# Temperature profiles, plumes and spectra in the surface layer of convective
boundary layers

Keith G. McNaughton111Kerikeri, New Zealand<EMAIL_ADDRESS>and Subharthi
Chowdhuri222Indian Institute of Tropical Meteorology, Pune, India

For correspondence<EMAIL_ADDRESS>
###### Abstract

We survey temperature patterns and heat transport in convective boundary
layers (CBLs) from the perspective that these are emergent properties of far-
from-equilibrium, complex dynamical systems. We use the term ‘plumes’ to
denote the temperature patterns, in much the same way that the term ‘eddies’
is used to describe patterns of motion in turbulent flows. We introduce a two-
temperature (2T) toy model to define the cross-sectional areas of plumes, and
so connect the scaling properties of temperature gradients, temperature
variance and heat transport to this area. We then examine temperature ($T$)
probability density functions and $w$-$T$ joint probability density functions,
$T$ spectra and $wT$ cospectra observed both within and above the surface
friction layer. Here $w$ is vertical velocity. We interpret these in terms of
the frequencies at which each value is observed along a horizontal transect,
and so with the cross-sectional areas of the plumes that give rise to them. In
our discussion of $T$ spectra and $wT$ cospectra we focus first on the self-
similarity property of the plumes and flux events above the SFL. The CBL
similarity parameters defined by McNaughton et al (Non-linear Processes in
Geophysics 14, 257-271, 2007) are used throughout. We interpret the $z^{1/2}$
dependence of the mixed length scale for wavenumbers in the $T$ spectra as
reflecting the cross-sectional areas of the plumes, and so with the $z^{-1/2}$
form of the temperature profile, where $z$ is observation height. That is,
mean temperature decreases with height as $z^{-1/2}$ because plume areas
decrease as $z^{-1/2}$. We introduce new scaling results for $T$ spectra and
$wT$ cospectra from within the surface friction layer (SFL), based on a new
analysis of data from the SLTEST experiment. We confirm earlier results
showing that the scaling behaviours of $T$ spectra and $wT$ cospectra change
for heights below $z/z_{s}<0.1$, where $z_{s}$ the height of the SFL, and come
to display properties associated with random diffusion. We conclude by
contrasting our interpretation of the role of buoyancy as a system-wide action
in CBL flows with that of Richardson (Proc. Roy. Soc. London A 87, 354-373,
1920), who proposed a local action and whose ideas inform the current
interpretation of the statistical fluid mechanics model of boundary-layer
flows.

##  1 Introduction

Turbulent transport in convective atmospheric boundary layers depends on the
forms, sizes and energies of the patterns of motion—‘coherent structures’ and
‘eddies’—found within them. These patterns can be regarded as emergent
properties of convective boundary layer (CBL) flows. CBL flows have much in
common with flows in Rayleigh-Bénard convection cells, particularly in
windless conditions, but CBLs are open systems that grow during the daytime as
warmer air is entrained through the capping inversion at their tops. Heat is
therefore introduced through at both the upper and lower boundaries of CBLs,
the eddies within them create patterns of temperature, or ‘plumes’, which are
also emergent properties of these flows. Our task is to describe these eddies
and plumes, and their inter-relationships. The problem is a difficult one, and
must necessarily be based on experiment since the governing equations cannot
be solved. Though experimental observations and results from simulations have
been analyzed in many ways, and much has been learned, the most systematic way
to begin is by empirical scaling analysis allied with the concurrent
development of a conceptual model. That is, we must learn to present these
observations in universal ways. Success in finding appropriate scaling
parameters will go hand-in-hand with understanding what controls the sizes of,
energy and scalar concentrations in, and fluxes carried by particular classes
of eddies and plumes. The forms of the eddies and plumes must be discovered by
other means.

In previous works we have focussed on the scaling properties of spectra and
cospectra, particularly of temperature ($T$) spectra and $wT$ cospectra from
CBLs (McNaughton et al, 2007; Laubach and McNaughton, 2009; Chowdhuri et al,
2019), where $w$ is vertical velocity. The reason is that spectra naturally
sort eddies by size, since the essential non-linearity of the governing
Navier-Stokes equations denies linear superposition of eddies, so eddies of
different kinds cannot coexist unless they are well-separated by size.
Implicit in this argument is that only a small number of distinct emergent
patterns of motion can satisfy all the feedback requirements inherent in this
flow system. Empirical results support this proposition (Perry and Abell,
1975; Perry et al, 1986; McNaughton et al, 2007).

The rationale for our basic set of scaling parameters is given in earlier
papers (McNaughton, 2004; McNaughton et al, 2007; Laubach and McNaughton,
2009). The set has two fundamental length parameters: $\lambda$, representing
the size of the largest turbulent structures in the CBL, and $z$, representing
observation height. It has two parameters to describe the flow of mechanical
energy through the CBL system: the outer dissipation rate, $\epsilon_{o}$,
which is constant with height above the surface friction layer (SFL) (Kaimal
et al., 1976; McNaughton et al, 2007), and the dissipation velocity,
$u_{\epsilon}$. The latter, with $z$, parameterizes the dissipation rate
within the SFL. In practical terms, $\lambda$ is the peak wavelength of the
streamwise velocity ($u$) spectrum, and $u_{\epsilon}=(kz\epsilon)^{1/3}$,
where $\epsilon$ is the dissipation rate at height $z$ near the ground, and
$k$ is the von Kármán constant. We also use the height of the SFL, $z_{s}$,
given by

$z_{s}=\frac{u_{\epsilon}^{3}}{k\epsilon_{o}}$ (1)

The SFL is characterized by the presence of a special class of attached shear
eddies. These develop upwards from the ground until their growth is terminated
by interaction with detached eddies of similar size from the outer Kolmogorov
cascade. Equation (1) represents the height at which inner and outer
dissipation rates match, as explained by McNaughton (2004) and McNaughton et
al (2007). Finally, we use the kinematic heat flux,
$H=\left<w^{\prime}\theta^{\prime}\right>$ where the primes indicate
deviations from the area means of $w$ and $\theta$, to construct a temperature
scale. Buoyancy effects of the heat flux are accounted through the dissipation
rates.

What complicates the picture is that the basic length scales, $\lambda$, $z$
and $z_{s}$, appear not just alone but also in combinations, as mixed length
scales and doubly-mixed length scales. A mixed scale is the geometric mean of
two component length scales, so taking the form $\ell_{1}^{1/2}\ell_{2}^{1/2}$
where $\ell_{1}$ and $\ell_{2}$ are the component length scales. Doubly-mixed
scales follow the same pattern, but with one of the component scales itself
being a mixed scale. We observe that mixed scales are found when one kind of
eddy or plume, with length scale $\ell_{1}$ exists within, and has sizes or
aggregation properties organized by larger eddies with length scale
$\ell_{2}$. It has been found that mixed and doubly-mixed scales often
describe turbulence processes near smooth walls (e.g., Alfredsson and
Johansson, 1984; DeGraaff and Eaton, 2000; Metzger et al., 2001; Buschmann et
al, 2009), with half powers appearing in every case, even while dimensional
consistency requires only that a scale length takes the form
$\ell_{1}^{\alpha}\ell_{2}^{1-\alpha}$, where $\alpha$ has any value. The
possibility of mixed scales puts the search for suitable scales beyond the
reach of simple dimensional analysis.

Figure 1: Dimensionless potential temperature gradient plotted against height,
with each axis scaled according to the Monin-Obukhov similarity model so that
$\phi_{\theta}=(zu_{*}/H)\partial\theta/\partial z$, where $u_{*}$ is the
friction velocity and $z$ is scaled on the Obukhov length $-L$, as defined in
(3). The $-1/2$ power law describes the data over at least a decade of $-z/L$.
Redrawn from Businger et al (1971).

In this paper we survey the scales found to be successful in collapsing
temperature ($T$) spectra and $w$-$T$ cospectra from above and within the SFL,
and interpret them in terms of the known properties of eddies and plumes found
in CBLs. We include a discussion of the temperature profile in this survey,
because its gradient follows a $-1/2$ power law for more than a decade, from
mast top down to deep within the SFL, as shown in Fig. 1. Power laws are of
particular interest here because they indicate a scale invariance that is
consistent with the self-similarity (collapse) of the $T$ spectra and $wT$
cospectra and so with a self-similarity of the underlying turbulent structures
over that range. We find the half power law to be particularly interesting now
that we have become familiar with mixed length scales for wavenumbers in our
spectra and cospectra. Generally, we seek to explain the ensemble-averaged
descriptions of these flows, as observed in experiments, in terms of the known
properties of the underlying, emergent structures that exist in space and time
in CBL flows. That is, we seek to extend our conceptual model of CBL
turbulence and its transport properties beyond what we have attempted
previously.

