantified, demonstrating the general applicability of the
techniques for future study of other boundary layer patterns.

Keywords — moist convection, coherent structures, structure characterisation,
methodology, convective triggering

##  1 Introduction

Coherent boundary layer structures carry perturbations of temperature,
moisture and vertical velocity necessary to trigger convective clouds, by
overcoming the boundary layer top inversion in a conditionally unstable
atmosphere. However, the degree to which the spatial distribution, morphology
and perturbations carried by the coherent structures affect how clouds form is
currently uncertain, as is which external drivers affect these properties of
the coherent structures. To study comprehensively the formation of clouds from
coherent boundary layer structures, we must first be able to identify and
measure the properties of these structures, which is the aim of this paper.

Coherent structures in the boundary layer carry so-called non-local (cannot
simply be calculated from local scalar values) counter-gradient transport in
the boundary layer [10], providing transport against the vertical mean
gradient of moisture and heat (in contrast to smaller turbulent eddies doing
down-gradient, diffusive, transport). Owing to the limited resolution
available in Global Circulation Models and Numerical Weather Prediction
models, it is necessary to parameterise the unresolved sub-grid processes that
provide vertical transport and lead to convective cloud formation. The
development of parametrisations of non-local transport has been key to
improving boundary layer parametrisations [18, 5]. Over the past two decades,
the Eddy-Diffusivity Mass-Flux (EDMF) approach to boundary layer
parametrisation [e.g. 46, 35, 28, 29, 36] has become popular: in this
approach, local turbulent transport and transport by coherent structures
(leading to the formation of convective clouds) are modelled separately.
However, our current lack of knowledge of how these coherent structures are
affected by external forcings (ambient wind, surface heterogeneity, cold-
pools, etc) and changes in these structures affect cloud formation, limits our
ability to refine models of the non-local transport, and thus in representing
the genesis (formation) of convective clouds in weather and climate
simulations.

As well as influencing the formation of individual clouds, these structures
capture the convective state of the atmosphere, through their spatial
organisation and by persisting sub-grid length-scales of motion over time (a
form of convective "memory"). Representation of these sub-grid forms of
organisation are largely absent in contemporary convection parameterisations,
however the importance of convective organisation in affecting, for example,
the radiative properties of the atmosphere, and the impact of limited
representation of these processes in models, is becoming increasingly clear
[3].

Prior work on characterising the morphology of coherent boundary layer
structures has focused primarily on measuring coherence in the boundary layer
as a whole, not looking at the properties of individual coherent structures,
but instead producing bulk length-scale estimates using spectral peaks in the
autocorrelation and covariance spectrum to measure spatial and angular
coherence (in the horizontal plane). Jonker et al. [20] found in cloud-free
Large-Eddy Simulations (LES) that the shortest correlation length-scale exists
in vertical velocity on the order of the boundary layer depth, whereas the
buoyancy providing field (potential temperature $\theta$ for dry LES, virtual
potential temperature $\theta_{v}$ when water vapour is included) typically
attains larger steady-state length-scales [20, 37, 39]. de Roode et al. [37]
in addition found that for passive tracers in dry and stratocumulus topped
boundary layers the vertical profile of horizontal length-scales is determined
by the surface to boundary layer top buoyancy flux ratio
$r=\frac{\overline{w^{\prime}\theta_{v}^{\prime}}_{T}}{\overline{w^{\prime}\theta_{v}^{\prime}}_{0}}$,
with minimum length-scales attained when this ratio was $r\approx-0.2$ for dry
convection (which is the classical buoyancy flux-ratio scale found for dry
convective boundary layers) and $r\approx-1$ in the case of stratocumulus.

In simulations where clouds are present the picture becomes more complicated
for the dynamic scalars [43, 37], apart from vertical velocity which stays
largely unchanged (restricted dynamically by the boundary layer depth). The
buoyancy ($\theta_{v}$) no longer necessarily reaches a steady-state length-
scale profile, and water-vapour and potential temperature produce meso-scale
variability (which appear to cancel in their contributions to $\theta_{v}$).
The length-scale characteristics found in LES have been supported through
similar analysis of observations [30, 23], with the relative scales of $q$ and
$\theta$ changing with Bowen ratio, producing narrower scales for the scalar
dominating buoyancy. Identifying and characterising coherent structures in
observations, Schumann and Moeng [43] found that coherence length-scale
typically peaks mid boundary layer, where the number of structures reaches a
minimum. Miao et al. [26] found the plume spacing and width to be
$\leavevmode\nobreak\ 0.7h_{BL}$ and $0.2h_{BL}$ respectively mid boundary
layer.

In place of studying correlation in inverse distance (or wave-length) space,
the present work studies correlation in real space using cumulants [see
history and review by 22]. Cumulants have been applied by Lohou et al. [24] to
study thermals in a daytime boundary layer with weak wind shear over land, and
characterise the influence of anisotropy on vertical transport. Schmidt and
Schumann [42] used cumulants to study coherence in the vertical rather than
the horizontal plane, and identified both large-scale plumes and transient
thermals in a convective boundary layer. They have also been used to expand
the prognostic equations [1] to predict correlations between fields rather
than the fields themselves. In modelling by Lohou et al. [24] and Schmidt and
Schumann [42] correlation is studied in real space, rather than inverse
distance, or wave-length space. In this work we develop this technique further
by utilising cumulants and producing vertical profiles of integral length-
scale and orientation of coherence in real space.

With respect to identifying individual coherent structures in the boundary
layer prior work focused on using limit values on vertical velocity or water
vapour concentration (either separately or in combination) to define object
masks [2, 15, 30, 43]. More recently Efstathiou et al. [13] developed a
masking technique to maximise the vertical transport carried by the selected
region of the boundary layer. Couvreux et al. [8] noted that object masks
based on the physical fields (vertical velocity, water vapour, temperature)
had the drawback that they poorly capture transport through the boundary layer
top inversion and into the cloud-layer, and proposed a technique based on a
surface-released decaying passive tracer to track the rising boundary layer
structures. This tracer technique has been used to identify coherent
structures in cloudy boundary layers [9, 32, 4]. The current work uses the
tracer in combination with the object-splitting technique described in Park et
al. [31], to identify individual structures that are characterised by a
prominent local maximum of the vertical velocity in the boundary layer.

The aim of this paper is to demonstrate the use of new techniques to
characterise the morphology of coherent structures in the boundary layer and
thereby provide the means to identify the properties of structures which
dominate the vertical transport. We first demonstrate that using measures of
spatial coherence on individual fields gives an incomplete picture of the
properties of coherent boundary layer structures, next move to characterise
individual coherent structures and later decompose the vertical transport by
the structure characteristics. Having the ability to measure the shape, size
and orientation of the coherent structures which dominate the vertical
transport will enable the study in further work of how and to what extent
these properties effect how clouds form and organise, and how external factors
effect the coherent structures. This in turn will enable the construction of
clearer process-understanding for the coupling of clouds to the boundary
layer, which ultimately can be used to build better representations of clouds
in weather and climate models.

As a means of investigating the extent to which the methods discussed herein
are able to unpick and quantify boundary layer transport and its influence on
clouds in different environmental conditions, we will use two simulations of
shallow convection, with and without shear, as a demonstration of large-scale
influence on coherent boundary layer structures and convective clouds. The
modelling setup for the simulations are discussed in Sec. 2. The methods used
to identify coherent boundary layer structures and quantify their properties
will be discussed in Sec. 3. Finally the application of these methods will be
shown in Sec. 4 and a discussion of this analysis will be given in Sec. 5.

##  2 Modelling setup

Simulations were carried out with the non-hydrostatic UCLA-LES Large-Eddy
Simulations model [47] with two-moment warm-rain microphysics scheme [48] on a
$$40\text{\,}\mathrm{k}\mathrm{m}$\times$40\text{\,}\mathrm{k}\mathrm{m}$\times$4\text{\,}\mathrm{k}\mathrm{m}$$
double-periodic domain with an isotropic grid-spacing of $(\Delta x,\Delta
y,\Delta z)=(25,25,25)$\text{\,}\mathrm{m}$$.

Figure 1:  Vertical profiles of horizontal mean wind at times
$t=$0\text{\,}\mathrm{h}$$ and $t=$6\text{\,}\mathrm{h}$$ in meridional ($v$)
and zonal ($u$) directions, together with wind direction $\phi$ (measured from
$x$-axis), for cases with and without shear. Mean wind direction in sub-cloud
layer after $t=$6\text{\,}\mathrm{h}$$ is at
$\phi\approx$210\text{\,}\mathrm{\SIUnitSymbolDegree}$$ in simulation with
shear.

The simulation setup is based on the Rain in Cumulus over the Ocean (RICO)
field study [34] and associated LES model inter-comparison study [51]. The
RICO setup is characterised by shallow cumuli developing from moisture-
dominated fluxes from the ocean surface, with the clouds constrained in growth
by a prescribed large-scale subsidence aloft and large-scale advection of
moisture out of the domain. In the original inter-comparison study, the
simulation settles into a quasi-steady state after a short
($\approx$2\text{\,}\mathrm{h}$$) rapid response to the initial condition,
after which convection slowly (over $\approx$20\text{\,}\mathrm{h}$$)
aggregates into larger cloud clusters by precipitation-induced cold pools. As
the process leading to formation of these cloud clusters is not the focus of
this study, we will be considering only the stage of cloud development before
these large clusters have developed (here using $t=$6\text{\,}\mathrm{h}$$).

In this work there are two key differences to the original RICO setup as
published in Seifert et al. [44]. Firstly, in order to study the effect of
ambient wind-shear on the coherent boundary layer structures, two simulations
were run, one with and one without shear (see profiles in Fig. 1). In the
former the wind-profile from RICO was left unchanged and in the latter the
meridional and zonal wind components were set to zero. Secondly, because the
near-surface horizontal velocity differs between the two simulations, the bulk
aerodynamic parameterisation of surface flux was replaced with a fixed
sensible ($F_{s}=$7\text{\,}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$$) and latent
heat flux ($F_{v}=$150\text{\,}\mathrm{W}\mathrm{/}\mathrm{m}^{2}$$) so that
the two conditions have the same fluxes provided from the surface. The surface
flux values were estimated from the original RICO simulation once near-
equilibrium conditions have been reached (at
$t\approx$6\text{\,}\mathrm{h}$$).

