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SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE
BOUNDARY LAYER
L. MAHRT?
College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, OR 97331,
U.S.A.
(Received in final form 29 October 1999)
Abstract. The concepts of vertical structure of the boundary layer over homogeneous surfaces are
discussed, including the roughness sublayer, surface layer (inertial layer) and outer layer. As an
interpretive literature survey, the vertical depth of the influence of surface heterogeneity, relative to
such vertical structure, is examined in terms of blending height theory, convective boundary-layer
scaling and internal boundary-layer theory.
These concepts are examined with data over different surface types. The rich variety of types of
surface heterogeneity and background flow preclude description of their influence on the boundary
layer by one single approach. Nonetheless, new scaling arguments offer promise for convective
conditions.
Internal boundary layers are found to form only in certain situations and even then may exhibit
more diffuse vertical structure compared to the textbook internal boundary layer. New scaling ar-
guments are developed to describe the potential for internal boundary-layer development in flow of
warm air over a cooler surface. These arguments explain some aspects of the observations but remain
incomplete.
Keywords: Heterogeneity, Surface fluxes, Convective boundary layer, Blending height, Internal
boundary layer.
1. Introduction
The atmospheric boundary layer is traditionally partitioned into different idealized
layers including the canopy layer, roughness sublayer, surface layer, outer layer
and the entrainment zone (Figure 1). This division has served the community well
and provides a framework for organizing our investigation of the boundary layer.
However, mixing in the convective boundary layer is generated by eddies on both
small and large scales and large eddy motions cross the boundaries of these layers
without any special recognition of a sharp change in physics. Although existing
concepts of the vertical structure of the boundary layer are even less applicable
over heterogeneous surfaces, such concepts provide a useful andnecessary starting
point.
To some extent, the earth’s surface is always heterogeneous. When is surface
heterogeneity important and when must the above concepts of vertical structure
?E-mail: mahrt@oce.orst.edu
Boundary-Layer Meteorology 96: 33–62, 2000.
© 2000 Kluwer Academic Publishers. Printed in the Netherlands.
34 L. MAHRT
Figure 1. Idealized layering of the boundary layer.
be modified? The answer is partly a matter of both horizontal scale and amplitude
of the surface heterogeneity. To address this query, the vertical structure of the
boundary layer over homogeneous surfaces is reviewed in Section 2. Section 3
reviews blending height theory and convective boundary-layer scaling in terms of
the impact of surface heterogeneity. Section 4 studies the influence of the scale
of the surface heterogeneity on the vertical structure of the boundary layer, and
Section 5 describes four data sets that are analyzed in Sections 6–7. Section 6
assesses the applicability of the blending height and convective boundary-layer
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 35
scaling. Section 7 evaluates existing concepts for the internal boundary layer over
heated and cooled surfaces.
2. Vertical Structure over Homogeneous Surfaces
2.1. ROUGHNESS SUBLAYER
Two definitions of the roughness sublayer are used in the literature. In the first
definition, the roughness sublayer is the layer immediately above the surface where
the individual surface roughness elements induce horizontal variability of the time-
averaged flow. This horizontal variation is due to spatial preference of wakes and
thermals semi-attached to vegetation elements (Figure 1). In this respect, the top
of the roughness sublayer (z∗) is the ‘blending height’ for the surface roughness
elements. However, in this study, the term blending height will be reserved for de-
scribing the height-dependence of the influence of surface heterogeneity on a scale
that is large compared to the scale of individual roughness elements. Horizontally-
averaged fluxes are required in the roughness sublayer since observations for a
fixed point, such as on a tower, would not be representative and would be altered if
the sensor is displaced horizontally even a few metres.
The roughness sublayer is sometimes also defined as the region below the ‘sur-
face layer’, where Monin–Obukhov (MO) similarity theory (ST) no longer applies.
This definition emphasizes that the flux-gradient relationship in the roughness sub-
layer is different from that predicted by MO similarity theory. For example, the
eddy diffusivity for momentum may be enhanced beyond that predicted by MO
similarity due to mixing associated with wakes behind the roughness elements. As
a result, a modified similarity theory is formulated for the roughness sublayer (Gar-
ratt, 1980; Cellier and Brunet 1992; Raupach, 1992; Physick and Garratt, 1995). In
principle, similarity theory in the roughness sublayer should be constructed from,
and applied to, horizontally-averaged flow.
The above two definitions are not necessarily equivalent. For example, Wieringa
(1993) defines an intermediate layer that is above the influence of individual rough-
ness elements on the time-averaged flow yet MO similarity does not apply. This
speculation is referred to as a horizontally homogeneous part of the roughness sub-
layer in Figure 1. The existence of the homogeneous part of the roughness sublayer
has not been clearly shown from observations over land. However, in the analogous
wave boundary layer over lakes and oceans, the time-averaged flow is horizontally
homogeneous close to the surface yet MO similarity does not apply because the
flux-gradient relationship depends on the surface wavelength. MO similarity theory
does apply in the overlying surface layer, located above this wave boundary layer.
Generally, the depth of the roughness sublayer is considered to be proportional
to the mechanical properties of the surface such as the momentum roughness length
(Wieringa, 1993; Parlange and Brutsaert, 1993), spacing between the roughness
36 L. MAHRT
elements (Raupach et al., 1980), height of the roughness elements (Garratt, 1980;
Wieringa, 1993; Raupach, 1994), and silhouette area of the roughness elements
(Lettau, 1969). In Raupach (1994), the depth of the roughness sublayer extends
a vertical distance of 2(hc−d) above the displacement height, d,wherehcis
the height of the roughness elements or canopy. Some of these formulations are
surveyed in Hess (1994) while a detailed treatment is provided by Raupach et al.
(1991). However, with heated weak wind situations, thermals may occur at pre-
ferred locations with respect to the individual roughness elements, as speculated in
Figure 1. Preferred locations for thermals emanating from the top of an individual
roughness element may extend the depth of the roughness sublayer beyond the
above predictions, which is consistent with the findings of Garratt (1978). Garratt
(1978) points out the potentially important role of thedifferent vertical distributions
of heat and momentum sources and sinks, which in turn vary between different
types of canopies.