##  2 Temperature gradients

We begin with mean temperature gradients in the atmospheric surface layer of
CBLs since they are important in themselves, and they open an important window
onto a self-similarity property of plumes near the ground. The temperature
gradient in much of the surface layer of CBLs can be written as

$\frac{zu_{*}}{H}\frac{\partial\overline{\theta}}{\partial
z}=-\alpha\left(\frac{z}{-L}\right)^{-1/2}$ (2)

using the scaling parameters proposed by the Monin-Obukhov similarity theory
(MOST) (Monin and Obukhov, 1954). We use the term ‘surface layer’ to mean the
layer below the mixed layer in CBLs, or roughly the bottom 10% of the CBL. Our
usage is compatible with MOST usage, where the surface layer is defined as the
layer where heat and momentum fluxes are sufficiently constant with height
that they can be represented by their surface values throughout the surface
layer. The SFL is a sub-layer of the surface layer in CBLs. In (2) $u_{*}$ is
the friction velocity, $H=\overline{w^{\prime}\theta^{\prime}}$ is the
kinematic heat flux at the ground, $\overline{\theta}$ is time-averaged
potential temperature, $\alpha$ is a constant and $L$ is the Obukhov length,
defined by

$-L=\frac{u_{*}^{3}T_{0}}{kgH}$ (3)

where $g$ is the acceleration due to gravity and $T_{0}$ is surface
temperature in degrees K. The $-1/2$ power in (2) was established empirically
and is not consistent with MOST itself, which predicts that the profile should
approach a $-1/3$ power law in the limit of $u_{*}=0$ in windless convection,
when $-z/L\rightarrow\infty$ (Businger et al, 1971). The comparison is shown
in Fig. 1. We note that Kader and Yaglom (1990), henceforth KY90, fit two
segments of $-1/3$ power laws to their data from several sources, in
conformity with their three-layer model of the surface layer and the
requirements of MOST, even though the $-1/2$ power law provides a better fit
over the full range $0.04<-z/L<4$ (KY90: Fig.1). Foken and Skeib (1983) report
that the data from the Tsimlyansk site in 1981 follow a $-1/2$ power law for
all $-z/L>0.06$.

The scaling used in (2) is not unique. Power laws indicate scale-free, self-
similar behavior of the underlying structures giving rise to the power law, so
temperature profiles can be scaled in whatever way best suits the context.
Thus the scales $u_{*}$ and $-L$ in (2) can be replaced by those that Laubach
and McNaughton (2009) found to give universal forms to $T$ spectra and $wT$
cospectra within the SFL. Using these scales we have

$\frac{z(z_{s}\epsilon_{o})^{1/3}}{H}\;\frac{\partial\overline{\theta}}{\partial
z}=-\alpha_{1}\left(\frac{z}{z_{s}}\right)^{-1/2}$ (4)

where $\alpha_{1}$ is another positive constant. Here
$u_{\epsilon}=(kz\epsilon)^{1/3}$ and is the value used to calculate $z_{s}$
by (1). The dissipation rate $\epsilon$ is measured deep within the SFL, so
$u_{\epsilon}$ is like $u_{*}$ but includes the effect of variations in
surface shear stress on dissipation rates near the ground, as first discussed
by Townsend (1961). Unlike $L$, $z_{s}$ does not go to zero in windless
convection.

We now have two equations, both describing the same temperature profile but
scaled in two different ways. Both can achieve some success if the parameters
of the two models are statistically correlated. Results from SLTEST (Chowdhuri
et al, 2019) do show a strong correlation between $z_{s}$ and $L$. If this is
generally true then both (2) and (4) can be expected to give fairly good
accounts of observed temperature profiles, despite their very different
foundations. The question is which scaling scheme gives less scatter and the
more ‘universal’ results?

We await direct experimental comparisons, but in the meanwhile we find it more
convenient to work with (4) because its parameters are related to the
geometrical properties of particular classes of eddies and plumes in CBLs, and
it is these properties that we investigate here. Before starting we rewrite
(4) as

$\frac{\partial\overline{\theta}}{\partial\zeta}=-\alpha_{1}\theta_{*}\zeta^{-3/2}$
(5)

where $\zeta=z/z_{s}$ is a dimensionless length, $\theta_{*}$ the temperature
scale $H/(z_{s}\epsilon_{o})^{1/3}$ and $\alpha_{1}$ a constant.

An interesting question is the plausibility of the profile forms (2) and (4)
above the SFL. KY90 discuss a three-layer model of the lower CBL, and they
find it surprising that (2) applies in their convective sublayer where $u_{*}$
is not a relevant parameter, reasoning that since $L$ depends on $u_{*}$ then
$L$ should have no significance there. Equally, our profile expressions, (4)
and (5), employ the length scale $z_{s}$, which we would not expect to be
relevant above the SFL. In our case the problem is resolved when we note that
the height variable in (5) can be rescaled by replacing $z_{s}$ with the outer
length scale, $\lambda$. We can then redefine the scaled height variable as
$\zeta=z/\lambda$, and redefine the temperature scale as
$\theta_{*}=H/(\lambda\epsilon_{o})^{1/3}$ in (5). With these we can rewrite
(5) in a form identical to (5) but with $\alpha_{1}$ redefined as
$\alpha_{2}=\alpha_{1}(z_{s}/\lambda)^{1/6}$. In this new version of (5) the
length scale $\lambda$ reflects the streamwise extent of the largest eddies
found in CBLs, so it is closely related to the height of the temperature
inversion layer that caps, and so marks the vertical height of CBLs, $z_{i}$.
The ratio of $z_{i}$ to $\lambda$ will depend on the aspect ratio of the large
cellular or roll structures in the CBL, and this relationship has not been
fully explored. We note that Kaimal et al. (1976) reported that
$\lambda=1.5z_{i}$ for thirteen runs from the five days before a lightening
strike terminated the Minnesota experiment. Overall, we see that (5) can be
written in either inner-scaled or outer-scaled versions by choosing to set
either $\eta=z/z_{s}$ or $\eta=z/\lambda$, respectively.

KY90 also present profiles of the standard deviation of temperature,
$\sigma_{\theta}$, observed during many experiments, both within and above
their ‘convective-dynamic sublayer’. The top of this sublayer corresponds well
to the top of our SFL, but our SFL extends right down to the ground. We will
return to this later, but here we note that KY90 plot their profiles within
and above the SFL in two segments using MOST scaling parameters, and find that

$\frac{u_{*}\sigma_{\theta}}{H}=\beta\left(\frac{z}{-L}\right)^{-1/3}$ (6)

in each, though the constant $\beta$ has a different value in each segment.
The $-1/3$ power is consistent with expectation from MOST in the limit of free
convection and with the earlier results of Priestley (1960); Businger et al
(1971) and Wyngaard and Coté (1972). This behavior extends down to
$-z/L\approx 0.05$. The side-step in the profile of KY90 is unexplained.

Once again, this profile can be written in terms of our favored parameters,
giving

$\sigma_{\theta}=\beta_{1}\theta_{*}\zeta^{-1/3}$ (7)

where $\zeta=z/\lambda$. This profile agrees with the variances found by
integrating the areas under scaled temperature spectra observed above the SFL
(McNaughton et al, 2007; Chowdhuri et al, 2019), and also within it according
to the new $T$ spectra presented below, which self-similarity extends down to
similarly low levels.

To summarize: temperature profiles follow a $z^{-1/2}$ power law through most
of the surface layer, while profiles of its standard deviation, $\sigma_{T}$,
follows a $-1/3$ power law over a similar range. Since both profiles follow
power-laws we infer that these profiles are associated with some kind of
scale-free, self-similar structures. In the lowest part of the SFL the
profiles depart from these power law forms and the temperature profile follows
a log law while the $\sigma_{\theta}$ profile becomes constant with height.
These behaviors indicates that heat transport there is effected by another
kind of self-similar eddy structure.

In searching for the mechanisms that underly these mean profiles we will
appeal to the geometric and scaling properties of various kinds of plumes, and
to do this we must have a working understanding of the nature of plumes.

##  3 Modeling plumes

We use the word ‘plume’ to mean any connected volume of air defined by its
temperature, so our usage goes beyond the narrow regions of ascending air that
common usage implies. Plumes may frequently take this form, but we include all
warm and cool updrafts and downdrafts, independent of their shape or size. We
restrict our discussion to heights above any viscous sublayer or roughness
sublayer immediately above the ground, and to below the mixed layer where mean
temperature becomes almost constant with height. That is, we deal with the
atmospheric surface layer as it is usually defined in boundary-layer
meteorology.

Plumes, like eddies, defy exact definition. They are recognizable ‘patterns of
scalar concentration’, just as eddies are ‘patterns of motion’, and they can
have discoverable scaling properties even while the patterns themselves are
more-or-less ill-defined. There is an intuitive connection between a plume or
an eddy and the concept of a strange attractor, since the actual shapes of
plumes and eddies will exhibit many variations on a theme. Since we find the
concept of plumes, and the shapes of plumes to be useful concepts, we now
introduce a toy model in which plume boundaries, and so plume shapes and
cross-sectional areas, can be rigorously defined.

###  3.1 A two-temperature plume model

Our 2T model is a toy model that allows temperatures to have just one of two
values at each point while allowing the velocity field to have all of its
observed properties. This is a great simplification of real temperature
fields, and it does create some problems, but this 2T model also leads to some
simple relationships that will help us to understand the similarity properties
of real plumes observed in real boundary layers.

We consider a plane horizontal flow in which there are two kinds of air, each
distinguished by its potential temperature, $\theta_{u}$ or $\theta_{d}$. Each
kind is organized into localized regions called plumes. Warmer plumes
originate somewhere below and move upwards on average, while cooler plumes
originate somewhere above and generally move downwards. We neglect molecular
diffusion, so the up-and down-plumes retain the distinct identities imparted
to them at their sources, even while they may be stretched and folded in
complicated ways. Warm plumes move upwards with a mean velocity
$\left<w_{u}\right>$, and cool plumes move downwards at $\left<w_{d}\right>$,
where the angle brackets indicate means over the areas of each kind of plume
measured on a horizontal plane at a chosen height $z$. A heat flux is
maintained by air being converted from $\theta_{d}$ to $\theta_{u}$ in a thin
layer adjacent to the ground, at heights below those of interest here. In the
real world this heating would occur within a viscous sub-layer or roughness
sub-layer adjacent to the ground, which layers are excluded from present
consideration.