As seen in the horizontal cross-sections of vertical velocity in Fig. 2, the
presence of ambient shear causes both boundary layer structures and clouds to
become organised into elongated structures instead of convective cells. This
is noticeable through the elongated line-like regions of high vertical
velocity in simulations with shear, causing the clouds (cloud-base is at
$z\approx$650\text{\,}\mathrm{m}$$) to organise into structures resembling
_cloud streets_ instead of (as in the case without shear) at the nodes of
boundary layer convective cells. The development of cloud streets under
conditions with ambient shear is consistent with prior studies [see reviews by
14, 52], as is the presence of sheet-like coherent structures attached to the
surface extending into the bulk of the boundary layer [as in 21].

Figure 2:  Horizontal cross-sections of vertical velocity through the boundary
layer middle ($z\approx$362.5\text{\,}\mathrm{m}$$, bottom) and at cloud-base
height ($z\approx$662.5\text{\,}\mathrm{m}$$, top) for simulations with shear
(right) and without (left). The presence of ambient shear is seen to clearly
break the geometry of convective cells and create elongated coherent boundary
structures.

##  3 Methods

Before the properties of coherent boundary layer structures that trigger
clouds can be measured, it is necessary to define exactly what we mean by a
coherent structure. In contrast to the cloud layer, where we can define a
coherent structure purely on the concentration of water droplets, in the
boundary layer there are at least three scalar fields that carry the
perturbation that eventually triggers a cloud: moisture, temperature and
vertical velocity.

We first investigate the length-scales of variability in these fields as a
bulk-property of the fluid. If these fields were to vary over similar length-
scales it would be relatively straightforward to define threshold criteria on
either of the scalar fields that would apply to all fields. Unfortunately this
is not the case (section 3.1) and so we instead develop a method which
identifies air with properties similar to that which triggers convective
clouds (section 3.2.1), by tracking air entering newly-formed convective
clouds.

###  3.1 Characteristic length-scales - cumulant analysis

As an alternative to moments, cumulants provide a means to summarise the
statistical correlation between one or more variables [22]. Similarly to
Tobias and Marston [50], where cumulants were used to identify and measure
coherent structures in 3D rotating Couette flow, we here utilise the second
cumulant (two-point correlation function), which for fields $\psi$ and
$\varphi$ at height $z$ [here $z_{1}=z_{2}=z$ in contrast to 50] is given by

$\begin{split}c_{\psi\varphi}(\xi,\nu,z)=&\\\
\frac{1}{L_{x}L_{y}}&\int_{0}^{L_{x}}\int_{0}^{L_{y}}\psi^{\prime}(x,y,z)\varphi^{\prime}(x+\xi,y+\nu,z)dxdy,\end{split}$

where $\psi^{\prime}$ and $\varphi^{\prime}$ are deviations from the
horizontal mean of $\psi$ and $\varphi$ respectively, and $L_{x}$ and $L_{y}$
are the lengths of the domain in the $x$\- and $y$-direction. The positions
are wrapped around in the $x$\- and $y$-direction exploiting the periodic
boundary conditions of the simulation.

An example of this method applied to the spatial correlation of vertical
velocity ($\psi=w$) and water vapour ($\varphi=q_{v}$) in the middle of the
boundary layer in a simulation with ambient shear is shown in Fig. 3. In cases
such as this where an external forcing is causing boundary layer and cloud
structures to develop in a preferential direction, the cumulant will show
increased correlation in this direction. To quantify this asymmetry we
identify a _principal_ and _perpendicular_ direction of coherence (measured in
terms of the angles $\theta_{p}$ and $\theta_{\perp}$) of the central part of
the cumulant. This central part $\hat{c}_{\psi\varphi}$ is defined as the
connected region at the origin with the same sign as at the origin of the
cumulant $c_{\psi\varphi}$. Treating $\hat{c}_{\psi\varphi}$ as a 2D mass-
distribution, we then estimate the orientation angle as the principal axis
(eigenvector with largest eigenvalue) of the moment of inertia tensor:

$\overline{\overline{I}}=\begin{bmatrix}\int\hat{c}_{\psi\varphi}(\xi,\nu)\nu^{2}\
d\xi d\nu&\int\hat{c}_{\psi\varphi}(\xi,\nu)\xi\nu\ d\xi d\nu\\\
\int\hat{c}_{\psi\varphi}(\xi,\nu)\xi\nu\ d\xi
d\nu&\int\hat{c}_{\psi\varphi}(\xi,\nu)\xi^{2}\ d\xi d\nu\end{bmatrix}.$ (1)

The cumulant can then be sampled in this horizontal plane along the principal
and perpendicular directions of coherence (as seen in Fig. 3 right) so that
the coherence can be quantified in these directions. The presence of ambient
shear is for example seen to cause elongation in the direction of the ambient
wind (this will be discussed in detail in section 4.1).

Once the direction of principal coherence ($\theta_{p}$) has been identified,
a characteristic length-scale ($L_{p}$) of coherence may be estimated in this
direction and similarly in the perpendicular direction ($L_{\bot}$). These
length-scales are computed through a cumulant-weighted integral of distance
($l=\sqrt{\xi^{2}+\nu^{2}}$) from the cumulant origin:

$\displaystyle L^{\psi,\varphi}_{\delta}$
$\displaystyle=\frac{\int_{-\frac{L}{2}}^{\frac{L}{2}}l\
\hat{c}^{\delta}_{\psi,\varphi}(l)\
dl}{\int_{-\frac{L}{2}}^{\frac{L}{2}}\hat{c}^{\delta}_{\psi,\varphi}(l)\ dl},$
(2)

where $\delta\in[p,\bot]$ (for either the principal or perpendicular length-
scale) and $L=min(L_{x},L_{y})$ (the minimum of the simulation domain length
in the $x$\- and $y$-direction), with the cumulant along a particular
direction given by

$\displaystyle\hat{c}^{\delta}_{\psi,\varphi}(l)$
$\displaystyle=\hat{c}_{\psi,\varphi}(\xi{=}l\cos(\theta_{\delta}),\nu{=}l\sin(\theta_{\delta}))$
(3)

evaluated at an arbitrary point using piece-wise linear interpolation. To
measure the degree of elongation we define the asymmetry ratio
$r_{a}=\frac{L_{p}}{L_{\bot}}$.

Figure 3:  a) Cumulant of vertical velocity and water vapour (i.e. the
horizontal moisture flux) in horizontal plane at $z=$300\text{\,}\mathrm{m}$$
with principal direction of coherence identified by the red dashed line, and
b) the same cumulant sampled along (in blue) and perpendicular (in orange) to
the principal axis with characteristic (integral) width indicated with
vertical lines. The elongated nature of the coherence as seen on the left is
quantified by a significantly larger ($\approx$1500\text{\,}\mathrm{m}$$ vs
$\approx$500\text{\,}\mathrm{m}$$) characteristic length along the direction
of shear. In both a) and b) the scaling of correlation magnitude is written
above.

As well as providing a means to quantify the length-scale of correlation (and
thus a single characteristic length-scale for all coherent boundary layer
structures), the shape of the cumulant can provide insight into the dynamical
structure of coherent structures by quantifying the relative spatial
distribution of different scalar fields. This is possible by studying two
different aspects of the cumulant produced from two different fields,
specifically the offset of the cumulant peak value from the origin and the
skewness of the distribution around the origin. An offset of the cumulant from
the origin indicates that the extreme values of two different scalar fields
are located spatially offset from each other and would suggest something is
driving a separation between two fields. Similarly, skewness in the cumulant
distribution indicates that the two different scalar fields have differently
spatially skewed distributions, e.g one field may appear spatially Gaussian,
but another may be skewed relative to this. The corollary to this is that a
2nd cumulant, between two different scalar fields, which is centered on and
symmetric around the origin, indicates that these two fields on average are
spatially distributed identically (in terms of skewness) around their
respective peak values and their peak values are, on average, co-located.

###  3.2 Object-based analysis

To gain a more comprehensive understanding of transport by coherent structures
in the boundary layer, we transition from looking at the boundary layer air in
a bulk sense, to studying transport by individual coherent structures that may
trigger clouds. This requires identifying the regions of the boundary layer
that contribute to transport into convective clouds, splitting these regions
into individual coherent structures and finally formulating methods to
quantify the properties of these structures.

####  3.2.1 Object identification

To quantify the characteristic properties of individual coherent structures
carrying out vertical transport, these structures must first be identified.
This was achieved by first producing a 3D mask to pick out the part of the
atmosphere thought to contain coherent structures, and thereafter splitting
this mask into individual 3D objects.

The 3D mask was produced from the concentration of a passive tracer ($\phi$)
decaying with a time-scale $\tau$, which was released from the surface [as
first used in 8]. Specifically the time evolution of the tracer is given by

$\frac{\partial\phi}{\partial t}=-\frac{\phi}{\tau}.$ (4)

The decay time-scale was set to
$\tau=$15\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$$ in this study as this
represents the typical overturning time-scale of boundary layer eddies in the
simulations used (see appendix A).

From the scalar $\phi$ a mask is created using its standard deviation in a
horizontal cross-section ($\sigma_{\phi}(z)$) and its local deviation from the
horizontal mean ($\phi^{\prime}(x,y,z)=\phi(x,y,z)-\overline{\phi(z)}$) by
requiring that the local deviation is $n$ standard deviations from the mean,
i.e. the mask $m(x,y,z)$ is given by

$m(x,y,z)=\left\\{\begin{matrix}1&\text{if}&\phi^{\prime}(x,y,z)>n\
\sigma_{\phi}(x,y,z),\\\ 0&\text{otherwise}\end{matrix}\right.$ (5)

here $n=2$ was used as this was found to produce closest agreement between the
properties of air entering clouds and those identified to belong to coherent
structures (see section 4.2). The choice of decay time-scale and limit value
for $n$ is similar to those
($\tau=$15\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$$ and $n=2.5$) identified by
Chinita et al. [7] to be optimal when studying shallow moist convection in the
BOMEX (Barbados Oceanographic and Meteorological Experiment) case [45].