2.2. SURFACE LAYER AND OUTER LAYER
MO similarity theory is applied in the surface layer (Figure 1) and assumes that the
underlying roughness sublayer is sufficiently thin, that the flux in the surface layer
is close to the spatially-averaged surface flux. In the surface layer, MOST is used
to predict the flux-gradient relationship in terms of the height above ground (or
displacement height) and the Obukhov length. The bulk formula based on vertically
integrated similarity theory is the primary method for predicting surface fluxes in
models. Sun et al. (1999) emphasize that the roughness length and aerodynamic
surface variables represent the influence of the surface characteristics on the flux-
gradient relationship only in the surface layer.
Above the surface layer, in the outer layer, the flux-gradient relationship is influ-
enced by the boundary-layer depth and additional scaling arguments are required
(Sorbjan, 1989). With weak winds and strong surface heating, the surface layer
may be separated from the outer mixed layer by a free convection regime (e.g.,
Holtslag and Nieuwstadt, 1986). If the boundary layer is shallow and/or the rough-
ness sublayer is deep, the surface layer may be squeezed from existence (Mahrt et
al., 1998). Then there is no layer where MOST applies since the influence of the
boundary-layer depth essentially extends downward to the roughness sublayer.
These concepts are based on homogeneous surfaces. Over heterogeneous sur-
faces, the surface layer may be eliminated, depending on the horizontal scale of
the surface heterogeneity. Then MOST is expected to break down. The vertical
extent of the influence of surface heterogeneity and the breakdown of surface layer
similarity theory are considered in the next two sections.
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 37
3. Scaling Approaches
If the surface heterogeneity is due only to the roughness elements, and therefore
influences only the roughness sublayer, the flow is considered to be homogeneous.
Studies of surface heterogeneity have implicitly assumed a distinct separation of
scales between the surface heterogeneity and the roughness elements, as will be
assumed here. This condition is often not met. For example, some forests are char-
acterized by clumping of trees. Is a clump of trees a surface roughness element or
should it be considered as very small-scale heterogeneity?
Surface spatial variations often occur simultaneously on a continuum of ho-
rizontal scales or consist of gradual changes without a well-defined scale, as
sometimes occur with natural changes of vegetation or soil moisture gradients
associated with recent precipitation patterns. In the next two sections, we assume
that surface heterogeneity occurs on a single well-defined scale. In Section 6, we
consider observed atmospheric flows where the surface heterogeneity may occur
simultaneously on a variety of horizontal scales.
3.1. BLENDING HEIGHT –LOCAL DIFFUSION
The blending height is viewed here as a scaling depth that describes the decrease
of the influence of surface heterogeneity with height. The blending height is not a
level where the influence of surface heterogeneity suddenly and completely van-
ishes but instead is the level where the influence of surface heterogeneity gradually
decreases below some threshold value. The blending height can be re-formulated in
terms of internal boundary layers and footprint theory (Mahrt, 1996). The blending
height has been studied mainly from numerical models, although the existence of
the blending height has been inferred from tethered balloon measurements (Grant,
1991). The various blending height derivations were intended more as qualitative
scaling arguments based in part on linear theory. Therefore, quantitative compar-
isons of the blending height formulations with more complex atmospheric flows
in this study may exceed the intended use by the original authors. One role of the
nonlinearity is evident in the response of the atmosphere to spatial variations of
the surface roughness. Schmid and Bünzli (1995) show that the effective rough-
ness length applied to the flow above the blending height is augmented due to the
fact that the effects of the rough-to-smooth and smooth-to-rough transitions on the
area-averaged momentum flux do not cancel, leading to significant ‘second order
roughness’.
Below the blending height, the turbulence may not be in equilibrium with the
local vertical gradient, which excludes application of MO similarity. MO similar-
ity can be applied above the blending height only if the blending height is low
compared to the boundary-layer depth. This condition is required in order that
the flux immediately above the blending height is close to the spatially-averaged
38 L. MAHRT
Figure 2. Scaling regimes based on blending height concepts (lower main part) and convective
boundary-layer scaling (upper part). The diagram is not to scale and the boundaries between different
regimes depend on stability and other boundary-layer variables. Here the surface layer is chosen
arbitrarily to be equal to 0.05 h so that the flux in the surface layer would be within 5% of the surface
value for a linear decrease of the flux with height to near zero at the boundary-layer top. The hatched
area denotes the scale regime where MO similarity is not valid. The vertical dashed line is defined
by Equation (11).
surface value and that the boundary-layer depth does not influence the flux-gradient
relationship.
Different formulations of the blending-height approach are discussed in Mason
(1988), Wood and Mason (1991), Claussen (1990), and Schmid and Bünzli(1995).
The general form of the blending height, zblend, can be expressed as
zblend =Cu∗
Up
Lhetero,(1)
where u∗is the friction velocity based on the horizontally-averaged momentum
flux, Uis the speed of the spatially averaged wind vector based on the flow at
the blending height, Lhetero is the horizontal scale of the surface heterogeneity, pis
usually 2 and Cis a nondimensional coefficient usually taken as unity. With p=2,
zblend becomes the drag coefficient times the scale of surface heterogeneity. The
diffusion height corresponds to p=1 (Wood and Mason, 1991; Claussen, 1990).
Observations do not normally provide winds at the time-dependent blending height
and winds from a fixed observational level(s) must be substituted. The influence of
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 39
surface heterogeneity is small above the blending height (Equation (1), region B in
Figure 2) and theoretically vanishes at the diffusion height.