We will use the term ‘up-plume’ to indicate a parcel of air whose temperature
is set near the ground and which moves generally upwards over time, and the
term ‘down-plume’ for a parcel of air whose temperature is set above and moves
generally downwards. We note that the up- and down- designations refer only to
their average motions, not the local motion at any particular place and time,
which may be in either direction.

###  3.2 Relationships in the 2T model

For simplicity we assume that air density is constant, independent of
temperature. The 2T model then leads to a number of basic relationships.

####  3.2.1 Continuity

Let the warmer and cooler plumes occupy fractional areas $f_{u}$ and $f_{d}$
on any horizontal plane at height $z$. These fractions comprise the whole, so

$f_{u}+f_{d}=1$ (8)

The condition that there be no net vertical transport of air is written as

$\left<w_{u}\right>f_{u}+\left<w_{d}\right>f_{d}=0$ (9)

where, for simplicity, we have neglected the effect of temperature on air
density.

####  3.2.2 Mean temperature gradient

The numbers of $\theta_{u}$ and $\theta_{d}$ values found in a sample taken
along a horizontal transect will be proportional to the area fractions of up-
and down-plumes, so the mean temperature on the horizontal plane at any value
of $z$ is

$\left<{\theta}\right>=f_{u}\theta_{u}+f_{d}\theta_{d}$ (10)

Using this with (8) we can write the mean temperature gradient as

$\frac{\textrm{d}\left<{\theta}\right>}{\textrm{d}z}=(\theta_{u}-\theta_{d})\frac{\textrm{d}f_{u}}{\textrm{d}z}$
(11)

or its twin

$\frac{\textrm{d}\left<{\theta}\right>}{\textrm{d}z}=-(\theta_{u}-\theta_{d})\frac{\textrm{d}f_{d}}{\textrm{d}z}$
(12)

We notice that the temperature difference $(\theta_{u}-\theta_{d})$ is a
constant, so the temperature gradient depends solely on how the fractional
area of up-plumes d$f_{u}/$d$z$, or its complement d$f_{d}/$d$z$, changes with
height. If warm plumes become faster and thinner as they rise then area-mean
temperature will decrease with height.

####  3.2.3 Plume mean velocity variance

Because there are two mean vertical velocities there will be a plume mean
velocity variance. As for temperature, the numbers of $w_{u}$ and $w_{d}$
values selected in a given sample will be in proportion to their area
fractions, $f_{u}$ and $f_{d}$, at a given height. Starting with

$\sigma_{<w>}^{2}=\left<{\left<w\right>^{2}}\right>-\left<{\left<w\right>}\right>^{2}\\\
$ (13)

we have

$\displaystyle\sigma_{\left<w\right>}^{2}=$ $\displaystyle
f_{u}\left<w_{u}\right>^{2}+f_{d}\left<w_{d}\right>^{2}$
$\displaystyle-(f_{u}\left<w_{u}\right>+f_{d}\left<w_{d}\right>)^{2}$ (14)

giving

$\sigma_{\left<w\right>}^{2}=f_{u}f_{d}\;(\left<w_{u}\right>-\left<w_{d}\right>)^{2}$
(15)

which can also be written as

$\sigma_{\left<w\right>}^{2}=-\left<w_{u}\right>\\!\left<w_{d}\right>$ (16)

by using (9).

We note that (16) gives the variance of the plume mean velocities, not the
total variance of vertical velocity, $\sigma_{w}^{2}$ . This is an important
distinction, and we have no guarantee that $\sigma_{w}$ and
$\sigma_{\left<w\right>}$ will be equal, or even that they will scale the same
way.

####  3.2.4 Temperature variance

Because the up- and down- plumes have different temperatures there is
temperature variance on the $(x,y)$ plane. Starting with

$\sigma_{\theta}^{2}=\left<{\theta^{2}}\right>-\left<{\theta}\right>^{2}\\\ $
(17)

we write

$\sigma_{\theta}^{2}=f_{u}\theta_{u}^{2}+f_{d}\theta_{d}^{2}-(f_{u}\theta_{u}+f_{d}\theta_{d})^{2}$
(18)

and so

$\sigma_{\theta}^{2}=f_{u}f_{d}\;(\theta_{u}-\theta_{d})^{2}$ (19)

Thus $\sigma_{\theta}^{2}$ is proportional to $f_{u}f_{d}$. We note that
$(\theta_{u}-\theta_{d})$ is independent of height, so it may be regarded as
the temperature scale in our 2T model. The product of the fractional areas,
$f_{u}f_{d}$, will depend on height scaled by an appropriate height scale.

####  3.2.5 The heat flux

The kinematic heat flux is given by

$\left<w^{\prime}\theta^{\prime}\right>=\left<w_{u}\right>f_{u}\theta_{u}+\left<w_{d}\right>f_{d}\theta_{d}$
(20)

where $\left<w^{\prime}\theta^{\prime}\right>$ is averaged over the whole
horizontal plane. We will use the symbol $H$ for this heat flux. Combining
(20) with (9) gives

$H=\left<w_{u}\right>f_{u}(\theta_{u}-\theta_{d})$ (21)

and its twin

$H=-\left<w_{d}\right>f_{d}(\theta_{u}-\theta_{d})$ (22)

Since $H$ and $(\theta_{u}-\theta_{d})$ are both constant with height we have
an inverse relationship between $\left<w_{u}\right>$ and $f_{u}$ and between
$\left<w_{d}\right>$ and $f_{d}$. On average the up-plumes get thinner and
faster as they ascend, while the down-plumes get thinner and faster as they
descend.

Multiplying left and right sides of the last two equations together we get

$H^{2}=-\left<w_{u}\right>\\!\left<w_{d}\right>f_{u}f_{d}(\theta_{u}-\theta_{d})^{2}$
(23)

which can be rewritten as

$H^{2}=\sigma^{2}_{\left<w\right>}\sigma^{2}_{\theta}$ (24)

using (16) and (19). The temperature variance scales on
$H^{2}/\sigma^{2}_{\left<w\right>}$.

###  3.3 Limitations of the 2T model

The 2T model is a toy model, designed to help us to understand the link
between plume geometry and mean vertical profiles of temperature and
temperature variance. Its principal value is that it allows us to talk of the
areas of plumes without worrying too much about how the boundaries of those
plumes should be defined. However, the 2T approximation has its limitations.

We can explore these by considering the temperature signal along a horizontal
transect, as represented by the 2T model, and comparing this with a more-
realistic representation. A plot of temperature along a 2T transect would look
like a comb with many teeth, with a bases at $\theta_{d}$ and all of the same
height, $\theta_{u}$, but with various widths and grouped into clusters of
various sizes and compositions. The total lengths of the teeth and the gaps
would be in ratio $f_{u}/f_{d}$ along a representative transect. A more
realistic model would allow molecular diffusion to operate, rounding the
corners of the teeth, thereby reducing the heights of smaller teeth. Despite
these changes the resulting groupings of fuzzy plumes would have almost the
same mean temperature as their 2T originals. On the other hand the temperature
variances would be reduced. For this reason we can have some confidence in
(11) but we must be wary of (19) and relationships that follow from it.

Despite its limitations, the 2T model allows us to talk of the lengths and
areas of plumes, and so to interpret the power-law temperature profile
geometrically, in terms of the self-similar tapering with height of plume
cross sections. We believe that this remains useful even while real plumes are
blurry composites of whole populations of smaller plumes, and even while the
plumes originating from near the ground and the top of the CBL have, in
reality, a wide range of temperatures. Also, the 2T model allows the profiles
of $\partial\overline{\theta}/\partial z$ and
$\partial\sigma_{\theta}/\partial z$ to depend differently on height, since
$\partial\overline{\theta}/\partial z\propto df_{u}/dz$, from (11), while the
second $\partial\sigma_{\theta}/\partial z\propto
d(f_{u}^{1/2}f_{d}^{1/2})/dz$, from (19). Observations show the profiles
exhibit different powers of $z$, even while this is contrary to the dictates
of MOST above the SFL, in what Tennekes called the ”local free convection
layer” (Tenneekes, 1970). We will adopt the 2T model as a useful, qualitative
guide as we seek to understand the mechanisms that underly the observed,
ensemble-mean properties of temperature and heat transport in the surface
layer.

Before using these ideas to interpret real observations we extend our
terminology to include composite plumes. We have used the terms ‘up-plume’ and
‘down-plume’ to designate plumes that originate either near the ground or far
overhead, so the up- and down prefixes describe their mean direction of
movement even while their local motion might be either direction at any
particular time and place. As we move towards a more realistic representation
of reality we will add the idea of composite plumes. These are defined as
aggregations of up- and down-plumes having an identifiable common boundary and
local bulk velocity. They are compatible with the 2T model, in that the up-
and down- identities of their parts remain intact, but they will have length
scales and concentration scales that reflect the common properties imparted to
them by the rest of the flow, which properties lie beyond the scope of the
simple 2T model.