The constructed 3D mask was observed to identify boundary layer air with
thermodynamic properties similar to air entering into recently formed clouds
(see section 12 for details), making it a suitable method to separate vertical
transport by local diffusive mixing (small eddies) from transport by larger,
non-local eddies carrying fluxes leading to cloud-formation.

The method of Park et al. [31] was used to identify individual objects from
the 3D mask. This method works by first labelling contiguous regions of the
mask as proto-objects. These proto-objects are then further subdivided based
on a second scalar field (here vertical velocity $w$) by assigning points to
local maxima in the second scalar and splitting where a boundary (the "col")
between two maxima has a relative value below a predefined threshold [$f=0.7$
was used for this "col"-factor, as in 31].

To measure the thermodynamic properties of air causing the formation of
clouds, individual clouds were identified from 2D column-integrated liquid
water path ($m_{lwp}$) with a threshold value of
$m_{lwp}>$0.01\text{\,}\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{2}$$ and
tracked by spatial overlap in consecutive time-steps using the method detailed
in Heus and Seifert [16]. This method identifies _active_ clouds as ones with
at least one buoyant core (identified from the virtual potential temperature
$\theta_{v}$) and _passive_ clouds as those without a buoyant core. In
addition, this cloud-tracking method splits clouds with multiple buoyant cores
into smaller sub-clouds with the non-buoyant regions defined as _outflow_. In
this work we only consider the properties of air entering single-core _active_
clouds [reaching the level of free convection, see 49] as these are likely to
have the strongest and clearest connection to boundary layer variations. To
facilitate selecting clouds which recently formed we also keep track of the
age of each cloud ($t_{age}$) by storing the time of appearance for each
tracked cloud.

####  3.2.2 Object characterisation

Once individual 3D coherent structures have been identified, a method is
needed to calculate characteristic properties of these objects. Here we detail
techniques to compute characteristic length-scales and orientation of each
coherent structure.

##### Topological measures - Minkowski functionals

Instead of attempting to fit a parameterised shape (for example an ellipsoid)
to each object with the intention of estimating an object’s scale (length,
width and thickness), we instead calculate a set of characteristic scales
using the so-called _Minkowski functionals_ [27] that measure the topology of
arbitrary structures in $N$-dimensional space (see review by Mecke [25] for
details). These have been used in other physical applications to characterise
for example dissipative structures in magnetohydrodynamic turbulence [53],
galaxy distribution [40] and cosmological structure formation [38, 41]. In
three dimensions the Minkowski functionals are

$\displaystyle V_{0}$ $\displaystyle=V=\int dV,$ (6) $\displaystyle V_{1}$
$\displaystyle=\frac{A}{6}=\frac{1}{6}\int dS,$ (7) $\displaystyle V_{2}$
$\displaystyle=\frac{H}{3\pi}=-\frac{1}{6\pi}\int\nabla\cdot\hat{\mathbf{n}}dS=\frac{1}{6\pi}\int(\kappa_{1}+\kappa_{2})dS,$
(8) $\displaystyle V_{3}$
$\displaystyle=\frac{1}{4\pi}\int(\kappa_{1}\kappa_{2})dS=\chi,$ (9)

where $\hat{\mathbf{n}}$ is the surface normal, $\kappa_{1}$, $\kappa_{2}$ are
the maximum and minimum local curvature, and $\chi$ the Euler characteristic
(essentially related to the number of holes through an object). To evaluate
these integrals numerically on the discrete output from the large-eddy
simulations, we use Crofton’s formula (see Appendix B), which provides
discrete approximations for terms (for example the surface normal) which are
otherwise difficult to evaluate on objects constructed from individual cubic
volumes of the simulation underlying grid. For reference, the Minkowski
functionals for a parameterised spheroid and ellipsoid will be shown, using
the analytical expressions (where available, and otherwise numerical
integration) for surface area and mean curvature given in Schmalzing et al.
[41] and Poelaert et al. [33].

From these functionals a characteristic _length_ ($L_{m}$), _width_ ($W_{m}$)
and _thickness_ ($T_{m}$) can be calculated as

$L_{m}=\frac{3V_{2}}{4V_{3}},W_{m}=\frac{2V_{1}}{\pi
V_{2}},T_{m}=\frac{V_{0}}{2V_{1}},$ (10)

where the normalisation is such that all of these measures equal the radius
when applied to a sphere. Note that for objects with one or more holes
$V_{3}=\chi\leq 0$ and so the length will not be evaluated for these objects.
The length, width and thickness may be further summarised by computing the
_filamentarity_ ($F_{m}$) and _planarity_ ($P_{m}$)

$F_{m}=\frac{L_{m}-W_{m}}{L_{m}+W_{m}},P_{m}=\frac{W_{m}-T_{m}}{W_{m}+T_{m}},$
(11)

which in turn indicate whether an object is more _stick_ -like (large
filamentarity) or _pancake_ -like (large planarity). The Minkowski functionals
thus enable the quantification of an object’s shape, making it possible to,
for example, investigate whether the objects that are the primary contributors
to vertical transport have a characteristic shape, which can be used to inform
the formulation of an integral model for representing this transport.

To provide a reference for the range of values which may be expected for
coherent structures in the atmospheric boundary layer, Fig. 4 is a
filamentarity vs planarity plot for a number of synthetically created,
numerically integrated, sheared and stretched spheroids. The length (through
the center of the structure) was kept constant across all shapes while varying
the width and shearing distance. Each data point is marked by an outline of
the shape’s structure by plotting the vertical cross-section through each
shape’s symmetry plane. As a means of reference for the numerically integrated
shapes the analytical functions for filamentarity and planarity are plotted
for a spheroid, by integrating the analytical forms of the Minkowski
functionals for a spheroid while varying the aspect ratio between one axis and
the two others. In addition to the spheroid reference lines, the deformation
of a spheroid through a general ellipsoid while keeping the volume and major
axis length constant is provided, including the aspect ratio of the two
remaining axis as $\alpha$. It can for example be seen that a prolate spheroid
with $\lambda=2$ becomes an oblate spheroid with $\lambda=1/4$ (the Minkowski
functionals are independent of orientation, which will be dicussed next).

Figure 4:  Filamentarity vs planarity plot for synthetically created 3D
thermal structures with centerline length $500\text{\,}\mathrm{m}$ and varying
shear and distance (left). For each structure the data point is marked with a
vertical cross-section (black). Two sets of reference lines are given using
analytical solutions to the Minkowski functionals of a spheroid (solid blue)
and ellipsoid (dotted blue). $\lambda$ parameterises the aspect ratio of the
spheroid semi- and major axis. $\gamma$ parameterisises the deformation of a
spheroid with a fixed value $\lambda$ and fixed major axis, by varying the
remaining two axis (with aspect ratio $\gamma$). The filamentarity-planarity
values effectively enable the projection of an object’s length ($L$),
thickness ($T$) and width ($W$) into two coordinates (right).

##### Object tilt and orientation

Figure 5:  Voxel-rendering of single coherent structure from sheared
simulation from two viewing angles, with object orientation angles $\theta$
(tilt from $z$-axis) and $\phi$ (xy-orientation from $x$-axis) and centerline
(red) indicated.

As the Minkowski functionals (detailed in the previous section) only provide
measures of scale, but not orientation, of individual objects, we introduce a
means of calculating the $xy$-plane orientation ($\phi$, measured from the
$x$-axis) and tilt ($\theta$, measured from the $z$-axis) of an individual
object (see Figure 5). The characteristic tilt and orientation for an
individual coherent structure is calculated by first forming a _centerline_
through the centroids of vertically adjacent slices of a given structure, and
then computing the area-weighted angular average of $z$-tilt and
$xy$-orientation along this centreline (see appendix C for details).

##  4 Results

In this section we analyse the sheared and non-sheared simulations using the
methods detailed above. The analysis first focuses on extracting
characteristic length-scales of different fields in the boundary layer as a
whole, without attempting to identify individual coherent structures. Later
the boundary layer air that is likely to be feeding convective clouds is
identified. And lastly, properties of _individual_ coherent structures are
studied with the aim of revealing the properties of coherent structures that
dominate the vertical moisture flux. We focus here on the moisture flux as the
simulations represent convection over the ocean and so have a low Bowen ratio.
Similar analysis may be carried out for heat flux in conditions of high Bowen
ratio.

###  4.1 Vertical profiles of characteristic horizontal scales

To study the vertical transport produced by coherent structures in the
boundary layer, we must first formulate how to define these structures.
Unfortunately the different scalar fields that are connected with transport
relevant to moist convection (vertical velocity, buoyancy, moisture and heat
content), vary on very different length-scales and these scales change with
height in the boundary layer. To demonstrate this variability, Figure 6 and 7
show the 2nd cumulant of vertical velocity with itself ($c_{w,w}$, the auto-
correlation) and the 2nd cumulant of vertical velocity and liquid water
potential temperature ($c_{w,\theta_{l}}$) in horizontal plane mid, boundary
layer ($z=$300\text{\,}\mathrm{m}$$).

Considering first the auto-correlation of vertical velocity ($c_{w,w}$), we
note that vertical velocity features are elongated with ambient shear and
axisymmetric without shear, as expected. The coherence length-scale (slashed
vertical line) is largest in the middle of the boundary layer where vertical
velocity peaks, before thermals are decelerated becoming negatively buoyant in
the relatively warm and dry layer below cloud.

