New models of the unstable atmospheric surface layer
K.G. McNaughton
School of GeoSciences,, The University of Edinburgh, Kings Buildings,
Edinburgh EH9 3JU, United Kingdom. e-mail: keith.mcnaughton@ed.ac.uk
Abstract
For the last 50 years Monin-Obukhov similarity theory has provided a foundation for nearly all models of theatmospheric surface layer. The theory was developed when turbulence was still thought to consist of small parcels of
fluid (eddies), with short lifetimes and erratic movements with respect to the mean flow. Such eddies would respond
only to local conditions in the flow, so larger-scale influences were not considered. Recently two new models have beendeveloped which assign much greater importance to the turbulence in the outer parts of convective boundary layers.
The new models of Hunt and Morrison (Euro. J. Mech. B - Fluids 19, 673-694, 2000) and McNaughton (BoundaryLayer Meteorol. 112, 199-221, 2004) both see the turbulent surface layer as being the direct product of eddies from theconvective outer layer interacting with the ground. Hunt and Morrison propose that eddies from the outer layer impinge
onto the surface where they are blocked by the surface and sheared by the mean wind, causing distortion. They use thelinearizing assumptions of rapid distortion theory to analyze this process. By contrast, McNaughton proposes that theeddies from the outer layer create variable shear across the whole surface layer, which shear powers the development of'TEAL' structures within the surface layer. These TEAL structures are attached to the ground, develop upwards by acascade process, and are the fundamental elements of a self-organizing system of eddies within the surface layer. The
TEAL model is incompatible with rapid distortion theory The paper points out some testable predictions of the newmodels.
Keywords: turbulence, Monin-Obukhov similarity, rapid distortion theory, self-organizing systems.

1. Introduction
In the 1940s turbulence was understood in terms ofPrandtl’s mixing length model. This model sawtransport by turbulence as analogous to transport by themolecules of a gas, but with whole parcels of fluid
replacing the molecules. Though large compared tomolecules, these ‘eddies’ were assumed to be very small
compared to the outer dimensions of the flow. Like
molecules, the eddies moved about erratically while
being carried along by the mean flow. During their brieflife they travelled a typical distance called the ‘mixing
length’, analogous to the mean free path of gas
molecules. Transport by such eddies was described bydiffusion equations, so the problem of modelling
turbulent transport became that of modelling the ‘eddy
diffusivity’. This depended on the mixing length and so
only on the local properties of the flow.
Using these concepts, Obukhov (1946) assumed thatbuoyancy forces could affect transport only by changing
the local properties of the flow: by changing the form
(structure) and intensities (velocity scale) of the eddies.
He represented the local influences by the friction
velocity u , kinematic virtual heat flux w' v', and the*buoyancy parameter g/ v. Though not a local parameter,height, z, was added because it was known to beimportant when modelling the mixing length inboundary layers. Dimensional analysis based on thesefour parameters lead to Monin-Obukhov similarity
theory.
This local view of turbulence is now known to be
wrong. Eddies, or coherent structures, can be large, evenspanning whole flows, and have dynamics that reflect
non-local influences. For example, large convective
eddies can span whole atmospheric boundary layers, so
the variations in near-surface winds can depend on thedepth of the whole boundary layer, zi . Monin-Obukhov
similarity theory, which neglects zi , can be correct only
if these large eddies do not affect vertical transport

processes near the ground. There is mounting evidence
that this is not so, and that the outer layer must be
considered when modelling transport in the surface layer(Högström, 1990; Khanna and Brasseur, 1997;
Johansson et al., 2002; McNaughton and Brunet, 2002).
The question now is how to understand this influence?
Here we compare two new models that address thisquestion. These models agree on the importance of theouter turbulence in determining the properties of thesurface layer, but they differ widely in their proposed
dynamics.