##  4 Experimental

In this paper we focus on observations made during the SLTEST experiment,
conducted in the Great Western Desert of Utah in 2005. The SLTEST site was
close to an ideal site for our experiment. Nine sonic anemometers were mounted
on a mast, facing North and spaced logarithmically over an 18-fold range of
heights, from 1.42 m to 25.7 m. The fetch area was a flat, unobstructed playa
surface stretching for 100 km to the North. The surface itself was
particularly smooth during the experiment in May and June of 2005, since
frequent rain in the winter and spring had eliminated the usual surface
cracking and suppressed the growth of salt crystals, giving a very uniform
surface of finely dispersed clay. This reduced the surface drag and resulted
in SFLs that were often only a few meters deep, so, with winds predominantly
from the North, the experiment yielded an unexpectedly large number of useable
observations from above the SFL. On other occasions the SFL was deeper so the
instruments recorded profiles within the SFL, as was envisaged in the original
experimental design. Further information is given by McNaughton et al (2007).

##  5 Plumes above the SFL

The $T$ and $w$ signals contain a great deal more information than just their
means, standard deviations and total heat fluxes, and this information can be
used to explore the properties of plumes and heat-flux events. A probability
density plot of temperature, for example, gives information on the relative
areas occupied by plumes of various temperatures, while $T$ spectra and $wT$
cospectra give information on the lengths of plumes and flux events, and on
their scaling properties. We now look at such information, starting with
observations from above the SFL, where $u_{\epsilon}$ is not a relevant
parameter because it represents the velocity scale of the shear eddies within
the SFL. Our set of similarity parameter then reduces to
{$z,\lambda,\epsilon_{o},H$}.

###  5.1 $T$ and $wT$ probability distributions

Figure 2: Probability distributions of temperatures at four heights above the
SFL site at 2100-2130 on 24 May 2005, when $\lambda=1230$ m. Here $T_{P}$ is
the temperature with peak probability. The distribution is strongly skewed
towards cooler temperatures because cooler, subsiding air occupies the greater
fraction of total area. Skewness increases with height as the area fractions
change: the area of cooler subsiding air increases while that of warmer rising
air increases.  Figure 3: Joint $w^{\prime}$-$T^{\prime}$ distribution at 3.0
m and 25.7 m plotted on the $w^{\prime}/\sigma_{w},T^{\prime}/\sigma_{T}$
plane and averaged over a selection of runs when both levels were above the
SFL and 900 m $<\lambda<$ 1300 m. The primes indicate deviations from the mean
at each level. Heat transport is highly organized, being dominated by warm
updrafts (I) and cool downdrafts (III) with little reverse transport in
quadrants II and IV.

Fig. 2 shows temperature PDFs at four heights, all above the SFL during a
particularly unstable run at SLTEST. These are based on time-series data
measured by fixed instrument. To interpret them we adopt Taylor’s frozen
turbulence hypothesis, by which the results are equivalent to observations
taken at equal steps along a streamwise transect, and so represent the
relative areas occupied by air of each temperature. Fig. 2 shows that
temperatures are not limited to just two values, as the 2T model would have
it, but have a continuous range of temperatures. Even so, we can retain the
general idea of up- and down-plumes by allowing real up- and down-plumes to
have a range of source temperatures. This being so, down-plumes will mostly be
represented on the left of the diagram and up-plumes on the right. We notice
that probabilities increase with height for $T-T_{P}<1$ and decrease with
height for $T-T_{P}>1$. That is to say, these empirical results support the
interpretation that up-plumes occupy an decreasing area fraction as height
increases while the area fraction of down-plumes increases. This is consistent
with what is suggested by (10), from the 2T model, and with the general
observation that temperature decreases with height in the surface layer of
CBLs.

The joint $w-T$ pdfs shown in Fig. 3 supports this interpretation. They show
the distribution of the heat flux on the $w^{\prime}$ and $T^{\prime}$ axes at
3.0 m and $25.7$ m averaged over a selection of runs when both levels were
above the SFL. It shows the joint probability distribution of $w^{\prime}$ and
$T^{\prime}$ weighted by the product $w^{\prime}T^{\prime}$ at each point. The
contributions of warm ejections (quadrant I) and sweeps plumes (quadrant III)
are quite distinct, and very little heat is transported in the ‘wrong’
direction, in quadrants II and IV. Heat transport is therefore highly
organized above the SFL.

In making these arguments we must accept that the frozen turbulence hypothesis
is not necessarily correct because eddies are not simply carried along in the
flow, as Taylor (1938) would have it, but the phase velocities of larger
eddies and plumes can be faster than the local mean velocity in wall-bounded
shear flows. Even so, the features of Fig. 2 that we rely on for our
interpretation would remain valid were a more exact representation of area-
sampled pdfs available.

###  5.2 $T$ spectra and $wT$ cospectra above the SFL

The mean profiles and PDFs considered so far tell us nothing about the lengths
of plumes. For that we turn to the analysis of $T$ spectra and $wT$ cospectra.
The surface layer above the SFL is a good place to begin because the power-law
forms of the $T(z)$ and $\sigma_{\theta}$ profiles above the SFL imply scale-
invariant properties for the underlying plumes there. McNaughton et al (2007);
Chowdhuri et al (2019) have identified scales that collapse the $T$ spectra
and $wT$ cospectra observed over a range of conditions onto ‘universal’
curves, so demonstrating self-similarity in the properties of the plumes that
give rise to them. These scales provide information on what controls the
behavior of the underlying plumes. In this they followed along the path
pioneered by Perry and Abell (1975), who studied eddies using these
techniques.

McNaughton et al (2007); Chowdhuri et al (2019) found that $T$ spectra and
$wT$ cospectra observed above the SFL collapse onto universal curves in three
wavenumber ranges, each with its own scales. We will call these the small-,
mid- and large-wavenumber ranges. The small-wavenumber range extends up to
$\kappa\lambda\sim 2\pi$ and the small-wavenumber range down to $\kappa z\sim
2\pi$. Spectra in the the mid- and large-wavenumber ranges appear to be
universal across sites, while the small-wavenumber spectra are somewhat
different at the very uniform SLTEST site and the more heterogeneous site of
the CAIPEEX experiment in India (Chowdhuri et al, 2019). This lack of
universality among sites is consistent with the well-known differences
observed for the largest flow structures in different flows, such as pipe,
channel and boundary layer flows (Monty et al., 2009). It seems that the
largest flow structures are influenced by the boundary conditions,
particularly as they influence the overall flow geometry, while the mid- and
small-scale structures depend on internal dynamics only and so have universal
properties.

Here we review the scaling results of McNaughton et al (2007); Chowdhuri et al
(2019) and extend our interpretation of them. The scales found to collapse the
$T$ spectra and $wT$ cospectra in each of the three ranges at SLTEST are given
in Table 1. The spectra themselves are given in the original reports. A novel
feature of the length scales is that many of them are mixed scales, or even
doubly-mixed scales. Here we adopt a qualitative explanation for mixed scales
based on our general understanding that eddies of all scales in the CBL are
parts of a single, coordinated flow system. Our hypothesis is that a mixed
length scale $\ell_{1}^{1/2}\ell_{2}^{1/2}$ implies that a larger-scale eddy
process, with length scale $\ell_{1}$, ‘organizes’ smaller plumes with length
scale $\ell_{2}$. For the word ‘organizes’ we have in mind such processes as
sweeping collections of up-plumes together into composite plumes. Doubly-mixed
length scales are interpreted similarly, with
$\ell_{1}^{1/2}\ell_{2}^{1/2}$-scale eddies organizing smaller plumes with
$\ell_{3}$-scale. We will apply this interpretation to the empirical scaling
results.

####  5.2.1 Mid wavenumbers

We start with mid-wavenumber ranges of $T$ spectra and $wT$ cospectra because
they account for most of the $T$ variance and $wT$ fluxes observed throughout
the surface layer, and they are found to be universal across sites. The mid
range includes the peak regions of these spectra and cospectra. More
practically, the search for the scales able to collapse mid-range spectra is
made simpler because peak wavenumbers can be scaled unambiguously before going
on to choose variance and covariance scales. These procedures are described by
McNaughton et al (2007); Laubach and McNaughton (2009), and the identified
scales are shown in Table 1.

A notable feature of the scales recorded in Table 1 is that many of them are
mixed scales, so that the properties of spectra depend on the “outer” length
scale $\lambda$ even quite near the ground. This is in good qualitative
agreement with the DNS simulation results of Fodor et al (2019), who found
Monin-Obukhov similarity theory inadequate when describing plume properties
and concluded that “updraft properties are not just determined locally, but
also by outer scales”. Our empirical studies of the scaling properties of
spectra confirm that observation. We now turn to a more detailed
interpretation of these scales.