Considering instead $c_{w,\theta_{l}}$, there is narrow length-scale of
positive correlation until $z\approx$200\text{\,}\mathrm{m}$$, embedded within
a larger-scale negative correlation. The positive correlation is provided by
buoyancy, in turn induced by sensible surface heat fluxes. However given that
RICO represents shallow convection over the ocean (making the Bowen ratio
small), the buoyancy becomes dominated by water vapour, and above
$z\approx$200\text{\,}\mathrm{m}$$, the correlation with potential temperature
becomes negative. This transition causes the correlation length-scale to
increase with height until $z\approx$200\text{\,}\mathrm{m}$$, above which the
larger-scale negative correlation takes over. In simulations with ambient
shear, the correlation between vertical velocity and temperature is not only
elongated in the direction of shear, but the correlation is asymmetric in the
direction of shear (evidenced by the skewness and offset of the correlation).
This means that potential temperature features are displaced in the downwind
direction relative to the vertical velocity, suggesting that the similarity
solutions that assume radially symmetric distributions of different scalar
fields (i.e. all scalar fields are assumed to be a function of a single radius
$r$), as most plume-based models do [e.g. 12], may not be valid in conditions
where shear is present.

Figure 6:  2nd cumulant of vertical velocity with vertical velocity at
increasing heights in the boundary layer in simulations without shear
(leftmost two columns) and with shear (rightmost two columns). At each height
the cumulant (left, measuring coherence as a function of distance) is
associated with the same cumulant sampled along the identified principle and
perpendicular direction of coherence (right), with the calculated coherence
length-scale indicated with slashed vertical lines. Note the magnitude of
coherence changes with height. For each subfigure the scaling of the
correlation magnitude is written above.  Figure 7:  2nd cumulant of vertical
velocity with liquid water potential temperature (panels as in Figure 6)
Figure 8:  Horizontal coherence length-scales of vertical velocity ($w$),
water vapour ($q_{v}$), liquid water potential temperature ($\theta_{l}$),
cloud water ($q_{c}$), virtual liquid water potential temperature
($\theta_{l,v}$) and the radiative passive tracer ($\phi$), for simulations
with shear (solid lines) and without shear (dashed lines).

To examine the variation in correlation length-scale with height, the
cumulant-based coherence calculation was carried out at every model level in
the boundary layer. Fig. 8 shows the autocorrelation length-scale as a
function of height for both the sheared and non-sheared simulations. In the
discussion below, subscript $S$ and $NS$ will be used to denote properties
extracted from the sheared and non-sheared simulations respectively. Comparing
first the characteristic length-scales across different fields, the vertical
velocity is consistently confined to narrower features, whereas moisture and
sensible heat organise on larger scales [consistent with 37]. The
characteristic length-scale of the radioactive tracer field is generally
larger than that for vertical velocity, possibly owing to the tracer
concentration retaining high values in over-turning vortices of thermals
within which the vertical velocity becomes negative.

In both simulations, with and without shear (though more prominent with
shear), the vertical velocity length-scale increases with height from the
surface as thermals are accelerated by the buoyancy provided through surface
fluxes, reaching a maximum scale of
$L_{w,w}^{p}\big{\rvert}_{NS}\approx$300\text{\,}\mathrm{m}$$ and
$L_{w,w}^{p}\big{\rvert}_{S}\approx$1300\text{\,}\mathrm{m}$$ at
$z\approx$400\text{\,}\mathrm{m}$$. Above this height, velocity scale
stagnates (no shear) or decreases (with shear) with height, as the buoyancy of
the rising coherent structures becomes negative and the structures begin
decelerating through the CIN-layer. This effect of buoyancy on the correlation
length-scale is more clearly seen when investigating the cross-correlations
with vertical velocity (see below). Comparing the moisture and virtual liquid
potential temperature (buoyancy) length-scales with the radioactive tracer, it
is notable that although the buoyancy is water-vapour dominated, the length-
scales of correlation of water vapour
($L_{q_{v},q_{v}}^{p}\big{\rvert}_{NS}\approx$600\text{\,}\mathrm{m}$$ and
$L_{q_{v},q_{v}}^{p}\big{\rvert}_{S}\approx$500\text{\,}\mathrm{m}$$ at
$z\approx$300\text{\,}\mathrm{m}$$) are significantly larger than those of
both the radioactive tracer
($L_{\phi,\phi}^{p}\big{\rvert}_{NS}\approx$400\text{\,}\mathrm{m}$$ and
$L_{\phi,\phi}^{p}\big{\rvert}_{S}\approx$300\text{\,}\mathrm{m}$$ at
$z\approx$300\text{\,}\mathrm{m}$$) and than the buoyancy features
($L_{\theta_{l,v},\theta_{l,v}}^{p}\big{\rvert}_{NS}\approx$500\text{\,}\mathrm{m}$$
and
$L_{\theta_{l,v},\theta_{l,v}}^{p}\big{\rvert}_{S}\approx$500\text{\,}\mathrm{m}$$
at $z\approx$300\text{\,}\mathrm{m}$$) in the lower-half of the boundary layer
(until $z\approx$300\text{\,}\mathrm{m}$$). This means that larger-scale water
vapour variability is producing buoyancy on a shorter length-scale, which in
turn is accelerating boundary layer air on an even shorter length-scale. This
is important for modelling, as the variability of the buoyancy scalar (here
water vapour) cannot simply be used to infer the scale of coherent rising
structures. Lastly, the maximum correlation length-scale of cloud water (a
measure of cloud-size) is on the order of
$L_{q_{c},q_{c}}^{p}\big{\rvert}_{NS}\approx$400\text{\,}\mathrm{m}$$ and
$L_{q_{c},q_{c}}^{p}\big{\rvert}_{S}\approx$300\text{\,}\mathrm{m}$$ at
$z\approx$650\text{\,}\mathrm{m}$$ (near cloud-base), showing most similarity
to the radioactive tracer coherence length-scale at cloud-base.

The direction of longest correlation distance (see Fig. 9) is for all fields
(including cloud water) oriented with the ambient wind-direction, which is
also the principle direction of shear, throughout the boundary layer and into
cloud-base. In the absence of shear, some fields demonstrate some degree of
turning with height, but the low degree of asymmetry ($r_{a}<2$ at all
heights) suggests that there is very little direction to the correlation and
on examining multiple timesteps the direction of correlation was seen to be a
transient feature, so that no preferential orientation can be discerned.

Figure 9:  Direction of longest coherence as angle from x-axis (for the
variables shown in Fig. 8) with the degree of asymmetry given by the marker
symbol. In the presence of ambient shear all fields are elongated in the
direction of shear. In particular, note that cloud structures (forming at
$z\approx 650$\mathrm{m}$$ are elongated in the same direction as boundary
layer structures.

Applying the same analysis to the vertical fluxes of heat, moisture, buoyancy
and radioactive tracer (Fig. 10) we see that these features are generally
narrower than the scalar being transported, suggesting that the scale of
vertical velocity is dominating the length-scales of vertical transport. All
flux fields show clear elongation by ambient shear. The heights at which the
correlation is negative is marked by minus-sign markers ("-") showing again
the transitions for heat (at $z\approx$200\text{\,}\mathrm{m}$$ the heat flux
changes sign) and buoyancy (at $z\approx$500\text{\,}\mathrm{m}$$ rising
structures are no longer buoyant) where the correlation-length for both
collapses. The moisture and radioactive tracer flux correlations
($c_{w,q_{v}}$ and $c_{w,\phi}$) have near-monotonic increases in size with
height until cloud-base is approached.

Figure 10:  Horizontal coherence length-scales, in simulations with (orange
lines) and without (blue lines) shear, of vertical velocity with water vapour,
liquid water potential temperature, buoyancy (virtual liquid water potential
temperature), cloud condensate and radioactive tracer effectively producing a
coherence length-scale for the vertical flux for each scalar field. Heights at
which anti-correlation occurs are marked with a minus-sign ("-").

###  4.2 Cross-correlation of scalar fields

Figure 11:  Joint distributions of total water vapour specific humidity and
liquid potential temperature (same as potential temperature in the sub-
saturated boundary layer below cloud) in horizontal cross-sections at
increasing heights in the boundary layer, together with properties of air
entering through cloud-base of newly formed
($t_{age}<$3\text{\,}\mathrm{min}$$) clouds in simulation with (bottom) and
without (top) shear, both without (left) and with (right) the radioactive
tracer mask applied. For each height and at cloud-base, the inner and
outermost contours identify the regions which cumulatively contain the
$10\text{\,}\mathrm{\char 37\relax}$ and $90\text{\,}\mathrm{\char 37\relax}$
highest point density, with the number of bins in $\theta_{l}$ and $q_{t}$
scaling by the number of points, $N_{bins}=(N_{x}\times N_{y})^{1/4}$. Grey
slashed lines are centred on peak of distributions for cloud-base entering
air. From a relatively moist and warm near-surface state the distribution
first dries and cools into the bulk of the boundary layer, then warms and
dries with height as air subsiding through the boundary layer top imparts a
stronger influence.

As shown above, different scalar fields show different length-scales of
coherence and we now turn to investigating the extent to which these fields
are correlated, not in space but in the distributions of their scalar values.
In Fig. 11 for the simulations with (right) and without shear (left), the
joint distribution of water vapour and potential temperature through
horizontal cross-sections at increasing heights are plotted. At each height a
bivariate histogram was created, the bins ranked by number of points and
contours drawn around the bins with 10% (inner contour, solid lines) and 90%
(outer contour, stippled lines) cumulative point count. Constructed in this
way the set of contours at each height identify the centre and width of the
joint distribution for the scalar fields visualised (forming a 2D box-and-
whisker plot). For reference, the boundary layer mean values for potential
temperature and moisture are included. In addition to the distributions in
horizontal cross-sections, the joint distribution for points immediately below
(grid-spacing $\Delta z=$25\text{\,}\mathrm{m}$$) newly-formed
($t_{age}<$3\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$$) clouds (identified by
cloud-tracking) is included.

The distributions have similar characteristics in the two simulations; near
the surface the boundary layer is warm and moist relative to the boundary
layer mean, through the super-adiabatic near-surface the distribution rapidly
becomes cooler and drier with height (until
$z\approx$100\text{\,}\mathrm{m}$$). With further increase in height, the
distribution is stretched to drier and warmer values as mixing with warm dry
air subsiding through the boundary layer top has an increasing influence.