The blending height increases linearly with horizontal scale of the surface het-
erogeneity so that the influence of larger scale surface features extends upwards
farther into the boundary layer. Consider a reference level, z, that might be the
first model level or the observational level. Equation (1) can be used to predict the
minimum horizontal scale of the surface feature, Lblend, required to significantly
influence the flow at level zby equating zwith zblend. Then, the minimum horizontal
scale for important influence of the surface heterogeneity becomes
Lblend =CblendzU
u∗2
.(2)
If the horizontal length scale of the surface feature, Lhetero, is less than Lblend,then
the surface feature does not induce significant horizontal heterogeneity at level z;
that is, level zis above the blending height (region B in Figure 2). From another
point-of-view, the impact of surface heterogeneity at height zis an increasing func-
tion of Lhetero/Lblend. This dependence will be incorporated in Section 3.5. Based
on observations in Section 6.1, Cblend is chosen to be roughly 0.6.
3.2. THERMAL BLENDING HEIGHT
The horizontal length scale, Lblend, decreases slowly with increasing instability
since u∗/U is predicted to increase slowly with increasing instability. However,
when surface heating is important, this weak dependence on instability under-
estimates the importance of instability (Section 6) and a more explicit stability-
dependence is required. Wood and Mason (1991) derive a thermal version of the
blending height based on a linearized thermodynamic budget where the perturb-
ation flow is induced by the surface heterogeneity. The thermodynamic budget
reduces to an approximate balance between the temperature advection and ver-
tical divergence of the perturbation heat flux where the perturbation heat flux is
due to surface heterogeneity. The resulting thermal blending height for unstable
conditions is
zwm ≡
Lheterow0θ0
sf c
U2
o
,(3)
where w0θ0
sf c is the absolute value of the spatially-averaged surface heat flux and
2ois the averaged potential temperature. In the derivation of this relationship, the
influence of the spatial variation of the heat flux cancels and the resulting blending
height for heat depends only on the background surface heat flux. This cancellation
occurs due to linearization; presumably the spatial variation of the heat flux would
be introduced in higher order analysis.
40 L. MAHRT
Equating the reference level zwith zwm, the minimum horizontal scale of the
surface heterogeneity that significantly influences the flow at level zgives
Lwm =CwmzU2
o
w0θ0
sf c
.(4)
Scales of surface heterogeneity larger than Lwm are predicted to significantly influ-
ence the flux at level z.With decreasing wind speed and increasing surface heating,
Lwm decreases and small scale surface features exert a stronger influence on the
flow at level z.ThevalueofCwm is estimated from observations in Section 6.1 to
be roughly 3.1 ×10−3.
3.3. BOUNDARY-LAYER CONVECTIVE SCALING
The blending height concept (Section 3.1) is based on shear-driven mixing where
the transport is more diffusive and local. In contrast, in the convective boundary
layer, large boundary-layer eddies can transfer information from the surface to the
top of the deep boundary layer on the time scale of a few tens of minutes and the
concept of a blending height is less applicable. For this case, Raupach and Finnigan
(1995) pose the impact of surface heterogeneity in terms of the basic time scales
of the convective boundary layer. The mixing time scale is defined as zi/w∗,where
ziis the depth of the convective boundary layer and w∗is the Deardorff convective
velocity scale. During one mixing time scale, the flow travels
LRau ≡CRau Uzi
w∗
,(5)
where CRau is a nondimensional coefficient estimated from observations in Section
6.1 to be roughly 0.8. For horizontal scales of surface heterogeneity smaller than
LRau, the influence of surface heterogeneity is confined to depths which are small
compared to the boundary-layer depth. The large convective boundary-layer eddies
horizontally mix the influence of the surface heterogeneity to the extent that it does
not significantly influence the bulk of the boundary layer.
This convective scaling does not address the height-dependence of the influence
of surface heterogeneity in that it predicts the influence of the surface heterogen-
eity on the boundary layer as a whole. Since the eddy length scale increases with
height in the lower part of the boundary layer, even in convective conditions, one
might expect that the influence of smaller scales of heterogeneity decreases with
height. Then, near the top of the boundary layer, only the largest scales of surface
heterogeneity are important, as with blending height arguments. We return to this
problem in Section 3.5.
An additional larger time scale is introduced into the convective scaling ap-
proach of Raupach and Finnigan (1995). The entrainment time scale,Te, describes
the period of growth of the convective boundary layer since its initial development
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 41
earlier in the morning. The corresponding length scale, TeU, defines the largest
horizontal scale where the surface heterogeneity influences the boundary layer. For
surface heterogeneity on scales significantly larger than TeU, the entire boundary
layer reaches approximate equilibrium with the local surface conditions.
3.4. THE THREE HORIZONTAL LENGTH SCALES
We can summarize the three horizontal length scales as follows:
1. Lblend is based on local diffusive mixing by shear-generated turbulence and
seems most applicable to near-neutral or stable conditions.
2. Lwm allows buoyancy-modification of the turbulence.
3. LRau represents the case of nonlocal bulk mixing due to buoyancy-driven
boundary-layer eddies on the scale of the boundary layer and seems most
applicable to the convective boundary layer.
All three length scales estimate the minimum horizontal scale of surface hetero-
geneity that significantly influences the flow at level z, or in the case of LRau,the
entire boundary layer. All three length scales are proportional to the wind speed
and inversely proportional to some measure of the strength of the turbulence.
3.5. OVERALL MODEL FORMAT
Toformulate a measure of the spatial variability of the flux, we define the standard
deviation of the flux, σF, divided by the spatially-averaged flux, F, both evaluated
at level z. Based on the above arguments, one can postulate
σF
F=fLhetero
LXGσTsf c
θo,(6)
where LXrefers to one of the horizontal length scales in the previous subsection.
The function, f(L
hetero/LX), is expected to increase with Lhetero/LXsince the tur-
bulence substantially reduces the influence of surface heterogeneity on horizontal
scales of LXand smaller. The function fcould consist of multiple terms where
LXdepends on both Lblend and LRau, in order to include the limiting cases of
near-neutral flows and the convective boundary layer. The dependence of σF/F
on the amplitude of surface heterogeneity, G, will be posed later in terms of the
surface radiation temperature. Over surfaces with complex vegetation, the surface
radiation temperature needs to be replaced with a more reliable indicator of surface
heterogeneity. Simplification of Equation (6) is pursued in Section 6 by analyzing
aircraft data. Examination of Equation (6) with data allows consideration of only
horizontal scales of heterogeneity that are smaller than the observational domain.