Table 1: Length and variance scales of $T$ spectra and $wT$ cospectra measured above the surface friction layer. This table summarizes the scales identified by McNaughton et al (2007) and Chowdhuri et al (2019).  Scales  |  Small wavenumbers |  Mid wavenumbers |  Large wavenumbers
---|---|---|---
$\kappa S_{ww}$ |  |  |
length | $\lambda$ | $z$ | $z$
variance  | $\displaystyle(\lambda\epsilon_{o})^{2/3}\left(\frac{z}{\lambda}\right)^{4/3}$ | $(z\epsilon_{o})^{2/3}$ | $(z\epsilon_{o})^{2/3}$
$\kappa S_{uu}$, $\kappa S_{vv}$ |  |  |
length | $\lambda$ | $\lambda$ | $\lambda$
variance  | $(\lambda\epsilon_{o})^{2/3}$ | $(\lambda\epsilon_{o})^{2/3}$ | $(\lambda\epsilon_{o})^{2/3}$
$\kappa S_{TT}$ |  |  |
length | $\lambda$ | $\lambda^{1/2}z^{1/2}$ | $\lambda^{1/4}z^{3/4}$
variance  | $\displaystyle H^{2}(\lambda\epsilon_{o})^{-2/3}\left(\frac{z}{\lambda}\right)^{-1/3}$ | $\displaystyle H^{2}(z\epsilon_{o})^{-2/3}$ | $H^{2}(z\epsilon_{o})^{-2/3}$
$\kappa S_{wT}$ |  |  |
length | $\lambda$ | $\lambda^{1/4}z^{3/4}$ | $z$
covariance  | $\displaystyle H\left(\frac{z}{\lambda}\right)^{1/2}$ | $\displaystyle H\left(\frac{z}{\lambda}\right)^{1/12}$ | $H$

The $T$ variance scale is $H^{2}(z\epsilon_{o})^{-2/3}$, which implies a plume
velocity scale $(z\epsilon_{o})^{1/3}$. This is also the $w$ variance scale,
and it identifies impinging outer Richardson eddies as the eddies active in
shaping the plumes at this scale (McNaughton et al, 2007). The length scale
here is the mixed length scale $\lambda^{1/2}z^{1/2}$, which we interpret as
signifying an interaction between the plumes created by the impinging outer
Richardson eddies and the larger, $\lambda$-scale eddies whose lower parts
extend down into the surface layer. We need more information before we can go
further, so we delay further interpretation until the end of this section.
However, we do note that the peaks of the $T$ spectra lie at
$\kappa\lambda^{1/2}z^{1/2}=5.5$ (Chowdhuri et al, 2019), so peak plumes are
long in the streamwise direction, and they elongate further as height
increases. This is consistent with the shapes of sheet plumes that comprise
the rising walls of warm air that converge towards the bases of the thermals
at the top of the surface layer.

We note that the $T$ variance scale, $H^{2}(z\epsilon_{o})^{-2/3}$, is not the
scale from the 2T model, $(\theta_{u}-\theta_{d})$, since the latter is
independent of $z$. The difference lies in the identity of the plumes above
the SFL. The properties of 2T up- and down-plumes are set at source, and their
identities are preserved as they rise or sink, on average, thereafter, however
much stretched, folded and tangled they may become. The lengths identified by
the position of the spectral peaks, on the other hand, represent the
aggregation properties of plumes as well as their individual properties. As a
result, the plumes whose temperature scales on $H^{2}(z\epsilon_{o})^{-2/3}$
are composite plumes with characteristic temperatures that decrease with
height as more and more clear air (down-plume) is added to the composite. Only
if there were no dissipation and we could focus on just the original filaments
of up-plumes would the variance scale be $(\theta_{u}-\theta_{d})$.

The $T$ variance scale can also be written as
$H(z_{s}\epsilon_{o})^{-1/3}$$\left(z/z_{s}\right)^{-1/3}$, and as
$H(\lambda\epsilon_{o})^{-1/3}\left(z/\lambda\right)^{-1/3}$. The first of
these is the temperature scale appropriate for plumes within the SFL, but
modified by the height function $(z/z_{s})^{-1/3}$ to express the diminishing
the fraction of ‘original’ up-plume air as height increases. The second
introduces the temperature scale of the thermals at the top of the surface
layer, $H(\lambda\epsilon_{o})^{-1/3}$ times the height function
$(z/\lambda)^{-1/3}$, which represent the same aggregation process towards the
final temperature of the thermals. This flexibility of interpretation reflects
the scale-free nature of the process itself: the whole up-plumes are embedded
in regions of the flow that converge laterally and accelerate upwards, and the
shapes of the embedded original plumes within the composites will be similar
to the envelope shapes of the composite plumes. This self-similar property is
consistent with the length-scale-independent, power-law form of the mean
temperature profile.

We can go further by focussing on the first interpretation: that the plume
properties reflect the action of impinging outer Richardson eddies, which
sweep together larger and larger aggregates of the original plumes rising from
within the SFL. If these impinging outer Richardson eddies acted in isolation
from eddies of other kinds then plume lengths would scale on $z$ alone, rather
than on $\lambda^{1/2}z^{1/2}$. The difference can be attributed to the up-
plumes all being embedded in regions of the flow where the horizontal flow
convergence, $\partial u/\partial x+\partial v/\partial y$, depends on
$(z/\lambda)^{-1/2}$. The absence of heat flux in quadrant IV of Fig. 3 tells
us that the up-plumes are all associated with the updrafts created by these
convergences, so we can directly associate the $(z/\lambda)^{-1/2}$ power law
of the convergences with a similar dependence of the total area of the up-
plumes. The length scale displayed by the mid-range of the $T$ spectrum is
consistent with the $-1/2$-power law followed by the mean temperature profile.

Scales for $wT$ cospectra are more complicated. The lengths of $wT$ flux
events scale on the doubly-mixed scale $\lambda^{-1/4}z^{-3/4}$ (because
wavenumbers scale on $\lambda^{1/4}z^{3/4}$), Table 1). Peaks of $wT$
cospectra are found to be at $\kappa\lambda^{1/4}z^{3/4}=1.2$, while the
wavenumbers of the peaks of $T$ spectra scale differently and are at
$\kappa\lambda^{1/2}z^{1/2}=5.5$ (Chowdhuri et al, 2019). Peak wavenumbers for
$wT$ flux events are therefore from 1.5 to 2.5 times larger than those for $T$
plumes as $z/\lambda$ increases from 0.01 to 0.1. Flux events are therefore
shorter than plumes, so we must conclude that heat transport is concentrated
in the faster-rising parts of plumes. A plausible explanation is that these
faster-rising parts of up-plumes lie closer to the roots of the thermals where
vertical velocities are greatest.

The doubly-mixed nature of the scale $\lambda^{1/4}z^{3/4}$ indicates that the
flux events themselves depends on both the $\lambda^{1/2}z^{1/2}$-scale eddies
organizing and driving the plumes and the $z^{1/2}$-scale length scale of the
up-plumes themselves. A $z^{1/2}$ dependence for the up-plumes is consistent
with the $\lambda^{1/2}z^{1/2}$ length scale of the $T$ spectra, and indeed
with the $z^{-1/2}$ dependence of the cross-sectional areas of the p-plumes as
indicated by the mean temperature profile. Overall, the length scale depends
on $z^{-3/4}$ because $z$ plays two roles: one as it relates to the cross-
sectional areas of the plumes, and the other as it related to what size of
impinging outer Richardson eddy is associated with the greatest vertical
velocities at each height. We know that there is an optimum size for these
impinging eddies because their kinetic energies increase with eddy size,
according to the Kolmogorov law, but increased eddy size also means that
blocking by the ground reduces the fraction of this energy allocated to the
vertical motions at small heights.

###  5.3 Small and large wavenumbers

The mid-range plumes aggregate as height increases, eventually to combine into
the $\lambda$-scale clusters we know as thermals. Between these lie cooler
areas of descending air with smaller temperature fluctuations, and these areas
increase with height. As a result the Fourier representations of temperature
signals change with height, with more variance in the small-wavenumber range
but less in the mid-wavenumber range as height increases (McNaughton et al,
2007; Chowdhuri et al, 2019). The same is true of the $wT$ covariance in
cospectra, so heat flux is transferred from the mid wavenumber to the small-
wavenumber ranges as height increases. The small-wavenumber parts of spectra
and cospectra collapse when wavenumbers are scaled on $\lambda$ (Table 1).

The mean temperature gradient is insensitive to these changes in spectral
representation. The same original plumes still taper in the same way, so the
combined area of these up-plumes continue to follow the $z^{-1/2}$ power law,
even while their spectral representation is sensitive to their changing
aggregation properties. In the small-wavenumber range the covariance increases
in proportion to the $(z/\lambda)^{1/2}$ increase at each wavenumber, which
$1/2$ power suggests a link to the $(z/\lambda)^{-1/2}$ decrease in the
envelope areas of clusters of original up-plumes. We have no formal analysis
to confirm this connection.

The large-wavenumber ranges of spectra and cospectra reflect the action of
detached outer Richardson eddies on the mid-range plumes, as indicated by the
doubly-mixed length scale of the $T$ spectrum in this range. This combines the
$\lambda^{1/2}z^{1/2}$ length scale of the plumes with the $z$ length scale of
the outer Richardson eddies. They carry this scale even while detached because
their population still reflects the influence of the ground. The Richardson
cascade is itself scale-free, and characterized by the $-5/3$ power-law
Komogorov spectrum. However, this spectrum is also characterized by the
absence of eddies with heights significantly greater than $z$, because such
large eddies increasingly interact with the ground (become attached) and so
behave differently. Detached outer Richardson eddies create fine structure in
plumes by stretching and folding them repeatedly, especially at their margins,
eventually creating plume filaments at scales where molecular dissipation
becomes important.

Detached outer Richardson eddies have no intrinsic sense of up and down, so
the direction of heat transport is set by the mean temperature gradient. Such
transport does not change the temperature profile itself. We have a scale-free
population of detached Richardson eddies acting on a scale-free temperature
profile, so the resulting heat flux is scale free, and so independent of
height. This independence is confirmed by observations, which show that $wT$
cospectra collapse over a range of heights when scaled on $H$ (Table 1). With
no flux divergence the mean temperature profile is unaffected. This model of
random transport down a pre-existing temperature gradient is consistent with
the ‘eddy diffusion’ model of Wyngaard and Coté (1972), which predicts that
the $wT$ cospectrum should follow a $-4/3$ power-law in this range, as is
observed (e.g. Chowdhuri et al, 2019).