In both simulations, the distribution of air entering newly formed clouds (in
red) coincides with the coldest and most humid part of the boundary layer
joint distribution. The cloud-base distribution contains a larger range of
moisture values in the simulation with shear, including drier parts of the
bulk of the boundary layer distribution. This may be because ambient shear
increases mixing into the rising coherent structures, carrying drier air in
the regions with high vertical momentum or because the lower overall virtual
potential temperature (buoyancy) of the boundary layer causes more air to be
buoyant enough to rise to the level of condensation.

In addition, the distributions in the two simulations differ through a
translation in potential temperature by approximately
$\approx$0.1\text{\,}\mathrm{K}$$, which although small is on the order of the
width of the distributions in both cases. This suggests that in the presence
of shear there is a stronger transport of heat out of the boundary layer and
into the cloud layer. This offset indicates that even in conditions where the
prescribed surface fluxes are exactly the same, the dynamics of boundary layer
transport can vary to a degree that alters the property of air that forms
clouds, and so thresholds based solely on the thermodynamic fields (e.g.
$\theta_{l}$ and $q_{t}$) are inadequate in identifying air that will form
clouds.

It is clear from these joint distributions that if the aim is to characterise
coherent structures which actually cause the formation of clouds, it is
inadequate to simply construct a conditional sampling based on threshold
values of the scalar fields causing transport. Instead a method that tracks
air transported from the surface layer and into cloud is required and to
achieve this, in the next section, we employ a radioactively decaying tracer.

###  4.3 Identifying cloud-feeding coherent structures

Figure 12:  Vertical moisture flux decomposed by height and object vertical
extent, decompositing by minimum height (left) and maximum height (right) of
each object in the simulation without shear, with mean flux vertical profiles
given as right marginal plots, and boundary layer mean flux and object number
distribution as top marginal plots. In the moisture flux profile the total
domain mean (orange) is shown together with the moisture flux contributed by
the selected objects (blue). The flux is carried, in aggregate, primarily by
objects extending from the surface and into the cloud-layer, providing flux
throughout the boundary layer.

We now turn to characterising the coherent structures that have the potential
to trigger clouds. This is done by first verifying that the air-mass selected
by conditional sampling using the decaying passive tracer (see 3.2.1 for
details) has thermodynamic properties similar to air entering newly formed
clouds. Secondly, we examine the vertical moisture transport by individual
coherent structures (identified by splitting the conditionally sampled air) as
a function of the vertical extent of each structure· This flux decomposition
will show that for coherent structures identified using the decaying passive
tracer, it is the structures which extend from near the surface, through
cloud-base and into cloud, which carry the bulk of the vertical moisture
transport, suggesting that the clouds manifest as billowing tree-crowns sat
atop trunks of continuous transport.

Examining first the thermodynamic properties of air sampled using the decaying
passive tracer, the joint distributions of the previous section are
conditionally sampled (Fig. 11 right), by requiring that the concentration of
decaying passive tracer is at least 2 standard deviations from the horizontal
mean (see Sec. 3.2.1 for details). Using this method for both the sheared and
non-sheared simulations, the distributions collapse down to align near-
perfectly with that of air entering through cloud-base, indicating that the
radioactive tracer is picking exactly the air that may trigger clouds. In
addition, for both cases the means of the distributions appear to be near-
linearly translated with height suggesting that these coherent cloud-
triggering structures mix with the bulk of the boundary layer at a similar
rate for both potential temperature and water vapour. Finally, the widths of
the joint distributions appear nearly unchanged with height. All of these
facts are encouraging for the prospect of parameterisation of the mixing into
coherent updrafts in the boundary layer, by which the properties of cloud-
triggering air may be predicted based on the surface fluxes (and other
external forcing factors).

Although the radioactive tracer method identifies air with the same
statistical properties as air that is triggering clouds, it is not guaranteed
that every volume of this boundary layer air will actually trigger a cloud.
Some structures may simply be too small to survive the journey to the
condensation level without being completely mixed into the bulk boundary layer
air. This can be addressed in further work by tracking boundary layer
structures and identifying which ones actually trigger clouds (beyond the
scope of this study). Here, we instead identify the structures that dominate
the vertical moisture transport.

We examine this next by decomposing the moisture flux at a given height by the
vertical extent (measured as the height of the bottom $z_{min}$ and the top
$z_{max}$) of coherent structures contributing to the flux in Figure 12. Both
the simulations with and without shear show the same structure, and so we only
show the situation without shear here. Note that an individual object of a
given vertical extent, will likely contribute to the flux at a number of
different heights and the figure simply shows at any given height how high
(Figure 13 right) or low (Figure 12 left) structures which contribute to the
flux extend.

Considering first the flux decomposition by the minimum height of each
coherent structure (Figure 12 left) we find that almost all flux is carried by
structures extending down to the surface
($z\approx$\mathrm{1}\mathrm{0}\mathrm{0}${m}$, even though this accounts for
only half of the number of coherent structures. Considering secondly the
maximum height of the coherent structures we find that $\approx 90\%$ of the
moisture flux through the boundary layer is transported by structures which
extend above cloud-base ($z_{cb}\approx$\mathrm{6}\mathrm{5}\mathrm{0}${m}$),
transporting moisture all the way from near the surface into and through
cloud-base. The flux is dominated by structures terminating at
$z\approx$\mathrm{9}\mathrm{0}\mathrm{0}${m}$ with a second peak at
$z\approx$\mathrm{1}\mathrm{8}\mathrm{0}\mathrm{0}${m}$, possibly accounting
for clouds reaching and passing the height of free convection [forced and
active clouds as denoted by 49]. From the above analysis we conclude that were
we to characterise coherent structures solely by the passive tracer we would
be characterising in part the properties of structures doing transport in
cloud. As we are concerned here primarily with the characteristics of
structures that effect transport below cloud, we will in later sections crop
structures at cloud-base so that only the sub-cloud part is considered.

###  4.4 Minkowski characteristics of coherent structures

Having identified the below-cloud coherent structures with correct
thermodynamic properties (for triggering clouds) and carrying the majority of
the vertical moisture, we next calculate characteristic properties of each
structure and decompose the boundary layer moisture flux by the object scales,
to ascertain the characteristics of structures dominating the flux. This flux
decomposition is visualised as a (boundary layer mean) moisture-flux weighted
probability density distribution (PDF) [computed using Gaussian Kernel Density
Estimation 19], with weights given by the per-object total moisture-flux and
the PDF computed over the object characteristics used in the decomposition. To
demonstrate the effect of external forcing the characteristics of structures
dominating the flux in the simulations with and without shear will be
contrasted. To do this we calculate a characteristic length, width and
thickness for each object using the so-called Minkowski functionals (detailed
in Sec. 3.2.2).

We first examine the boundary layer moisture flux when decomposed by the width
and thickness of the coherent structures (Figure 13 left) with contours in red
and blue depicting the distribution in the cases with and without shear
respectively. The scatter-plot, depicting the width and thickness of each
structure, shows that the majority of structures are small and nearly axially
symmetric (appear near the unit line). However, once we consider the
contribution to the vertical moisture flux the larger and more asymmetric
structures dominate. This asymmetry is more pronounced when ambient shear is
present, shifting the width/thickness ratio from
$\approx$\mathrm{1}\mathrm{5}\mathrm{0}${m}/$\mathrm{2}\mathrm{0}\mathrm{0}${m}=3:4$
to
$\approx$\mathrm{1}\mathrm{0}\mathrm{0}${m}/$\mathrm{2}\mathrm{0}\mathrm{0}${m}=1:2$.

Figure 13:  Minkowski functional derived width vs thickness (left) and
filamentarity vs planarity (right) of individual objects (points) for the
cases with (red) and without (blue) shear. The contours depict the
distribution of vertical moisture flux with these object characteristics
(computed as a flux-weighted probability density functioned constructed using
Gaussian Kernel Density Estimation). In the filamentarity vs planarity plot a
spheroid is given as reference reference (solid black) with independent axis
parameterised by $\lambda$ and deformation of a spheroid through a general
ellipsoid with fixed major axis and varying aspect between the remaining two
axes ($\alpha$).

This asymmetry can be succinctly captured by computing the filamentarity and
planarity (measuring how _pencil_ -like or _disc_ -like object each is) as
seen in Figure 13 right, showing that ambient shear causes the coherent
structures to be stretched planar, which can be seen by a
$\approx$100\text{\,}\mathrm{\char 37\relax}$$ increase in planarity from
$P_{NS}\approx 0.15$ and $P_{S}=0.35$. As a reference (in black), the
filamentarity and planarity of an ellipsoid with varying elongation
(parameterised as the aspect ratio between one axis and the remaining two) the
coherent structures can be seen to move from being more cylindrical to sheet-
like in shape. Here only structures with Euler characteristic $\chi=1$
(objects without holes) are considered (as the length is otherwise not
defined). Although objects with holes do contribute a significant fraction of
the moisture flux ($\approx 20\%$ and $\approx 70\%$ for the cases without and
with shear respectively) this fraction is reduced when the passive tracer
threshold is increased (to $\approx 4\%$ and $\approx 30\%$ for $\phi=3$) and
the relative stretching between the two cases becomes more pronounced (not
shown here).

###  4.5 Object orientation

Figure 14:  Vertical moisture flux decomposed (as in Figure 13) by xy-
orientation angle ($\phi$) against z-axis slope angle ($\theta$) of coherent
structures present at $t=$6\text{\,}\mathrm{h}$$ in simulations with (red) and
without shear (blue), together with distributions in each along plot margins

In addition to knowing the characteristic length-scales and shape of
individual coherent structures, it is instructive to determine the tilt and
orientation of each object to be able to formulate an integral model to
represent transport by coherent structures. In Figure 14, the vertical
moisture flux has been decomposed by the tilt and orientation angles of all
sub-cloud objects present at $t=$6\text{\,}\mathrm{h}$$ in both the simulation
with and without shear. This shows the presence of ambient shear
($\approx$2\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}$$ change in wind-
speed over the sub-cloud layer) caused the mean tilt of individual objects to
change from
$\theta_{no,shear}\approx$15\text{\,}\mathrm{\SIUnitSymbolDegree}$$ (near-
vertical, given the near symmetrical angular spread in orientation angle) to
$\theta_{shear}\approx$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$ and changed
the structures from having no preferential horizontal orientation, to being
oriented with direction of wind shear
($\phi_{shear}\approx$210\text{\,}\mathrm{\SIUnitSymbolDegree}$)$.