2. The 'top-down' model
The top-down model proposes that turbulence in theatmospheric surface layer is created from eddies from theouter part of the boundary layer that impinge onto theground. The model was proposed by Hunt and Morrison(2000), with further contributions by Hunt and Carlotti(2001), Högström et al. (2002) and Drobinski et al.
(2004). This model is illustrated in Figure 1.
This model divides the surface layer into two parts: an
upper part called the shear surface layer; and a lower partcalled the eddy surface layer. In the shear surface layer theeddies moving down from above are blocked by theground while being sheared by the mean wind. Nearer
the ground, in the eddy surface layer, these distorted
large eddies scrape along the ground, creatingintermittent internal boundary layers which then break
down into substructures. The large-scale ramp structures
observed by Hommema and Adrian (2003) are interpreted
as this breakdown process in action. Turbulence in thesurface shear layer achieves its distinctive form only inflows with Reynolds number high enough that velocity
spectra display clear inertial subranges. A shear surface
layer appears in the atmospheric boundary layer, but may
be absent in laboratory flows with lower Reynolds
numbers.
This conceptual model is analyzed using rapid
distortion theory (RDT). The starting point is thefrictionless Navier-Stokes equations

wavenumbers, up to k1 ~ 1/Λs , the spectra reflect the
motions of large eddies that fill the whole boundary layer
and have horizontal lengths ~Λs . They fill the boundary
layer and so cannot move vertically or impinge onto theground. Eddies of intermediate size can move about andcan impinge onto the ground. The distortion of theseeddies is analysed using RDT. Their final break-up
occurs in the eddy surface layer. It is by non-linear
processes that are beyond the scope of RDT. These
eddies affect the spectra at wavenumbers k1 > 1/z.
The general forms of various spectra predicted for theeddy shear layer are shown in Fig 3. The horizontal
velocity spectra k1 E11 (k1 ) and k1 E22 (k1 ) have flat
middle sections where E11 , E22 ~ k–1 .

3. The 'bottom-up' model

Figure 1. Schematic representation of turbulence
processes in a convective boundary according to the
Hunt 'top down' model. Here zi ≈ 1 - 2 km, hs ≈ 100 200 m, he ≈ 10 - 20 m. Adapted from Hunt and Carlotti(2001).
∂u i

∂ ui

∂Ui

∂u i

∂p
∂xi

(1)
∂t
∂xj
∂x j
∂ xj
and the continuity equation
∂u i
(2)
=0
∂xi
where symbols have their conventional meanings and thevelocities have been split into mean, Ui , and fluctuating,ui , components. These equations govern the dynamics ofall eddies larger than a few millimetres. RDT linearizes(1) by neglecting the term before the '=' sign. Thisequation can describe the blocking and shearing of large
eddies on time scales short compared to those of internal
eddy processes, but it can not describe the transfer ofenergy from one scale of motion to another by non-linear
processes, such as eddy break-up, amalgamation or
growth. RDT is justified when eddies are 'rapidly'transformed by interaction with the surface, which is tosay when the mean strain rate (the inverse of the timescale associated with the mean shear, S = (∂Ui /∂x)-1 ), isshort compared to the evolutionary time scale of theeddies, s. It does not apply to eddies of the Richardson
cascade, whose internal rate of strain is s = (k1 2 )1/3 .
Here k1 is the streamwise wavenumber associated withthe eddy. In the neutral surface layer (∂Ui /∂x) ~ u /kz and*= u 3 /kz, where k is von Kármán's constant. This*means that RDT can apply only to eddies whose heights
are large compared to height above ground.
Mathematical development of the top-down model is
described in the cited papers. What concerns us here are
the testable predictions arising from this analysis,
especially the predictions of the forms of velocity spectrain the surface layers. These spectra behave differently ineach of three wavenumber ranges. At the smallest
=U j