##  6 Plumes within the SFL

Plumes within the SFL are more complicated. We know that the power-law form of
the mean temperature gradient extends down into the SFL, but follows a log law
very near the ground (Kader and Yaglom, 1990). We therefore do not expect
plumes to have a single, self-similar form throughout the SFL, but one that
changes with height. The nature of that change is our focus here.

Though many experiments have been conducted within the SFL, most published
results are of limited value for our work. Only the results of Laubach and
McNaughton (2009), have been reported using non-dimensionalizations based on
our preferred parameter set, {$\lambda,z,\epsilon_{o},u_{\epsilon},H$}; and
even those results are problematic because they were from experiments not
originally designed for our kind of analysis. Here we base our interpretations
on results from a new analysis of the SLTEST data. We first describe how data
were selected, then introduce the new results as we go along.

###  6.1 Data selection and processing

A difficulty encountered when using our parameter set is that calculation of
the outer dissipation rate, $\epsilon_{o}$ must be based on observations made
well above the SFL, often at many tens of meters above ground, while the
dissipation velocity, $u_{\epsilon}$ must be based on observations made very
near the ground, at small $z/z_{s}$. That is, instruments must be mounted at
two widely-spaced heights if both parameters are to be determined directly.
The SLTEST experiment was designed with this in mind, and instruments were
arranged so that they would span the SFL on many occasions. Laubach and
McNaughton (2009) used a limited range of instrument heights—just one at their
principal site—so $z/z_{s}$ was essentially an inverse scale variable in their
analysis, while $z_{s}$ was calculated using an empirical relationship for
$\epsilon_{o}$ taken from KY90. We have repeated their analysis using SLTEST
data.

For the present analysis we first selected all daytime runs from SLTEST where
wind was from Northerly directions, and for each of these we calculated trial
values of $\epsilon_{o}$ and $u_{\epsilon}$ by assuming that the highest sonic
anemometer lay well above the top of the SFL and the lowest very near its
bottom, and so calculated trial values of $z_{s}$ for each run. Runs were
accepted for further analysis when 7.1 m $<z_{s}<12.8$ m, which values of
$z_{s}$ validate our assumptions that the top sonic anemometer was at
$z/z_{s}>2$ and the bottom one at $z/z_{s}<0.2$. By this means 32 runs were
selected for further analysis, and for each of these the sonic anemometers
were well spread across the SFL. This makes $z/z_{s}$ a true scaled height
variable in our analysis. A limitation of the SLTEST results is that even an
18-fold range of anemometer heights is rather small, so we have a rather
narrow range of $z_{s}$ values to work with. While some of the SLTEST results
differ from those reported by Laubach and McNaughton (2009) some also agree,
as we point out. The earlier results are still very valuable, being from two
experiments where $z_{s}$ varied widely even while having a limited range of
$z_{s}$ values.

We present our spectral results on wavenumber axes rather than frequency axes
even while the basic data are from time-series observations made by fixed
instruments. This choice is to facilitate interpretation in terms of the
geometries of eddies and plumes. We adopt Taylor’s frozen turbulence
hypothesis and use mean wind speed at each level to make the conversion from
frequency to wavenumber, in the standard way. In earlier papers we performed
similar analyses of specta above the SFL, and for these we used the single
mean velocity at the top of the mast to make the conversion at all levels,
noting that wind speed varies only slowly with height above the SFL
(McNaughton et al, 2007; Chowdhuri et al, 2019). Neither procedure is ideal
since plumes of different sizes move at different speeds in sheared
atmospheric surface layers (Davison, 1974; Cheng et al, 2017). Use of Taylor’s
hypothesis leads to compression of the wavenumber axis of the $T$ spectra and
$wT$ cospectra. Even so, the scales that collapse these spectra and cospectra
are unaffected, and so too are our interpretations of those scales.

Figure 4: Temperature probability distributions at four heights at SLTEST.
Temperature signals were first filtered using a Fourier low-pass filter set at
$\kappa z=1$ to remove high-frequency fluctuations. Data are from the 32 runs
whose selection is described in the text, except that $\epsilon_{o}$, and so
$z_{s}$ could not be measured directly for runs where $z/z_{s}<0.04$. Runs for
this case were selected by assuming $z_{s}=-L$, and further selected so that
$-L<100$ m to ensure that the flow regime was always CBL.

###  6.2 $T$ and $wT$ probability distributions

Figure 5: Accumulated heat fluxes, $w^{\prime}T^{\prime}/H$ mapped onto the
($w^{\prime}/\sigma_{w}$, $T^{\prime}/\sigma_{T}$) plane. The red contours
represent two Gaussian signals with the same flux. Run details are as for Fig.
4.

Fig. 4 shows the $T^{\prime}$ probability distribution at SLTEST at four
height ranges, calculated for just the fluctuations in the mid-range of the
spectrum, obtained by first low-pass-filtering the signal with cut-off
wavenumber $\kappa z=1$. Fig. 5 shows the joint distribution of the mid-range
heat flux mapped on $w^{\prime}/\sigma_{w}$ and $T^{\prime}/\sigma_{T}$ axes.
Starting with Figs. 2 and 3, which show results from above the SFL, then
moving progressively down through the SFL in Figs. 4 and 5, we see a
continuous progression from well-organized transport above the SFL to near-
Gaussian behavior at its bottom. It seems that plume temperatures have a
Gaussian distribution at their source at the ground, in strong contrast to the
2T model, and that this initial distribution is generally maintained as the
up-plumes rise. The temperature distribution of the down-plumes can be
interpreted as reflecting the combination of a decreasing area fraction of
original down-plume air, and its mixing with up-plumes having a Gaussian
temperature distribution, with the vertical velocities of the composites
becoming increasingly random as height diminishes. (We recall that the up- and
down-plume terminology is based on the origin of the plume, not the current
direction of movement.) This progression of plume properties is generally
consistent with expectation based on the mean profiles, but to go further we
must assemble more evidence. We now turn to the $T$ spectra and $wT$
cospectra.

###  6.3 $T$ spectra and $wT$ cospectra within the SFL

Figure 6: $T$ spectra from within the SFL at the SLTEST site. The variance
scale is that which collapses peak heights above the SFL, though the length
scale is now doubly-mixed. The scaled spectrum begins to change only at the
lowest levels, $z/z_{s}<0.14$. Figure 7: $wT$ cospectra from within the SFL at
the SLTEST site. The doubly-mixed plume length scale in (a) is chosen to
collapse the positions of the main peaks, while in (b) the simple mixed scale
$z$ is chosen to collapse the position of the emerging minor peak at larger
wavenumbers.

$T$ spectra for the 32 selected runs from SLTEST are shown in Fig. 6. The peak
positions collapse when lengths are scaled on the doubly-mixed scale
$\lambda^{1/4}z_{s}^{1/4}z^{1/2}$, in agreement with the collapse shown in
Laubach and McNaughton (2009, Fig. 10). This behavior is therefore consistent
across three experimental sites. Fig. 6 also shows that spectrum heights
collapse at all wavenumbers in most of the SFL when variance is scaled on
$H^{2}(z\epsilon_{o})^{-2/3}$, in agreement with the $z^{1/3}$ power law for
the $\sigma_{\theta}$ profile described above. The same scale collapsed the
mid ranges of $T$ spectra above the SFL (Table 1). Only the lowest level shown
in Fig. 6, for $0.09<z/z_{s}<0.14$, deviates significantly. That is, plume
self-similarity does not extend right down to the ground.

We note that plume length scales are different above and within the SFL: the
appropriate scale above the SFL being the mixed scale $\lambda^{1/2}z^{1/2}$
(Table 1), but the doubly-mixed length scale $\lambda^{1/4}z_{s}^{1/4}z^{1/2}$
within it. These scales both depend on $z^{1/2}$ so the profiles can match at
and match at $z=z_{s}$ without any break in slope. We have already interpreted
the mixed scale above the SFL as reflecting the scale of plumes created by the
action of $z$-scale impinging outer Richardson eddies. The consistent
explanation for the doubly-mixed scale is that we have plumes created by the
action of $z$-scale impinging outer Richardson eddies, but embedded within and
shaped by the converging flow field of the larger and more powerful,
$\lambda^{1/2}z_{s}^{1/2}$-scale eddies. If so the impinging outer Richardson
eddies that act on the $z$-scale plumes have a fixed length at all heights
within the SFL. This is consistent with our conceptual model of the SFL, in
which the only impinging outer Richardson eddies able to penetrate the SFL are
those large enough to survive direct interaction with the more-powerful shear
eddies within the SFL.

An interesting feature of our results is the small shoulders lying a decade
and more to the right of the main peaks in both $T$ spectra and, more
prominently, in the $wT$ cospectra shown in Fig. 7. These shoulders indicate
emerging peaks that become more prominent near the ground and lie at the
beginnings of the large-wavenumber parts of the $wT$ cospectra. Fig. 7 shows
the same cospectra in both panels, but with wavenumbers scaled in two
different ways. The doubly-mixed length scale
$\lambda^{1/4}z_{s}^{1/4}z^{1/2}$ gives better collapse of the positions of
the main peaks, but simple $z$ scaling gives better collapse of the subsidiary
peaks. At larger wavenumbers the whole $wT$ cospectrum scales on $z$. We also
notice that the main peak diminishes with height as the minor peak grows. All
of these features support our interpretation that two kinds of eddies, and so
two kinds of plumes, transport heat within the SFL. These finding agree with
those of Laubach and McNaughton (2009, Fig. 12), so the results are again
consistent across sites.