This direction of orientation in the presence of ambient wind was found to
coincide with the principle direction of coherence identified using cumulants
(subsection 4.1). Being able to quantify the stretching and tilting of
individual structures is an improvement on measure of elongation provided by
using cumulants as we are able to separate out cumulant elongation due to
individual structures from that caused by structures being spatially organised
into linear features.

##  5 Discussion

The two methods presented here, the first focusing on bulk-measures of
coherence in the boundary layer (using cumulants) and the second on
identifying and quantifying morphological properties of flux dominating
individual coherent structures (using Minkowski functionals and
tilt/orientation calculation), have complementary strengths.

The principal difference between the two approaches is that the cumulant-based
method produces a length-scale estimate through considering the spatial
coherence throughout the fluid, whereas identifying and characterising
individual objects gives an estimate of scale for individual coherent
structures. Two individual objects in close proximity will increase the
cumulant length-scale estimate, meaning that the cumulant length-scale
conflates the object size and spatial organisation of objects (specifically
the inter-object distance). In addition, in cases where there are multiple
populations of coherent structures, the cumulant method will conflate these
into one integrated measure, and the relevant details of each separate
population may be lost. In particular, in the upper part of the boundary layer
which is characterised by ascending buoyant thermals and descending
entrainment flows, the cumulant analysis will not necessarily give a "clean"
description of either of these.

Another aspect by which the two approaches differ is how the cumulant method
estimates length-scales in the horizontal plane (the method requires
translational symmetry to study coherence as a function of displacement and so
cannot be applied vertically without picking a reference height), whereas the
Minkowski length-scales of individual objects are not constrained in the
orientation in which these length-scales are calculated. This means that the
two measures of length cannot be directly compared, as the Minkowski length-
scales are not measured in the horizontal plane and necessitate the
calculation of object orientation to interpret the length-scales calculated.
It further means that the cumulant-based measure of scale cannot separate out
the orientation of individual structures from the orientation of spatial
organisation in general. The degree of tilt could though be measured with
cumulants by computing correlation between different heights with distance in
the horizontal plane (not shown here).

Both techniques demonstrate that the spatial characteristics of coherent
boundary layer structures are affected by the ambient wind. With the cumulant-
based technique this manifests as an elongation across all scalar fields in
the direction of shear. Through decomposition of the vertical moisture flux by
the vertical extent of each structure, we showed that the majority of the flux
is carried by structures which extend from the near-surface and into the cloud
layer. This suggests that these structures are more plume-like (providing
near-continuous flux as a function of height) rather than thermal-like. The
ambient wind causes these flux-dominating coherent structures from being near
axisymmetric to being stretched planar and tilted in the direction of ambient
wind.

These findings suggest that models representing non-local transport by
coherent boundary layer structures, should choose the plume model as the
fundamental starting point. The effect of ambient shear appears to be to
organise vertical transport into coherent structures which are tilted and
stretched planar (both of which increase entrainment by increasing the surface
area), and at the same time organised into linear features (which may decrease
entrainment by limiting the dry air reaching individual plumes).

##  6 Conclusions and further work

This paper has demonstrated two methods by which to characterise the shape,
size and orientation of coherent structures. The first method quantifies the
horizontal orientation and length-scale of coherence between any two scalar
fields, and through this makes it possible to measure the coherence in the
boundary layer as a whole. The second method identifies cloud-feeding coherent
structures using a decaying passive tracer and is able to quantify length-
scales and orientation (both vertical and horizontal) for each of these
objects, allowing for a more instructive decomposition where the non-local
transport by individual coherent structures can be studied.

We have demonstrated the use of cumulants to measure characteristic length-
scales for different scalar fields and fluxes of these fields in the bulk of
the boundary layer. This showed that in the absence of shear, vertical
velocity features were significantly narrower
($\approx$200\text{\,}\mathrm{m}$$) than the moisture and potential
temperature fields ($\approx$1000\text{\,}\mathrm{m}$$), but of a similar
scale to a surface-released passive tracer
($\approx$300\text{\,}\mathrm{m}$$). This method is also able to quantify the
elongation of spatial coherence and calculate the elongation direction,
showing how the presence of ambient wind shear causes elongation (along with
the wind-direction in this case) of the vertical velocity field and a less-
pronounced change to the heat and moisture fields.

We additionally demonstrated how a surface-released radioactive tracer may be
used to identify air with thermodynamic properties statistically similar to
air entering through cloud-base of newly-formed clouds. This allowed for the
identification of individual cloud-feeding coherent structures. By decomposing
the vertical flux by the vertical extent of each structure it was found that
the vertical flux is dominated by structures extending from the surface,
through cloud-base and into cloud. This insight suggests that flux-dominating
structures are more plume-like (carrying transport from the surface and
throughout the boundary layer) rather than thermal-like. The part of these
structures carrying out transport in the boundary layer were then
characterised using Minkowski functionals (producing a characteristic length,
width and thickness for each object) and a technique for calculating an
object’s tilt and horizontal orientation. With these methods it was showed
that although the majority of coherent structures are closest to a sphere in
shape, the structures which dominate the vertical flux in the boundary layer
are generally thicker than they are wide ($4:3$ ratio); an asymmetry which
increases in the presence of ambient shear ($2:1$) as the structures are
stretched planar. In absence of shear the structures do generally exhibit some
degree of tilt ($\theta\approx$10\text{\,}\mathrm{\SIUnitSymbolDegree}$$), but
this has no particular orientation, whereas in the presence of shear objects
were more tilted ($\theta\approx$40\text{\,}\mathrm{\SIUnitSymbolDegree}$$)
and all in the direction of the ambient wind. These findings cannot be made
through a bulk-estimate of length-scale of coherence, as was done in previous
work, but requires identifying and measuring individual coherent structures.

These findings suggest that when formulating models to represent transport by
cloud-feeding coherent boundary layer structures, the most appropriate model
may be a plume, at least for the kind of (shallow moist convective) boundary
layer studied here. In the presence of shear, the planar stretching and
tilting [6, possibly increasing the entrainment of dry air] of these
structures may be necessary to consider when constructing a parameterisation
of boundary layer transport. Further study can explore the generality of this
conclusion to other boundary layer configurations, e.g. land-based,
heterogeneously forced, deep convective etc. Similarly the techniques
presented herein may be used to study the relationship between the spatial
characteristics of coherent boundary layer structures and the spatial
characteristics of the clouds they form.

This work used the ARCHER UK National Supercomputing Service
(www.archer.ac.uk) and was funded through the NERC/Met Office Joint Programme
"Understanding and Representing Atmospheric Convection across Scales
(ParaCon)", grant NE/N013840/1. The software implementation of the techniques
presented here are available in Denby and Liberzon [11].

##  Appendix A Time-scale of convective overturning in boundary layer

Using the sub-cloud characteristic velocity scale $w_{*}$ [as in 17, but
corrected for the contribution to buoyancy from water vapour] and the boundary
layer depth $z_{BL}$ we can calculate a sub-cloud convective overturning time-
scale $\tau_{BL}$ as

$\tau_{BL}=\frac{z_{BL}}{w_{*}}$ (12)

The sub-cloud convective velocity scale is given as

$w_{*}=\left(\left.\overline{w^{\prime}b^{\prime}}\right|_{0}z_{BL}\right)^{1/3},$
(13)

with buoyancy flux

$\left.\overline{w^{\prime}b^{\prime}}\right|_{0}=\frac{g}{T_{v,0}}\left.\overline{w^{\prime}\theta_{v}^{\prime}}\right|_{0},$
(14)

and virtual potential temperature flux (approximately) given by [37]

$\left.\overline{w^{\prime}\theta_{v}^{\prime}}\right|_{0}=\left.\overline{w^{\prime}\theta^{\prime}}\right|_{0}+\frac{R_{v}}{R_{d}}\overline{\theta}_{0}\left.\overline{w^{\prime}q^{\prime}}\right|_{0},$
(15)

where $z_{BL}$ is the boundary layer depth, $g$ the gravitational acceleration
and $|_{0}$ denotes surface values. With surface fluxes for sensible
$\left.\overline{w^{\prime}\theta^{\prime}}\right|_{0}=\frac{F_{s}}{\rho_{0}c_{p,d}}$
and latent heat
$\overline{\theta}_{0}\left.\overline{w^{\prime}q^{\prime}}\right|_{0}=\frac{F_{v}}{\rho_{0}L_{v}}$,
surface moisture $q_{v,0}=$15\text{\,}\mathrm{g}\text{\,}{\mathrm{kg}}^{-1}$$,
surface temperature $T_{0}=$300\text{\,}\mathrm{K}$$ and diagnosed boundary
layer depth $z_{BL}=$650\text{\,}\mathrm{m}$$ (here taken as the cloud-base
height) the convective overturning time-scale becomes
$\tau_{BL}=$16\text{\,}\mathrm{m}\mathrm{i}\mathrm{n}$$. The constants used
above are those for density of dry air
$\rho_{0}=$1.2\text{\,}\mathrm{kg}\text{\,}{\mathrm{}}^{-3}\text{\,}\mathrm{m}$$,
latent heat of vaporisation
$L_{v}=$2.5\text{\times}{10}^{6}\text{\,}\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}$$,
specific heat capacity of dry air
$c_{p,d}=$1005\text{\,}\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{K}}^{-1}$$,
gas constants for dry air
$R_{d}=287$287\text{\,}\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{K}}^{-1}$$
and water vapour
$R_{v}=$461\text{\,}\mathrm{J}\text{\,}{\mathrm{kg}}^{-1}\text{\,}{\mathrm{K}}^{-1}$$.