+u j

+u j

=–

The bottom-up model proposes two kinds ofturbulence whose interaction defines the top of the surface
layer. The outer turbulence is created by buoyancy andits largest eddies span the whole boundary layer. In
typical daytime conditions eddies from the outer layer do
impinge onto the surface layer, where they are blocked
and sheared, but the main effect of this is to transfer
energy from the vertical wind components to thehorizontal components at the top of the surface layer.
This creates a variable horizontal wind at the top of thesurface layer and so variable shear across the wholesurface layer. This is superimposed on the mean shear
and acts in exactly the same way as the mean shear. This
unsteady shear drives production of turbulence within thesurface layer which has entirely different characteristics.
In this model the turbulence within the surface layer isa self-organizing system of eddies that transfer
momentum down to the ground. It is 'bottom-up' in thesense that instabilities form at the ground and initiate
structures that grow upwards until they reach the top of
the surface layer. These eddies draw their power from thelocal shear, so their transport properties depend on boththe mean and the variable components of that shear.
The model is described by McNaughton (2004b, c).
Its basic element is a coherent structure called the'Theodorsen ejection amplifier-like', or TEAL structure.
Each structure is initiated by an upward squirt of air,called an ejection. The oncoming flow lifts over, andcurls around this creating a vortex with a hairpin-shaped
core. This creates another, larger ejection from within its
arc. A TEAL structure can thus initiate another, larger
TEAL structure, so we can have TEAL cascades.
Growing TEAL structures, compete for space bydistorting each other. Only the best formed and mostsymmetrical at each stage produce ejections able toinitiate a further cycle. The fragments of unsuccessful
TEAL structures join the down-scale Richardson cascade
to dissipation. TEAL cascades are 'inverse' cascades, inthat they are in the inverse direction to the down-scale
Richardson cascade.
The TEAL model predicts that the height of the layer
dominated by TEAL structures is given by zs ~ u 3 /kε o,*where o is the dissipation rate in the outer layer. Herethe upwards development of the largest TEAL cascades
is checked by strong interactions with eddies of similar
scale formed as part of the outer Richardson cascade.
This defines the height of the surface layer. The layer

Figure 2. Schematic representation of turbulence
processes in a convective boundary layer according to the
TEAL model of McNaughton (2004b). Here zi ≈ 1 - 2
km, hs ≈ 1|L| - 2|L|, where L is the Obukhov length.
where the interaction occurs (Fig. 2) has a thickness
comparable to the size of the interacting structures, whichis to say, it is about as thick as the surface layer itself.
Buoyant production of turbulent kinetic energy roughly
equals dissipation in the outer layer, so o depends on
the surface heat flux and zs becomes correlated with the
Obukhov length, |L|. When zi /|L| is large zs is typically
about one or two times |L| (McNaughton, 2004b). Thus
the depth of the surface layer in the bottom-up model canvary from a few meters to the whole depth of theboundary layer. Typically it is 10 to 100m deep, so the'surface layer' of the bottom-up model corresponds to the'eddy surface layer' of the top-down model.

enough to be discriminated by careful experiments.
Fig. 3 shows k1 velocity spectra in the (eddy) surface
layer as predicted by the two models. The predicted
spectra for vertical velocity, E33 , are the same, but
predictions for the horizontal spectra are significantly
different. The top-down model predicts spectra thatstretch continuously from the smallest wavenumbers to
the largest ones, and predicts that E11 , E22 ~ k1 -1 . That
is to say, the spectra plotted in Fig. 3 as k1 E11 andk1 E22 have flat mid sections. This is a central predictionof the top-down model.
By contrast, the bottom-up model predicts that k1 E11
spectra have two parts—an 'inner' part at larger
wavenumbers reflecting the TEAL structures and theirbreak-down products, and an 'outer' part at smaller
wavenumbers representing the modulation of the inner
turbulence by the outer eddies. These spectral parts
simply overlap, and the heights and frequencies of theouter and inner peaks are governed by different
parameters. These predictions are also illustrated in Fig.
3.
In Fig. 3 the heights of the two peaks of the bottom-up
k1 E11 spectra are drawn with equal heights. This is often
observed in the surface layer, and it makes flat thelinking spectrum between two peaks. This flat section isindistinguishable from the k-1 spectrum predicted by the
top-down model. Such observations do not discriminate
between the two models. A stronger test is to compare
slopes of k1 E11 spectra at several heights. The top-down
model predicts mid regions of k1 E11 spectra to be flat at
all heights within the surface layer, while the bottom-up
model predicts the slope of this linking region to
decrease progressively with height, moving towards
negative values as the outer peak grows relative to theinner peak. Lauren et al. (1999) find slopes to increase