We note that Smedman et al. (2007) also reported $wT$ cospectra with two
peaks, both in positions consistent with our peaks, but at small $-z/L$ their
larger-wavenumber peak is much bigger than that shown in Fig. 7. They
interpret its emergence as indicating a transition to a distinct Unstable-
Very-Close-to-Neutral (UVCN) flow regime. We agree, but note that our runs are
selected so as to avoid sampling this near-neutral regime. The defining
characteristic of a CBL regime is the presence of an outer layer deep enough
to support an outer Richardson cascade. The thickness of this outer layer
reduces as the heat flux decreases because the SFL deepens as neutrality is
approached. The remaining outer layer, of depth $(z_{i}-z_{s})$, must
eventually become too shallow to accommodate an outer Richardson cascade,
whereupon the CBL regime will transition to a near-neutral-boundary-layer
(NNBL) regime. We use NNBL rather than UVCN to describe this regime since
there is evidence that it persists into slightly stable conditions, where it
is described as the ‘strong turbulence regime’ of stable boundary layers.
Smedman et al. (2007) had no direct way to discriminate CBL and NNBL regimes,
so their results reflect a mixture of both. Laubach and McNaughton (2009)
found a similar $z$-scaled peak emerged at small $-z/L$, but their observation
height was fixed so this always occurred in near-neutral conditions, and NNBL
results were not excluded.

###  6.4 Plumes at the bottom of the SFL

The parameterization of CBLs adopted in this paper is based on an idea
introduced by McNaughton (2004). It is that there exists a special kind of
eddies, called shear eddies, whose fundamental role is to transport momentum
to the ground at small heights where blocking by the ground limits the
vertical motions of, and so the transport effectiveness of, larger eddies.
Near the ground these shear eddies comprise a self-similar population of
attached eddies whose heights scale on $z$. Shear eddies transport momentum
down to the ground in stages by the action of ever-smaller shear eddies. That
this process is rather inefficient is shown by the high dissipation rates
observed near the ground.

The surface shear stress is steady in laboratory boundary layers, so the
dissipation rate can be written as $\epsilon=u_{*}^{3}/k$, where $k$ is the
von Kármán constant. In CBLs the surface shear stress is variable under the
action of energetic large eddies, which Townsend (1961) called ‘inactive’
turbulence, and this variability enhances the dissipation rate within the SFL.
We have defined a new velocity scale, $u_{\epsilon}$, to account for this
based on the energy of the shear eddies rather than their ability to transfer
momentum in the streamwise direction. Thus $\epsilon=u_{\epsilon}^{3}/kz$ for
$z<<z_{s}$. We expect this ‘dissipation velocity’ to be the appropriate scale
velocity for scalar transport very near the ground.

Figs 6 and 7 show spectra in the layer where shear eddies dominate scalar
transport, which layer we characterize here as the ‘bottom of the SFL’. It
corresponds to the ‘dynamic sublayer’ in the three-sublayer scheme for the
surface layer proposed by KH90. It is a very shallow layer, occupying only the
bottom few percent of the SFL. The 32 runs we selected for analysis when
preparing Figs 6 and 7 do not provide adequate information in this layer, so
we do not confirm directly that the amplitude of the temperature spectrum
scales on $H^{2}u_{\epsilon}^{-2}$. Instead we present qualitative arguments
for a model that is consistent with observations from the literature, and with
the conceptual model described in the sections above.

We draw a firm distinction between shear eddies themselves and the plumes they
create. Shear eddies are “patterns of motion”, characterized by their ability
to transport momentum. They are highly-organized if transient patterns whose
form is dictated by their function. They are also dynamical structures in
which motion can be transferred from one part to another, and from one eddy to
another, by pressure forces. Plumes, on the other hand, are “patterns of
scalar concentration” where identities are permanently attached to particular
parcels of air. There can therefore be no ongoing association between eddies
and plumes, which is to say Reynolds’ analogy is a false analogy. Momentum and
heat are not transported by the same eddies. Why then do mean velocity and
mean temperature profiles both have the same logarithmic form near the ground?

Near the ground plumes see the velocity field of the shear eddies as an un-
coordinated churning action, and this action depends on the $z$ and
$u_{\epsilon}$ scales of the shear eddies. The churning will effectively
randomize the forms of plumes and their spatial distribution, while imparting
to them the $z$ and $u_{\epsilon}$ scales of the shear eddies. Wind velocity
displays a log profile because the shear eddies comprise a self-similar
population of attached (scaling on $z$) plumes. The plumes have similar self-
similar properties, but with the difference that their self-similarity is
inherent in the randomness of their forms, not in their order. We then have

$\frac{u_{\epsilon}z}{H}\frac{\partial\overline{\theta}}{\partial
z}=\frac{1}{k_{\theta}}$ (25)

where $k_{\theta}$ is the temperature counterpart of the von Kármán constant
for the wind profile. In comparison, the standard von Kármán constant is
defined by

$\frac{z}{u_{*}}\frac{\partial\overline{u}}{\partial z}=\frac{1}{k}$ (26)

We write these equations using overbars to indicate time averages because the
values of $k$ and $k_{\theta}$ are almost always calculated using time-series
observations made by fixed instruments. It is not possible to compare values
of $k_{\theta}$ and $k$ quantitatively because $u_{\epsilon}$ has not been
used to scale temperature profiles in published studies, but we do not expect
$k_{\theta}$ to be equal to $k$ since the underlying transport mechanisms are
different. Different constants have been reported, but all such reports have
used scaling procedures different to ours. A recent example is the study by
Cheng et al (2020).

Both observation and mechanistic arguments therefore agree that plumes have a
Gaussian distribution of temperatures at source, even as they emerge from the
log layer, and in this respect real up-plumes are quite unlike the single-
temperature up-plumes of the 2T model. Even so, and like in the 2T model,
their initial temperature signature seems to be maintained as they rise
subsequently. This is evident in the slowly-changing distribution of
temperatures in quadrant I of Fig. 5. The remarkable thing is that the
transition from randomness to organized up-plumes with predominantly upwards
local vertical velocity occurs at such a low level, with the transition from
randomness starting at $z/z_{s}\approx 0.04$, or even less, and well
progressed by $z/z_{s}=0.1$. It is this early transition in transport
mechanism that answers the question why the log layer for scalars is so much
shallower for temperature than for velocity (Bradshaw and Huang, 1995).

##  7 General Discussion

This paper has given an account of the scaling properties of temperature
profiles and plumes, and of $T$ spectra and $wT$ cospectra in terms of known
properties of the eddies and plumes that comprise turbulent flow in the
surface layer of convective boundary layers. The 2T model has contributed to
this by allowing us to talk of the cross-sectional areas of plumes, even if
only by analogy with idealised plumes defined in a system where such areas can
be strictly defined. The distinction between $\sigma_{<w>}$ and $\sigma_{w}$
is important in this interpretation, particularly within the upper SFL where
$\sigma_{w}$ scales on $u_{\epsilon}^{2}$ while $\sigma_{<w>}$ has no simple
scale because shear and impinging outer Richardson eddies both contribute to
the velocities of composite plumes there.

###  7.1 The role of buoyancy

Our account is based on concepts that are, in some respects, foreign to the
statistical fluid mechanics (SFM) understanding of turbulence in CBLs. In
particular, buoyancy plays a very different role in our conceptual model of
convection. To explain how such a difference can be possible we first review
the origins of the SFM model, which model encompasses the work of Reynolds
(1895) and Richardson (1920), who laid down the concepts, Monin and Obukhov
(1954) who formalized them in a useful way, and many later authors who have
developed empirical relationships based on these concepts.

The origins of SFM lie in the work of Reynolds (1895) who, given the
intractability of the Navier-Stokes equations, decided that turbulence could
only be addressed statistically. He developed the set of Reynolds-averaged
Navier-Stokes (RANS) equations, but formal rigour in this development was
achieved much later, by Kolmogorov. The problem was that Reynolds’ volume
averages, also taken to be averages over large populations of eddies, could
not properly be formed in flows whose largest eddies span the flow
(McNaughton, 2012). One can not then define elementary things like mean
velocity gradients in terms of Reynolds’ volume-averaged variables. Kolmogorov
solved this by redefining the volume averages as ensemble averages, but in
this formulation the RANS equations can contain no information on the dynamics
of flows. To form an ensemble average one takes a series of ‘snapshots’ of a
flow, either from separate flow realizations (experiments or simulations) or
from a single flow but separated widely enough in time to ensure no
statistical connection between the snapshots. This very procedure denies all
dynamical interpretations since dynamics is about causal connections between
successive states of the flow. Ensemble averaging also smears out any
transient spatial patterns in the flow so that the forms of eddies and plumes
can not be represented by ensemble-averages. Much recent work has attempted to
re-introduce information on flow structures by melding ideas from SFM with
ideas taken from the study of physical flow structures, but we have taken an
independent path based only on ideas taken from the study of complex dynamical
systems (CDS).

Our conceptual departure from SFM is most noticeable in our treatment of
buoyancy. Richardson (1920) extended the RANS equations by adding a buoyancy
term to Reynolds’ original equations for neutral flows. But Richardson
explicitly excluded large eddies from his system and, with the eddy diffusion
model in mind, adopted the only alternative and assumed that the energy was
transferred to the small, attached eddies associated with the temperature
gradient near the ground. His ‘buoyant production’ was therefore a local
phenomenon, with its action located near the ground where temperature
gradients are largest. This interpretation has persisted (e.g. Monin and
Yaglom, 1971; Wyngaard, 2010) despite the impossibility of any such dynamical
interpretation of individual, ensemble-averaged terms in the RANS equations.
The RANS energy equation, of which Richardson is part author, simply expresses
a balance between the divergence of the flux of mechanical energy and the
local dissipation rate, and it can be derived from that principle alone,
without need to interpret the dynamical origins of particular terms. Though
the magnitude of the ‘buoyant production’ term in the RANS energy equation can
be significant, there is no argument to locate this ‘production’ in the
smaller eddies.