42 L. MAHRT
4. Scale Regimes
Based on the horizontal length scales derived in the previous section, we now for-
mulate different scale regimes for the surface heterogeneity based on the vertical
extent of their influence. The various scaling regimes are schematically illustrated
in Figure 2. The terminology at the bottom of the figure is modified from the survey
of Mahrt (1996) while the notation at the top follows Raupach and Finnigan (1995).
The influence of heterogeneity on the scale of the individual roughness elements is
confined to the roughness sublayer (Section 2), noted by “E” in Figure 2. We now
discuss the vertical depth of the influence of surface heterogeneity on the micro-,
meso- and macro-scales.
4.1. MICROSCALE HETEROGENEITY
If the blending height is low compared to the depth of the boundary layer,
zblend << h, (7)
then the flux immediately above the blending height is close to the spatially-
averaged surface value and Monin–Obukhov similarity theory can be used to
predict the surface flux. This case is referred to as microscale surface heterogeneity
(left hatched region in Figure 2) and from Equations (1) and (7) corresponds to
Lhetero << Cblend(U/u∗)2h. (8)
For the thermal blending height (Equation (4)), the corresponding condition is
Lhetero << Cwm
U2
o
w0θ0
sf c
h. (9)
Convective scaling does not explicitly address the influence of surface heterogen-
eity in the surface layer, although one would anticipate that Lhetero <L
Rau is a
necessary condition in order that MO similarity theory is applicable.
The above inequalities can be used to assess the suitability of the tile approach
used to partition subgrid surfaces in numerical models, also known as the mosaic or
flux aggregation approach (see Avissar and Pielke, 1989, or additional references
in Mahrt, 1996). Such an approach was formulated for momentum transfer over
heterogeneous surfaces in terms of an effective roughness length (Taylor, 1987).
Each grid surface area is divided into smaller subgrid areas corresponding to dif-
ferent surface types while the variables at the first model level, presumably within
the surface layer, are assumed to equal their grid-averaged values, unaffected by
subgrid surface heterogeneity. Such an assumption can be justified as a rough ap-
proximation only if the blending height is below the first model level, which in turn
must be below the top of the surface layer. One could derive restrictions on the scale
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 43
of the surface heterogeneity for the tile approach by replacing the boundary-layer
depth in Equations (7)–(9) with the height of the first model level. For larger scale
surface heterogeneity (higher blending height), the tile approach formally breaks
down. Then one must compare the errors resulting from this breakdown against
errors resulting from complete neglect of subgrid surface heterogeneity.
4.2. MESOSCALE OR INTERMEDIATE-SCALE HETEROGENEITY
If the equalities in Section 4.1 are not met, then the influence of the surface het-
erogeneity extends upward to a significant fraction of the boundary-layer depth
(region C in Figure 2). Then the flow immediately above the blending height is too
high to estimate the surface flux (region B, Figure 2). For convective scaling, the
mesoscale regime is defined as LRau <L
hetero <T
eU. In terms of blending height,
the condition is h < Lblend <hwhere may be chosen as 0.05, for example, to
approximate the depth of the surface layer.
In this scaling regime, MO similarity theory cannot be used to predict the
spatially-averaged surface fluxes. The blending height is sufficiently high that there
is no longer a level above the blending height where the flux-gradient relationship
depends on z/L alone and where the spatially-averaged flux is close to the surface
value. This failure of similarity theory is probably responsible for the failure of the
emergence of a unifying approach for predicting surface fluxesovermesoscale het-
erogeneity in terms of effective roughness lengths or transfer coefficients. Different
studies yield different predictions, depending on the details of the surface condi-
tions and assumptions invoked. For mesoscale heterogeneity, no rigorous approach
is available.
4.3. MACROSCALE OR LARGE-SCALE HETEROGENEITY
If the predicted blending height is greater than the boundary-layer top,
zblend >h, (10)
then the local surface feature controls the entire boundary layer. This condition
occurs when the horizontal scale of surface heterogeneity exceeds
Lhetero >C
blendhU
u∗2
,(11)
corresponding to region D in Figure 2. In this region, the boundary layer establishes
equilibrium with the local surface type and similarity theory can be applied without
concern for the change of surface conditions far upstream.
This boundary-layer equilibrium regime occurs with convective boundary-layer
scaling if
Lhetero TeU. (12)
44 L. MAHRT
This regime is referred to as macroscale heterogeneity. For small wind speed in
Equation (12) or small boundary-layer depth in Equation (11), macroscale hetero-
geneity may occur on scales that are normally referred to as mesoscale. Note that
the above discussions refer to mesoscale and macroscale organization of the tur-
bulent fluxes and do not address the direct vertical transport by mesoscale vertical
motions.
5. Data Sets
To assess the impact of surface heterogeneity on fluxes in the atmospheric bound-
ary layer, we will analyze data collected by the Twin Otter of the Canadian National
Research Council in three different field programs. The measurements were made
at approximately 35 m above the surface, which roughly corresponds to the top of
the surface layer.
5.1. SGP
The primary data set for this study is repeated aircraft passes at approximately
35 m over the El Reno track during 10 flight days of the Southern Great Plains
Experiment (SGP). The flight on 2 July 1997 is selected as a case study since it
contains the most complete data set for this field program. On this day, the winds
were from the southwest andcontained asignificant wind component parallel to the
east-west track, allowing examination of the impact of surface heterogeneity and
advection. On most of the days, the wind was mainly perpendicular to the track.
The 13-km El Reno track consists of mixed farming in the west, senescent wheat
and bare fields in the central part where evapotranspiration was smaller and green
grasslands in the eastern part where the evapotranspiration was larger. The time
series from repeated aircraft passes have been stretched to account for variable air-
craft ground speed and have been aligned using ground markers based on remotely
sensed NDVI (Normalized Difference of Vegetation Index) and surface radiation
temperature. Fluxes, variances and means are computed from aircraft data for
windows of 1-km width using unweighted means. The 1-km window generally cap-
tured almost all of the turbulent flux. The window is sequentially marched through
the record by increments of 250 m. For some calculations, variables are first aver-
aged over all of the aircraft passes for each window position. The number of passes
within each flight ranged from 3 to 12. Boundary-layer depth was estimated from
aircraft soundings that were executed at variable intervals during the flight periods.