4. Comparing models
The top-down and bottom-up models are based onquite different conceptual models of eddy processes in thesurface layer. Theoretical arguments are incapable ofdeciding which, if either, is correct. Absolute theoretical
proofs will remain beyond reach until the full NavierStokes equations can be solved. The RDT analysis of
the top-down model is attractive if the non-linear term in(1) is indeed unimportant, but not otherwise. The TEAL
cascade mechanism of the bottom-up model is attractive,
but only if TEAL structures exist and behave as
described in the model. The two models are also
incompatible. The top down model proposes that theouter eddies penetrate the surface layer, break up andcontribute their energy directly to the Richardson cascadein the surface layer. The TEAL model proposes that theouter eddies do not penetrate the surface layer but simply
modulate the shear across it.
Despite these differences in concept we should not
expect gross differences in model predictions because theconstruction of both models has been guided byexperimental results. Even so, some of their predictions
are different enough that meaningful comparisons can be
made. In particular, the predicted forms of the horizontal
velocity spectra in the (eddy) surface layer, are different

Figure 3. Spectra of wind components in the atmosphericeddy surface layer according to the top-down and bottomup models. Axes are logarithmic.

with height, but this single result needs confirmation by
results from a more extensive site.
Transverse velocity spectra, k1 E22 , can also becompared. The few such spectra published show outerand inner peaks to be spread more widely than in k1 E11
spectra. The inner peak is at a wavenumber about twicethat in the k1 E11 spectrum. This has been ascribed tothe different effects that streamwise aggregation of
attached eddies has on the k1 E22 and k1 E11 spectra(McNaughton, 2004a). Observations show the outer peakof k1 E22 to be typically at about half the wavenumber ofthe corresponding outer k1 E11 peak (Nicholls andReadings, 1981; McNaughton and Brunet, 2000; Honget. al., 2004). This, presumably, is because the outerconvective cells are partly aligned with the mean wind.
The predictions of the top-down model are not sensitive
to the separation of the shoulders of these spectra, so it
again predicts a flat mid spectrum. The bottom-up
model, on the other hand, predicts that a spectral gap
will open up to reveal the separate natures of the two
peaks. This gap would be larger when peak separation is
maximized by taking observations near to the groundwithin a deep boundary layer. The few published results
do show a gap, but confirmation is needed.
Cross-wind spectra can be observed by aircraft flying
across the mean wind, and these can also be comparedwith model predictions. Unfortunately k2 E11 (k-2 ), andk2 E22 (k-2 ) spectra are rarely published, but those that areshow the outer and inner spectral peaks to be spread
more widely in these than in spectra taken along thewind. The wavenumber of the inner peak of the k2 E11 (k2 ) spectrum is at a wavenumber about a decade larger
than in the along-stream spectrum k1 E11 (k-1 ). The
explanation given by the bottom-up model is that thestreamwise alignment of attached eddies in the surface
layer does not affect k2 spectra (McNaughton, 2004a).
The k2 E11 (k-2 ) spectrum of Nicholls and Readings(1981) shows increased separation and a distinct spectral
gap in the k2 E11 (k2 ) spectrum. This is a single resultthat requires confirmation.

5. Conclusion
Two new models have been proposed to explain how
turbulence processes in the outer part of the boundary
layer can affect turbulent transport near the ground. Bothmodels introduce outer-layer parameters, so neither iscompatible with Monin-Obukhov similarity theory.
However, the two models are also incompatible witheach other.
This brief paper has outlined the nature of the twomodels and their incompatibly. It also points out thatthe two models can be discriminated experimentally bycomparing the turbulent spectra they predict withexperimental observations. The most useful observations
are also those least reported: transverse velocity spectra
measured from an aircraft flying along the wind, or from astationary tower using Taylor's frozen turbulence
hypothesis, k1 E22 (k1 ); and cross-wind spectra ofhorizontal velocity components which can be observed
from aircraft, k2 E11 (k2 ) and k2 E22 (k2 ). The top-down
model of Hunt and Morrison (2000) and followers
predicts these spectra to be continuous with flat middle
sections, while the 'bottom-up model of McNaughton(2004b) predicts that separate peaks with a spectral gap

between will be apparent when the two peaks are widely
separated. For optimal discrimination spectra should bemeasured at low heights within a deep convective
boundary layer.

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