In CBLs mechanical energy flows into the system at the largest scale and
passes down progressively, from one kind of flow structure to the next, to
finally be dissipated as heat at the smallest scale, as shown in Fig. 8. The
largest flow structures have the greatest energy densities and the longest
lifetimes, so smaller eddies and smaller plumes are simply swept along within
them, moving in whatever direction and at whatever velocity the larger eddies
dictate. In particular, plumes do not accelerate upwards under the action of
local buoyancy forces. It is, however, intrinsic to the system-wide
organization of the flow that the largest eddies accommodate the buoyancy of
smaller plumes by aggregating them and incorporating them into the largest-
scale patterns in the flow. We can then best understand the role of buoyancy
in the context of all of the other inputs of mechanical energy into the CBL
system.

Figure 8: The flow of mechanical energy in CBLs. Arrows represent energy
transfers from one kind of eddy to the next. The energy flowing from the
large, CBL spanning structures to the shear eddies has been divided into three
streams, A, B and C. The first of these supports the mean transfer of shear
stress to the ground while the second and third supports the fluctuations. The
relative sizes of these flows change with instability. Adapted from McNaughton
(2012).

###  7.2 Flow of mechanical energy in CBLs

Fig. 8 is a schematic diagram of the flows of mechanical energy in the CBL
system. It is based on the diagram drawn by McNaughton (2012), but here the
large, semi-permanent coherent structures and the largest turbulent eddies are
combined and described as large, CBL-spanning flow structures. This change
acknowledges the importance of the changing form of the large eddies with
changing instability, as described by Jayaraman and Brasseur (2018). An extra
energy flow pathway has been added so the mean and fluctuating effects of
these large flow structures can be represented separately. Energy now flows to
the shear eddies by three pathways, marked A, B and C in Fig. 8. The first
component, A, supports the transfer of mean shear stress to the ground while
component B supports the fluctuations in shear stress created by the large
coherent structures, and component C supports the smaller-scale fluctuations
in surface shear stress created by outer Richardson eddies impinging onto the
ground. All three will contribute to dissipation near the ground, and so to
the value of $u_{\epsilon}$. Taken alone, an increase in $(F_{A}+F_{B}+F_{C})$
will therefore cause an increase in $u_{\epsilon}$ and so an increase in
$z_{s}$, where the $F$s are the fluxes of mechanical energy. However, we know
that the form of the large CBL-spanning flow structures do change with
stability—from long roll vortices parallel to the wind in near-neutral
conditions to cellular convective cells when the CBL is very unstable—so
pathways B and C increase in importance as instability increases. The sum of
these two energy flows is just $z_{i}\epsilon_{o}$, where $z_{i}$ is the
height of the CBL, since the velocity scale of the large eddies can be
parameterized as $(z_{i}\epsilon_{o})^{2/3}$ and $\epsilon_{o}$ also
parameterizes energy flow down the outer Richardson cascade (McNaughton et al,
2007). The effect is that the ratio $(F_{B}+F_{C})/(F_{A}+F_{B}+F_{C})$
increases as the large CBL-spanning flow structures become more cellular, so
$z_{s}$ decreases as this ratio increases. Indeed, the ratio acts as a
stability parameter for CBLs.

In Fig. 8 the large structures pass energy continuously down to smaller and
smaller structures in which the form of the source eddies quickly becomes
irrelevant. This leads to universal $T$ spectra for the mid- and large-
wavenumber ranges of the smaller plumes, and to the parameterization of their
velocity scales in terms of the energy flowing through the eddies that shape
them. For this reason the outer dissipation rate, $\epsilon_{o}$, and the
dissipation velocity, $u_{\epsilon}$, are key parameters in our scaling
scheme. This perspective is consistent with the importance of the dissipative
flux in the theory of far-from-equilibrium systems (Evans and Morriss, 2007).

###  7.3 Stability parameters

In our work $z/z_{s}$ plays a role somewhat analogous to the role of $-z/L$ in
MOST, and indeed the two are correlated (Chowdhuri et al, 2019). We can use
Fig. 8 to explore the source of this correlation since both $z_{s}$ and $L$
can be derived from it. As we have seen $z_{s}$ is given by (1), where
$u_{\epsilon}$ depends on the mechanical energy dissipated within the SFL, and
so on the sum of the energies flowing down pathways A, B, and C. If we were to
delete the energy flow pathways B and C then the same equation (8) would lead
to the Obukhov length, $-L$. Without pathways B and C the dissipation
velocity, $u_{\epsilon}$, becomes $u_{*}$, and $gH/T_{0}$ replaces
$\epsilon_{o}$ since all of the gravitational energy must now pass directly to
outer dissipation. This is so because buoyancy forces act vertically, so none
of their energy can go to supporting the mean momentum flux. With these
changes $z_{s}=-L$. Though this leads us to expect correlation between $z_{s}$
and $L$, it also leads us to expect that the Monin-Obukhov model will break
down in windless convection, when $u_{*}$ goes to zero. Several adjustments to
the standard Monin-Obukhov model have been proposed to avoid this. Businger
(1973) proposed that $u_{*}$ should have a lower limit $w_{*}$, where $w_{*}$
is the Deardorff convective velocity scale. Beljaars (1994) redefined $L$
using $u_{*}^{2}+\beta w_{*}^{2}$ in place of $u_{*}^{2}$, so adding a proxy
for the missing energy flow in all conditions. Others, (e.g. Stull, 1988),
have redefined $u_{*}^{2}$ as
$(\overline{u^{\prime}w^{\prime}}^{2}+\overline{v^{\prime}w^{\prime}}^{2})^{1/2}$,
which is clearly an _ad hoc_ adjustment since an ensemble-averaged
$\overline{v^{\prime}w^{\prime}}$ must be zero, but in practice this also
seems to compensate for the missing energy flow (e.g. Wilson, 2008). The other
change—setting $\epsilon_{o}$ equal to $gH/T_{0}$—has some empirical support
in the work of Kader and Yaglom (1990), but their empirical relationship,
equivalent to $\epsilon_{o}=1.1\;gH/T_{0}$, implies that the contribution of
entrained kinetic energy to $\epsilon_{o}$ outweighs work done against
buoyancy in the capping inversion in moderately and strongly convective
conditions. This is disturbing in itself, and it must fail in less convective
conditions when gravitational potential energy becomes the minor contributor
to the outer budget of mechanical energy. Of the two relationships necessary
to support a correlation between $z_{s}$ and $-L$, we expect the
$u_{\epsilon}=u_{*}$ approximation to become more accurate in nearer-neutral
conditions but the $\epsilon_{o}=gH/T_{0}$ approximation to become worse.

##  8 Conclusions

In this paper we have surveyed ensemble-averaged observations of temperature
profiles, $T$ spectra and $wT$ cospectra and probability plots from within
CBLs, and given interpretations of them based on the conceptual model of eddy
and plume processes previously introduced by McNaughton et al (2007); Laubach
and McNaughton (2009). This survey has been wider than the previous accounts,
and it includes number of new results and interpretations. In particular, we
now include the cross-sectional areas of plumes in our interpretations of
empirical results. We have been able to identify the $z^{-1/2}$ power law for
the mean temperature gradients above the SFL with the narrowing of rising warm
plumes as they rise, and to connect this power law with the half power in the
$\lambda^{1/2}z^{1/2}$ mixed length scale that collapses the positions of the
peaks of $T$ spectra above the SFL. We have also presented results from a new
analysis of $T$ spectra from within the SFL at the SLTEST experimental site.
These confirm some of the results reported by Laubach and McNaughton (2009),
and they support a model where plumes near the ground are effectively
randomized by the action of shear eddies, but above that, by $z/z_{s}\approx
0.1$, these are swept together and given form by the action of $z_{s}$-scale
impinging outer Richardson eddies, in which they become embedded. This process
also dictates their areas and so the form of the temperature profile further
from the ground. Quadrant plots of $T$ probability distributions and joint
$w$-$T$ probability distributions support this interpretation.

A key feature of our survey is that it uses concepts that are consistent with
our general understanding of the CBL as a dissipative complex dynamical
system. In particular, we identify eddies and plumes as emergent structures
that embody the high levels of organization. Emergence is a defining
characteristic of all far-from-equilibrium, dissipative systems, whose high
level of organization is sustained by a constant flow of energy. This energy
takes the form of mechanical energy in dynamical systems, and our similarity
scheme is based on the parameters that characterize this energy flow. The same
perspective leads to an understanding of buoyancy in CBL flows that is very
different to that introduced a century ago, by Richardson (1920).

Even so, the new model raises many questions. We still cannot say why mixed
scales always involve $1/2$ powers. We need a new experiment to characterize
the temperature profile using the new similarity parameters. We do not know
how to model entrainment in a way that will connect energy flows within the
CBL to the synoptic-scale flow. This task of parameterizing entrainment is
made more complicated by the expectation that the entrainment rate will depend
on both the form and the energy of the large structures within the CBL. These
are matters for the future. Even so, we are optimistic that the present
approach can be extended usefully to the near-neutral and stable regimes in
atmospheric boundary layers.

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