For two of the flights, the boundary-layer depth was not well-defined, leaving eight
flights for the analysis below.
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 45
5.2. CODE, BOREAS AND RASEX
We will also analyze data from two flights in the California Ozone Deposition
Experiment (CODE), each consisting of eight repeated passes at 35 m by the NRC
TwinOtter over irrigated and nonirrigated surfaces (Pedersen et al., 1995; Mahrt et
al., 1994). Land-lake contrasts will be examined with NRC Twin Otter data taken
in the Boreal Ecological and Atmospheric Study (BOREAS) described in Sun et
al. (1997). We choose the case where the wind was most aligned with the aircraft
track.
Internal boundary layers are also studied using data collected during the Risø
Air Sea Experiment (RASEX) with an instrumented mast, 2 km off the Danish
coast (Mahrt et al., 1998) and an instrumented tower at the shore. As a case study,
we have chosen 24 October 1994 where a stationary stable boundary layer formed
for a 5-hour period of offshore flow of warm air over cooler water. Fluxes, variances
and means are computed from the tower data in terms of deviations from a 10-
minute mean. The fluxes and variances are then averaged over a one-hour period.
6. Evaluation of Length Scales
Determination of the blending height from data is problematic, partly because
the influence of surface features gradually fades with height and other mesoscale
features gradually become dominant. For example, in the uppermost part of the
boundary layer in the Southern Great Plains Experiment, transient spatial variations
of the boundary-layer depth and entrainment led to substantial spatial variation of
measured fluxes along the aircraft track in the upper part of the boundary layer.
These variations partially mask any remaining influence of surface heterogeneity
on the upper part of the boundary layer. Therefore this study concentrates on the
low level passes.
The influence of transient motions and large random flux sampling errors can
be reduced by first averaging over all of the passes for a given flight for each 1-km
segment (Section 5). Unfortunately, this averaging incorporates different boundary-
layer conditions within the averaging period associated with diurnal variation
during the flight period of several hours. Nonetheless, the relationship between
the atmospheric response σF/F and the horizontal length scales (Section 4) is
improved by first averaging variables over the passes. These relationships probably
would improve further had more repeated passes been available. The following
reports results only for flight-averaged values.
6.1. FLUX CORRELATION WITH SURFACE TEMPERATURE
The influence of surface heterogeneity on the boundary layer can be assessed in
terms of the correlation between the heat flux at level zand the surface radiation
temperature. Here we have computed the correlation between the spatial variation
46 L. MAHRT
of the flight-averaged heat flux and the flight-averaged surface radiation temper-
ature for the 1-km overlapping windows. Considering all of the measurement
difficulties and complexity of real air-surface interactions, the relationship between
the heat flux-surface temperature correlation and the various length scales (Lblend,
Lwm and LRau) is promising (Figure 3), even though the scatter is large. When
these length scales are large, the impact of smaller scale surface heterogeneity on
the heat flux at level zis reduced and the correlation between the heat flux and the
surface radiation temperature decreases. Within the accuracy of the data, a clear
distinction between the predictive value of the length scales cannot be determined.
To estimate the nondimensional coefficients for the horizontal length scales
(LX=Lblend,L
wm or LRau) from the observations, we assume that the surface
heterogeneity exerts a significant influence on the fluxes at the aircraft level (near
the top of the surface layer) when the correlation between the flux and the sur-
face radiation temperature exceed 0.7 (50% or more variance-explained). This
threshold value was estimated from Figure 3 by subjectively drawing a curved line
through the data points. The value of the horizontal length scale corresponding to
this threshold, L∗
X, is then equated to the estimated scale of the dominant surface
heterogeneity, so that
L∗
X=Lhetero.(13)
The length scale for the surface heterogeneity, Lhetero, is not easily determined since
surface heterogeneity occurs simultaneously on a variety of scales. Based on sur-
face radiation temperature and NDVI, the principal scale of surface heterogeneity
appears to be about 4 km although significant variations occur on smaller scales.
Here, we chose 3 km for the horizontal scale of the surface heterogeneity and then
estimate the nondimensional coefficients from Equation (13) and the definitions of
LX. The numerical values of the coefficients were reported in Section 3 after the
defining relationships for LX. Because of the various uncertainties, the values of
the nondimensional coefficients are only order-of-magnitude estimates. Note that
the calibrated values of LXestimate the minimum horizontal scale of the surface
heterogeneity required for the most important influences on the fluxes at level
z.Thatis,LXis a scaling variable, not a cutoff value, and smaller scale surface
heterogeneity will still exert some influence on the flow at level z. To include all
of the significant influences on the fluxes at level z, one should decrease the values
of the nondimensional coefficients for the length scales by an order of magnitude.
The above estimates are prejudiced against smaller scales since the fluxes had to
be averaged over a one-kilometre window. Averaging over significantly smaller
windows would be vulnerable to random flux errors.
6.2. RELATIVE FLUX VARIATION
The prediction of the magnitude of the spatial variation of the flux at a given level,
z, is of considerable importance for a number of practical applications, including
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 47
Figure 3. The correlation between the heat flux and surface radiation temperature as a function of the
horizontal length scale: (a) Lblend,(b)Lwm,and(c)LRau.
48 L. MAHRT
scaling up of area-averaged fluxes. The spatial variation of the heat flux will be
represented by the spatial standard deviation of the flux computed from the over-
lapping 1-km windows that are averaged over all of the passes in the flight (σF).
The relative standard deviation is computed by dividing this standard deviation by
the track-averaged flux for the flight (F).