##  Appendix B Crofton’s formula for discrete integrals

Numerically evaluating the integrals of the Minkowski functionals in 3D
(equations 6 to 9) is non-trivial on discrete 3D masks as these structures are
necessarily blocky and so constructing, for example, the surface normal at a
corner is poorly defined.

Instead of approximating the surface normals, the integral in the Minkowski
functionals can be approximated discretely using Crofton’s formula [41]. This
amounts to counting the number of vertices ($N_{0}$), edges ($N_{1}$), faces
($N_{2}$) and cells ($N_{3}$) on both the interior and exterior of a given
object mask. With these the Minkowski functionals in 3D are given as

$\displaystyle V_{0}$ $\displaystyle=N_{3},\
V_{1}=2\frac{N_{2}-3N_{3}}{9\Delta x},\
V_{2}=2\frac{N_{1}-2N_{2}+3N_{3}}{9\Delta x^{2}},$ $\displaystyle V_{3}$
$\displaystyle=\frac{N_{0}-N_{1}+N_{2}-N_{3}}{\Delta x^{3}}.$

It can be shown for shapes where analytical forms for the Minkowski
functionals exist that these approximate definitions converge to the true
values when $\Delta x\to\infty$. Note that the above approximations assume the
underlying grid to have isotropic grid spacing ($\Delta x=\Delta y=\Delta z$).

##  Appendix C Calculation of slope and orientation of individual objects

The $xy$-orientation angle ($\phi$ measured from the $x$-axis) and tilt angle
$\theta$ (measured from the $z$-axis) are calculated from characteristic slope
scales $\overline{\Delta x},\overline{\Delta y},\overline{\Delta z}$:

$\displaystyle\phi=\arctan 2(\overline{\Delta y},\overline{\Delta x}),\
\theta=\arctan 2(\overline{\Delta l_{xy}},\overline{\Delta z}),\ \Delta
l_{xy}=\sqrt{\overline{\Delta x}^{2}+\overline{\Delta y}^{2}},$

which are evaluated as area-weighted changes in the centroid position

$\displaystyle\overline{\Delta x}=\frac{\Delta
z}{V}\sum_{k}A(z_{k})\frac{x^{c}(z_{k+1})-x^{c}(z_{k-1})}{2},\
\overline{\Delta y}=\frac{\Delta
z}{V}\sum_{k}A(z_{k})\frac{y^{c}(z_{k+1})-y^{c}(z_{k-1})}{2},\
\overline{\Delta z}=\frac{\Delta
z}{V}\sum_{k}A(z_{k})\frac{z^{c}(z_{k+1})-z^{c}(z_{k-1})}{2},$

where $x_{c}(k)$, $y_{c}(k)$ and $A(k)$ are the centroid x and y position, and
area at height-index $k$ respectively, and $V$ the volume, given by

$\displaystyle A(z_{k})=\sum_{i,j}m(x_{i},y_{j},z_{k})\Delta x\Delta y,\
V=\sum_{i,j,k}m(x_{i},y_{j},z_{k})\Delta x\Delta y\Delta z$ $\displaystyle
x^{c}(z_{k})=\frac{\sum_{i,j}x_{i}\
m(x_{i},y_{j},z_{k})}{\sum_{i,j}m(x_{i},y_{j},z_{k})},\
y^{c}(z_{k})=\frac{\sum_{i,j}y_{i}\
m(x_{i},y_{j},z_{k})}{\sum_{i,j}m(x_{i},y_{j},z_{k})}$

for an individual object defined by the mask $m$ and grid-spacing $\Delta
x,\Delta y,\Delta z$.