The relative standard deviation of the surface heat flux (σH/H ) shows no ob-
vious relationship with the blending height predictions, Lblend (Equation (2)) and
Lwm (Equation (3)). The failure of these horizontal length scales for the present
data set is due to: (a) the importance of the depth of the convective boundary layer,
(b) lack of information on the amplitude of the surface heterogeneity, and (c) a
number of additional complications noted below. The influence of the value of the
boundary-layer depth on σH/H is strong for boundary-layer depths greater than
about 400 m. Smaller boundary-layer depths correspond to transient cases earlier
in the morning when the convective eddies are not well-developed.
The relative standard deviation of theheat flux tends to decrease with increasing
LRau although the scatter is quite large (Figure 4a). With large values of LRau,the
influence of small-scale surface heterogeneity is more effectively eliminated by
large boundary-layer eddies. However, the large scatter in Figure 4a suggests that
other factors are important. To include a measure of the amplitude of the surface
heterogeneity, we pursue Equation (6) using LRau as the principal horizontal length
scale. Assuming linear dependencies between the relative standard deviation of the
heat flux and the scaling variables, Equation (6) is written as
σH
H∝Lhetero
LRau
σTsf c
θo
.(14)
For bookkeeping convenience, we define
LRT ≡CRT
LRauθo
σTsf c
,(15)
in which case
σH
H∝Lhetero
LRT
.(16)
The value of the nondimensional coefficient, CRT , is estimated from the method
in Section 6.1 to be 4.3 ×10−3. For the present data, the spatial variation of the
heat flux seems to be better related to LRT (Figure 4b) than LRau (Figure 4a) al-
though large scatter remains. The relative spatial variation of the heat flux depends
significantly on the amplitude of the surface heterogeneity. The amplitude of the
surface heterogeneity varies from day to day partly due to the precipitation history
along the flight track. After recent rains, the surface heterogeneity is less since the
moisture flux over the bare soil/senescent wheat region is almost as large as that
over the grassland.
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 49
Figure 4. Relative standard deviation of the spatial variation of the heat flux for the eight flights in
SGP as a function of: (a) LRau and (b) LRT and (c) the relative variation of the momentum flux as a
function of LRT .
50 L. MAHRT
The remaining scatter in Figure 4b could be due partly to the influence of non-
stationarity, errors in estimation of the surface fluxes and boundary-layer depth,
and the influence of transient mesoscale disturbances not removed by compositing
over the passes. The dependence on wind direction with respect to the surface
heterogeneity was found to be small. The influence of partial cloud cover was small
for flight-averaged values.
Consider the practical problem of estimating the required horizontal resolution
for remotely sensed surface variables used to infer spatial variation of surface
fluxes. Such inferences can be based on empirical models which are calibrated
in terms of aircraft flux data. The value of LRT determines the smallest scale of
surface heterogeneity, which significantly influences the flow at the aircraft level
(approximate top of the surface layer). Finer spatial resolution of surface character-
istics would not be needed, especially considering the approximate nature of such
models.
6.3. MOISTURE FLUX
The surface moisture flux seems to vary in a less organized fashion compared to the
surface heat flux. The relative standard deviation of the moisture flux for the flight-
averaged flow is 0.19, as compared to 0.34 for the surface heat flux. Over the drier
regions, the moisture flux decreases less than the heat flux increases so that the sum
of H+LE is greater. Here, Hand LE are the sensible and latent heat fluxes in
Wm
−2. We speculate that this difference is due to horizontal convergence of heat
and moisture below the aircraft level over warmer surfaces. However, the horizontal
convergence could not be evaluated from the data with sufficient accuracy.
The relationship of the surface moisture flux to LRau and the other length scales
is weaker than that for the surface heat flux. The heat flux may respond to the
surface heterogeneity in a more organized fashion because the temperature is more
dynamically active compared to moisture and, therefore, more strongly coupled
with updrafts and downdrafts. Furthermore, the moisture flux even near the surface
is often significantly influenced by entrainment.
6.4. MOMENTUM FLUX
The relative spatial variation of the momentum flux, σF/F ,whereFis the mag-
nitude of the stress vector, decreases rapidly with increasing values of all of the
heterogeneity length scales LX. As an example, this decrease for LRT is shown in
Figure 4c. The relative flux variation, σF/F , is much larger for momentum than
that for heat and moisture at small values of LRT . Apparently bluff roughness
elements such as occasional clumps of trees or buildings and edges of fields, exert
a strong influence on the momentum transfer even when such obstacles do not
significantly influence the heat and moisture flux.
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 51
7. Internal Boundary Layers
How does the formation of internal boundary layers affect the applicability of the
blending height arguments and convective boundary-layer scaling? To address this
question, it is necessary to distinguish between mesoscale and local internal bound-
ary layers (Garratt, 1990). Mesoscale internal boundary layers are associated with
large-scale surface features and eventually engulf the entire pre-existing boundary
layer. Mesoscale convective internal boundary layers due to flow of cool air over
a warm surface are often well-defined with a capping inversion, which slopes up-
ward in the downstream direction. Flow of marine air over a heated land surface
is a classic example of the mesoscale internal boundary layer (see references in
Garratt, 1990). Flow of warm air over cooler water on the scale of 100 km can lead
to a stable internal boundary layer with well-defined vertical structure (Garratt and
Ryan, 1989).
On smaller scales, this well-defined state is not necessarily achieved, as will
be discussed below. These small-scale local internal boundary layers grow to a
maximum depth which is small compared to the upstream boundary-layer depth,
hIBL h, (17)
and then encounter a new surface type and lose surface support. Here hIBL is the
top of the internal boundary layer, assuming that it is definable. Using traditional
scaling arguments, the maximum value of the internal boundary-layer depth is
estimated in Mahrt (1996) to be
max(hIBL)=CIBL σw
ULhetero,(18)
where CIBL is thought to be O(0.1) as also found in Section 7.3, although more ob-
servations are needed. This is a scaling estimate only and does not account for the
decrease of the growth rate of the internal boundary layer in the downstream direc-
tion. Local internal boundary layers are defined to be those cases where max(hIBL)
is small compared to the boundary-layer depth, which in turn requires that the scale
of the surface heterogeneity is sufficiently small that
Lhetero U
CIBLσw
h. (19)
If σwis formulated in terms of the convective velocity scale w∗, then the right hand
side of Equation (19) is proportional to LRau and scales of surface heterogeneity
that are small compared to LRau correspond to local internal boundary layers.