## References

  * Ait-Chaalal et al. [2016] Ait-Chaalal, F., Schneider, T., Meyer, B. and Marston, J. B. (2016) Cumulant expansions for atmospheric flows.  New Journal of Physics, 18.
  * Berg and Stull [2004] Berg, L. K. and Stull, R. B. (2004) Parameterization of joint frequency distributions of potential temperature and water vapor mixing ratio in the daytime convective boundary layer.  Journal of the Atmospheric Sciences, 61, 813–828.
  * Bony et al. [2017] Bony, S., Stevens, B., Ament, F., Bigorre, S., Chazette, P., Crewell, S., Delanoë, J., Emanuel, K., Farrell, D., Flamant, C., Gross, S., Hirsch, L., Karstensen, J., Mayer, B., Nuijens, L., Ruppert, J. H., Sandu, I., Siebesma, P., Speich, S., Szczap, F., Totems, J., Vogel, R., Wendisch, M., Wirth, M., Siebesma, P., Crewell, S., Wendisch, M., Totems, J., Delanoë, J., Sandu, I., Nuijens, L., Speich, S., Bigorre, S., Chazette, P., Stevens, B., Gross, S., Ament, F., Farrell, D., Bony, S., Hirsch, L., Mayer, B., Ruppert, J. H., Flamant, C., Karstensen, J., Wirth, M., Emanuel, K. and Szczap, F. (2017) EUREC4A: A Field Campaign to Elucidate the Couplings Between Clouds, Convection and Circulation.  Surveys in Geophysics, 38, 1529–1568.  URL: http://link.springer.com/10.1007/s10712-017-9428-0.
  * Brient et al. [2019] Brient, F., Couvreux, F., Villefranque, N., Rio, C. and Honnert, R. (2019) Object-Oriented Identification of Coherent Structures in Large Eddy Simulations: Importance of Downdrafts in Stratocumulus.  Geophysical Research Letters, 46, 2854–2864.
  * Brown and Grant [1997] Brown, A. R. and Grant, A. L. (1997) Non-local mixing of momentum in the convective boundary layer.  Boundary-Layer Meteorology, 84, 1–22.
  * Bursik [2001] Bursik, M. (2001) Effect of wind on the rise height of volcanic plumes.  Geophysical Research Letters, 28, 3621–3624.  URL: http://onlinelibrary.wiley.com/doi/10.1029/2001GL013393/full.
  * Chinita et al. [2018] Chinita, M. J., Matheou, G. and Teixeira, J. (2018) A joint probability density-based decomposition of turbulence in the atmospheric boundary layer.  Monthly Weather Review, 146, 503–523.
  * Couvreux et al. [2010] Couvreux, F., Hourdin, F. and Rio, C. (2010) Resolved Versus Parametrized Boundary-Layer Plumes. Part I: A Parametrization-Oriented Conditional Sampling in Large-Eddy Simulations.  Boundary-Layer Meteorology, 134, 441–458.  URL: http://link.springer.com/10.1007/s10546-009-9456-5.
  * Dawe and Austin [2012] Dawe, J. T. and Austin, P. H. (2012) Statistical analysis of an les shallow cumulus cloud ensemble using a cloud tracking algorithm.  Atmospheric Chemistry and Physics, 12, 1101–1119.
  * Deardorff [1966] Deardorff, J. W. (1966) The Counter-Gradient Heat Flux in the Lower Atmosphere and in the Laboratory.  Journal of the Atmospheric Sciences, 23, 503–506.  URL: http://journals.ametsoc.org/doi/abs/10.1175/1520-0469%281966%29023%3C0503%3ATCGHFI%3E2.0.CO%3B2.
  * Denby and Liberzon [2020] Denby, L. and Liberzon, A. (2020) GENESIS toolkit for analysing atmospheric coherent structures.  URL: https://zenodo.org/record/3956423.
  * Devenish et al. [2010] Devenish, B. J., Rooney, G. G., Webster, H. N. and Thomson, D. J. (2010) The entrainment rate for buoyant plumes in a crossflow.  Boundary-Layer Meteorology, 134, 411–439.
  * Efstathiou et al. [2020] Efstathiou, G. A., Thuburn, J. and Beare, R. J. (2020) Diagnosing Coherent Structures in the Convective Boundary Layer by Optimizing Their Vertical Turbulent Scalar Transfer.  Boundary-Layer Meteorology, 174, 119–144.  URL: https://doi.org/10.1007/s10546-019-00480-1.
  * Etling and Brown [1993] Etling, D. and Brown, R. A. (1993) Roll vortices in the planetary boundary layer: A review.  Boundary-Layer Meteorology, 65, 215–248.
  * Grant and Brown [1999] Grant, A. L. M. and Brown, A. R. (1999) A similarity hypothesis for shallow-cumulus transports.  Quarterly Journal of the Royal Meteorological Society, 125, 1913–1936.  URL: http://doi.wiley.com/10.1002/qj.49712555802.
  * Heus and Seifert [2013] Heus, T. and Seifert, a. (2013) Automated tracking of shallow cumulus clouds in large domain, long duration large eddy simulations.  Geoscientific Model Development, 6, 1261–1273.
  * Holtslag and Nieuwstadt [1986] Holtslag, A. A. and Nieuwstadt, F. T. (1986) Scaling the atmospheric boundary layer.  Boundary-Layer Meteorology, 36, 201–209.
  * Holtslag and Moeng [1991] Holtslag, A. A. M. and Moeng, C.-H. (1991) Eddy Diffusivity and Countergradient Transport in the Convective Atmospheric Boundary Layer.  Journal of the Atmospheric Sciences, 48, 1690–1698.  URL: https://journals.ametsoc.org/jas/article/48/14/1690/22553/Eddy-Diffusivity-and-Countergradient-Transport-in.
  * [19] Jones, E., Oliphant, T., Peterson, P. and Others () SciPy: Open source scientific tools for Python.  URL: http://www.scipy.org/.
  * Jonker et al. [1999] Jonker, H. J. J., Duynkerke, P. G. and Cuijpers, J. W. M. (1999) Mesoscale Fluctuations in Scalars Generated by Boundary Layer Convection.  Journal of the Atmospheric Sciences, 56, 801–808.  URL: http://journals.ametsoc.org/doi/abs/10.1175/1520-0469%281999%29056%3C0801%3AMFISGB%3E2.0.CO%3B2.
  * Khanna and Brasseur [1998] Khanna, S. and Brasseur, J. G. (1998) Three-dimensional buoyancy- and shear-induced local structure of the atmospheric boundary layer.  Journal of the Atmospheric Sciences, 55, 710–743.
  * Lauritzen [2007] Lauritzen, S. L. (2007) Thiele: Pioneer in Statistics.  URL: http://www.oxfordscholarship.com/oso/public/content/maths/9780198509721/toc.html.
  * Lenschow and Sun [2007] Lenschow, D. H. and Sun, J. (2007) The spectral composition of fluxes and variances over land and sea out to the mesoscale.  Boundary-Layer Meteorology, 125, 63–84.
  * Lohou et al. [2000] Lohou, F., Druilhet, A., Campistron, B., Redelspergers, J. L. and Saïd, F. (2000) Numerical study of the impact of coherent structures on vertical transfers in the atmospheric boundary layer.  Boundary-Layer Meteorology, 97, 361–383.
  * Mecke [2000] Mecke, K. R. (2000) Additivity, Convexity, and Beyond: Applications of Minkowski Functionals in Statistical Physics.  In Statistical Physics and Spatial Statistics, 111–184. Berlin, Heidelberg: Springer Berlin Heidelberg.  URL: http://link.springer.com/10.1007/3-540-45043-2_6.
  * Miao et al. [2006] Miao, Q., Geerts, B. and LeMone, M. (2006) Vertical velocity and buoyancy characteristics of coherent echo plumes in the convective boundary layer, detected by a profiling airborne radar.  Journal of Applied Meteorology and Climatology, 45, 838–855.
  * Minkowski [1903] Minkowski, H. (1903) Volumen und Oberfläche.  Mathematische Annalen, 57, 447–495.  URL: http://link.springer.com/10.1007/978-3-322-90190-3_7http://link.springer.com/10.1007/BF01445180.
  * Neggers [2009a] Neggers, R. A. (2009a) A dual mass flux framework for boundary layer convection. Part II: Clouds.  Journal of the Atmospheric Sciences, 66, 1489–1506.
  * Neggers [2009b] — (2009b) A dual mass flux framework for boundary layer convection. Part II: Clouds.  Journal of the Atmospheric Sciences, 66, 1489–1506.
  * Nicholls and Lemone [1980] Nicholls, S. and Lemone, M. A. (1980) The fair weather boundary layer in GATE: the relationship of subcloud fluxes and structure to the distribution and enhancement of cumulus clouds.
  * Park et al. [2018] Park, S. B., Böing, S. and Gentine, P. (2018) Role of surface friction on shallow nonprecipitating convection.  Journal of the Atmospheric Sciences, 75, 163–178.
  * Park et al. [2016] Park, S. B., Gentine, P., Schneider, K. and Farge, M. (2016) Coherent structures in the boundary and cloud layers: Role of updrafts, subsiding shells, and environmental subsidence.  Journal of the Atmospheric Sciences, 73, 1789–1814.
  * Poelaert et al. [2011] Poelaert, D., Schniewind, J. and Janssens, F. (2011) Surface Area and Curvature of the general Ellipsoid.  1–13.  URL: http://arxiv.org/abs/1104.5145.
  * Rauber et al. [2007] Rauber, R. M., Ochs, H. T., Di Girolamo, L., Göke, S., Snodgrass, E., Stevens, B., Knight, C., Jensen, J. B., Lenschow, D. H., Rilling, R. A., Rogers, D. C., Stith, J. L., Albrecht, B. a., Zuidema, P., Blyth, a. M., Fairall, C. W., Brewer, W. A., Tucker, S., Lasher-Trapp, S. G., Mayol-Bracero, O. L., Vali, G., Geerts, B., Anderson, J. R., Baker, B. A., Lawson, R. P., Bandy, A. R., Thornton, D. C., Burnet, E., Brenguier, J.-L., Gomes, L., Brown, P. R. A., Chuang, P., Cotton, W. R., Gerber, H., Heikes, B. G., Hudson, J. G., Kollias, P., Krueger, S. K., Nuijens, L., O’Sullivan, D. W., Siebesma, A. P. and Twohy, C. H. (2007) Rain in Shallow Cumulus Over the Ocean: The RICO Campaign.  Bulletin of the American Meteorological Society, 88, 1912–1928.  URL: http://journals.ametsoc.org/doi/abs/10.1175/BAMS-88-12-1912.
  * Rio and Hourdin [2008] Rio, C. and Hourdin, F. (2008) A thermal plume model for the convective boundary layer: Representation of cumulus clouds.  Journal of the Atmospheric Sciences, 65, 407–425.
  * Rio et al. [2010] Rio, C., Hourdin, F., Couvreux, F. and Jam, A. (2010) Resolved versus parametrized boundary-layer plumes. Part II: Continuous formulations of mixing rates for mass-flux schemes.  Boundary-Layer Meteorology, 135, 469–483.
  * 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, 403–421.
  * Sahni et al. [1998] Sahni, V., Sathyaprakash, B. S. and Shandarin, S. F. (1998) Shapefinders: A New Shape Diagnostic for Large-Scale Structure.  The Astrophysical Journal, 495, L5–L8.
  * Salesky et al. [2017] Salesky, S. T., Chamecki, M. and Bou-Zeid, E. (2017) On the Nature of the Transition Between Roll and Cellular Organization in the Convective Boundary Layer.  Boundary-Layer Meteorology, 163, 41–68.
  * Schmalzing and Buchert [1997] Schmalzing, J. and Buchert, T. (1997) Beyond Genus Statistics: A Unifying Approach to the Morphology of Cosmic Structure.  The Astrophysical Journal, 482, L1–L4.
  * Schmalzing et al. [1999] Schmalzing, J., Buchert, T., Melott, A. L., Sahni, V., Sathyaprakash, B. S. and Shandarin, S. F. (1999) Disentangling the Cosmic Web. I. Morphology of Isodensity Contours.  The Astrophysical Journal, 526, 568–578.
  * Schmidt and Schumann [1989] Schmidt, H. and Schumann, U. (1989) Coherent structure of the convective boundary layer derived from large-eddy simulations.  Journal of Fluid Mechanics, 200, 511–562.
  * Schumann and Moeng [1991] Schumann, U. and Moeng, C.-H. (1991) Plume Fluxes in Clear and Cloudy Convective Boundary Layers.  Journal of the Atmospheric Sciences, 48, 1746–1757.  URL: https://journals.ametsoc.org/jas/article/48/15/1746/22406/Plume-Fluxes-in-Clear-and-Cloudy-Convective.
  * 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, 1918–1937.  URL: http://doi.wiley.com/10.1002/2015MS000489.
  * 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., Sanchez, E., Stevens, B. and Stevens, D. E. (2003) A Large Eddy Simulation Intercomparison Study of Shallow Cumulus Convection.  Journal of the Atmospheric Sciences, 60, 1201–1219.  URL: https://journals.ametsoc.org/jas/article/60/10/1201/103861/A-Large-Eddy-Simulation-Intercomparison-Study-of.
  * Siebesma et al. [2007] Siebesma, A. P., Soares, P. M. M. and Teixeira, J. (2007) A Combined Eddy-Diffusivity Mass-Flux Approach for the Convective Boundary Layer.  Journal of the Atmospheric Sciences, 64, 1230–1248.
  * Stevens et al. [2005] Stevens, B., Moeng, C.-H., Ackerman, A. S., Bretherton, C. S., Chlond, A., de Roode, S., Edwards, J., Golaz, J.-C., Jiang, H., Khairoutdinov, M., Kirkpatrick, M. P., Lewellen, D. C., Lock, A., Müller, F., Stevens, D. E., Whelan, E. and Zhu, P. (2005) Evaluation of Large-Eddy Simulations via Observations of Nocturnal Marine Stratocumulus.  Monthly Weather Review, 133, 1443–1462.
  * Stevens and Seifert [2008] Stevens, B. and Seifert, A. (2008) Understanding macrophysical outcomes of microphysical choices in simulations of shallow cumulus convection.  Journal of the Meteorological Society of Japan, 86A, 143–162.
  * Stull [1985] Stull, R. B. (1985) A Fair-Weather Cumulus Cloud Classification Scheme for Mixed-Layer Studies.  Journal of Climate and Applied Meteorology, 24, 49–56.  URL: http://journals.ametsoc.org/doi/10.1175/1520-0450(1985)024%3C0049:AFWCCC%3E2.0.CO;2.
  * Tobias and Marston [2017] Tobias, S. M. and Marston, J. B. (2017) Three-dimensional rotating Couette flow via the generalised quasilinear approximation.  Journal of Fluid Mechanics, 810, 412–428.  URL: https://www.cambridge.org/core/product/identifier/S0022112016007278/type/journal_article.
  * VanZanten et al. [2011] VanZanten, M. C., Stevens, B., Nuijens, L., Siebesma, A. P., Ackerman, A. S., Burnet, F., Cheng, A., Couvreux, F., Jiang, H., Khairoutdinov, M., Kogan, Y., Lewellen, D. C., Mechem, D., Nakamura, K., Noda, A., Shipway, B. J., Slawinska, J., Wang, S. and Wyszogrodzki, A. (2011) Controls on precipitation and cloudiness in simulations of trade-wind cumulus as observed during RICO.  Journal of Advances in Modeling Earth Systems, 3.
  * Young et al. [2002] Young, G. S., Kristovich, D. A. R., Hjelmfelt, M. R. and Foster, R. C. (2002) Supplement to Rolls, Streets, Waves, and More.  Bulletin of the American Meteorological Society, 83, 1001–1001.  URL: http://journals.ametsoc.org/doi/abs/10.1175/BAMS-83-7-Young.
  * Zhdankin et al. [2014] Zhdankin, V., Boldyrev, S., Perez, J. C. and Tobias, S. M. (2014) Energy dissipation in magnetohydrodynamic turbulence: Coherent structures or "nanoflares"?  Astrophysical Journal, 795.