52 L. MAHRT
7.1. COOL TO WARMER
Consider an example of flow of cool air from a partially irrigated cool region
over a hot unirrigated surface, as observed during CODE Flight 19 (Section 5).
In this example, the heat flux was near zero over the partially irrigated surface
and quite large over the dry surface, about 0.2 K m s−1. Aircraft passes through
the sloping top of the internal boundary layer over the dry surface indicate that
the instantaneous top of the internal boundary layer is sometimes well-defined in
terms of a jump in potential temperature (Figure 5a). This example corresponds to
a slope of the top of the internal boundary layer of about 14%. However most of
the passes encountered a more diffuse zone of temperature change.
Since the position of the top varies in time, apparently in response to individual
thermals and fluctuations in wind speed, the time-averaged flow is characterized
by more gradual temperature changes compared to the instantaneous temperature
distribution. During periods of weak winds, the thermals over the heated surface
rise more vertically while during wind gusts the thermals are more tilted in the
downstream direction. As a result, the time-averaged horizontal structure is smooth
and no sharp changes are encountered. In this sense, the top of the internal bound-
ary layer represents the upper limit of the measurable influence of thermals during
the averaging period. For the present case, this upper limit rises steeply for the flow
composited over all of the passes (Figure 5b). We will consider this composite
to be an estimate of the time-averaged flow. Pockets of slightly warmer air are
even encountered just upstream from the surface discontinuity on three of the eight
passes (also evident in the composited temperature in Figure 5b), perhaps resulting
from eddies which temporarily reversed the horizontal flow. Ignoring this upstream
anticipation of the surface discontinuity, the heated internal boundary layer grows
with a slope of roughly 45% although uncertainties in the location of the top of
the internal boundary layer and alignment errors become significant for such steep
slopes.
We conclude that the top of the internal boundary layer for the time-averaged
flow: (1) is not associated with a sharp temperature change, (2) may rise with any
slope including nearly vertical orientation in weak-wind, heated conditions, and (3)
rises more vertically with greater averaging time which increases the probability of
capturing vertically oriented thermals.
The internal boundary layer forming in the flow of cool air from Candle Lake
over the heated land surface also initiates an internal boundary layer which rises
steeply (Figure 6). On other days when the ambient wind is weak, a divergent land
breeze leads to unstable internal boundary layers over land on both sides of the
lake, both of which rise steeply (Figure 5 in Sun et al., 1997).
Traditionally, the time-averaged internal boundary layer consists of a thin
thermal equilibrium layer at the surface (Brutsaert, 1982; Garratt, 1990; de Bruin
et al., 1991), where the potential temperature horizontally varies only slowly (Fig-
ure 5b), and a much thicker transition layer between the equilibrium layer and the
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 53
Figure 5. (a) An example of an instantaneous sharp horizontal change of temperature due to a single
aircraft crossing the sloped top of the internal boundary layer during Flight 19 of CODE. (b) Averages
over the eight passes.
54 L. MAHRT
Figure 6. Instantaneous air and surface temperature and vertical velocity in a stable internal boundary
layer over Candle Lake, where the wind is easterly and along the flight track at 6 m s−1. The lake is
slightly warmer at the west end, where the lake is shallow.
top of the internal boundary layer. The top of the instantaneous internal boundary
layer varies within the transition layer for the time-averaged flow. Based on po-
tential temperature for the time-averaged flow (Figure 5), the inferred depth of the
equilibrium layer for this data increases downstream with a slope of only about
2%. The concept of an equilibrium layer is not precise since the turbulence in the
equilibrium layer interacts with non-equilibrium turbulence above the equilibrium
layer.
7.2. WARM TO COOLER
Now consider flow from a warm surface to a cooler surface of limited width where a
stable internal boundary layer may or may not develop. McNaughton and Laubach
(1998) show that convective eddies advected over a cooler surface modulate the
cool stable air near the surface. Presumably, if the surface temperature change is
sufficiently small, mixing by the advected large eddies may prevent formation of
a stable internal boundary layer over the cool surface. As an instructive scaling
argument, consider cooler air which attempts to form in a layer near the surface
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 55
TABLE I
Evaluation of R. Field program (described in Section 5), change in surface radiation tem-
perature, upstream air temperature – downstream surface radiation temperature 1Tsf c
(◦C), upstream boundary-layer depth h(m), upstream standard deviation of the vertical
velocity σw, the ratio R, and ratio of downstream to upstream σw.
Field prog. 1Tsf c (◦C) 1θ (◦C) h(m) σw(m s−1)Rσ
wratio
Candle Lake 9.0 8.5 1500 0.6 0.7 0.22
CODE flt 13 28.0 2 900 0.5 3.75 0.35
RASEX Unavail. 1.5 500 0.4 2.0 1.0
SGP flt 13 9.5 ≈0 550 0.75 Large 0.90
with a temperature deficit of 1θ,where1θ is estimated as the upstream air temper-
ature minus the surface radiation temperature of the new surface. How far can the
advected large eddies with vertical velocities σwlift cool surface air upward? Here,
σwis chosen to be the standard deviation of the vertical velocity in the boundary
layer upstream. Based on solutions of the vertical equation of motion, Mahrt (1979)
found that the depth of lifting by large eddy updrafts can be estimated in terms of
the vertical scale ασ 2
w/g(1θ /2o). The coefficient αis probably large compared to
unity since the vertical motions of the most energetic updrafts and downdrafts are
significantly larger than σw.
Furthermore, the temperature deficit of air lifted from near the surface is less
than 1θ since the near-surface air will be warmer than the surface radiation tem-
perature. If the predicted depth of mixing is comparable to, or greater than, the
depth of the upstream boundary layer, then the cool air will be mixed throughout
the boundary layer and a new internal boundary layer does not form.
The ratio of the depth of upward transport of cool air to the upstream boundary-
layer depth is
R≡ασ2
w
hg 1θ /2 o
.(20)
If this ratio is large, then the advected large eddies from the boundary-layer up-
stream have enough kinetic energy to mix the new cool air throughout the boundary
layer. If this ratio is small, then a new cool internal boundary layer is more likely
to form. This approach ignores changes in the roughness of the surface and the
difficulty of defining the surface radiation temperature over complex vegetation. To
make Rof O(1) with typical development of internal boundary layers, αis chosen
to be 103. With this choice, how large must R(Equation (20)) become before a
stable boundary layer fails to develop? The ratio Rwas evaluated from the four
data sets described in Section 5 and then recorded in Table I.
56 L. MAHRT
As one quantitative measure of the strength of the stable internal boundary layer,
we define the ratio of the upstream standard deviation of the vertical motion to the
downstream standard deviation of vertical velocity within the modified flow. This
measure was better defined than those based on temperature. The σwratio is close
to unity for two of the three cases where Ris greater than unity. Other factors are
important as discussed below.
The Candle Lake case (Figure 6) is characterized by the smallest value of R,the
smallest value of the σwratio, and the best-defined internal boundary layer in terms
of the horizontal variation of mean variables observed by the aircraft. As warm air
from land flows over the cooler lake surface, the standard deviation of the vertical
velocity eventually decreases by almost a factor of five. Note that the change of
surface radiation temperature is not large compared to the other experiments partly
because the surface radiation temperature over the forested land is reduced by the
influence of the cool ground surface and shaded canopy elements and is therefore
not representative of the surface heating (Sun and Mahrt, 1995). However, the dif-
ference between the upstream air temperature and downstream surface temperature
is substantially larger than in the other experiments.
As anticipated from the study of McNaughton and Laubach (1998), large
boundary-layer eddies advected from the heated land surface influence the flow
at the upstream east end of the lake (Figure 6), but this influence decays rapidly in
the downstream direction. Ignoring the locally warmer air over the land near the
shore, the temperature decreases gradually in the downstream direction until about
7 km offshore when the temperature suddenly drops. This drop is probably due to
penetration of the aircraft through the slowly thickening stable internal boundary
layer. Considering that the flight level is about 30 m above the water surface, the
corresponding upward slope of the internal boundary layer averages only about
0.4%. This corresponds to CIBL =0.15 in Equation (18). The other aircraft pass
on this day shows intersection of the internal boundary layer at approximately the
same location as in Figure 6 although the horizontal temperature gradient upstream
from this intersection is greater.
Repeated flights were flown over Candle Lake on four other days when the lake
surface was considerably cooler than the air. Occasionally, the aircraft appeared to
penetrate a well-defined top of the cool stable internal boundary layer. However,
on most of the aircraft crossings, the temperature decrease was more gradual.
In CODE, the change of surface radiation temperature is much larger but the
upwind air-ground temperature difference is also large. As a result, the difference
between the upwind air temperature over the hot dry surface and downwind surface
temperature of the irrigated surface is significantly less than that for the Candle
Lake case. Nonetheless, this difference is significant and Ris relatively small.
This leads to a reasonably well-defined internal boundary layer and a substantial
decrease of σw(Table I).
The towers in RASEX capture flow of warm air from land over cooler water.
The internal boundary layer is complicated by flow acceleration from 6 m s−1,at
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 57
the land mast on the coast, to about 9 m s−1at the sea mast, 2 km downstream from
the coast. This acceleration is probably at least partly due to the decreased rough-
ness of the water surface. As a result of this acceleration, σwdoes not decrease in
the downwind direction in spite of increasing stability.
To various degrees, the above flow cases are complicated by:
(a) A roughness change that would act to accelerate the flow in the two cases of
flow over the water;
(b) An adverse component of the hydrostatic pressure gradient associated with
decreasing temperature in the downstream direction (Doran et al., 1995; Sun
et al., 1998);
(c) Acceleration of the flow immediately above the cool internal boundary layer,
that was originally within the upstream boundary layer, but is now decoupled
from the surface stress (Smedman et al, 1995; Mahrt, 1999).
These three influences are not included in R. Therefore, Ris, by itself,
incomplete.
As a final example, consider flow from a warm region of nontranspiring senes-
cent wheat and bare fields to a cooler region of transpiring green grass examined
with aircraft passes in SGP (Section 5). Although, the surface radiation temper-
ature decreases significantly between the senescent wheat and the grassland, the
air temperature upstream is approximately the same as the downstream surface
radiation temperature, making Rundeterminably large. An internal boundary layer
does not form in terms of the mean flow. Therefore, this case is considered to be
an “adjusting boundary layer” instead of an internal boundary layer, as discussed
in the next subsection.
7.3. ADJUSTING BOUNDARY LAYERS
An internal boundary layer may not develop in flow over small amplitude surface
heterogeneity. Instead, the original boundary layer gradually adjusts in the down-
stream direction without decoupling of the upstream boundary layer from the new
surface or modification of the mean vertical structure associated with formation of
internal boundary layers. To contrast the adjusting boundary layer with formation
of a new internal boundary layer, we consult the approximate stationary budget of
an arbitrary variable, φ, downstream from the surface change
u∂φ
∂x ≈−
∂w0φ0
∂z ,(21)
where the horizontal turbulent flux divergence has been neglected. The vertical
flux divergence in an internal boundary layer is large because the flux generally
decreases from its surface value to a small value at hIBL such that the right hand
side of Equation (21) scales as w0φ0
sf c /hIBL. This large vertical flux divergence
corresponds to large horizontal gradients in the mean flow. However in the adjust-
ing boundary layer, the vertical flux divergence is governed by the boundary-layer
58 L. MAHRT
depth and scales as w0φ0
sf c /h. This flux divergence is much weaker compared
to the internal boundary layer and therefore must correspond to weak horizontal
gradients. This distinction becomes more obscure in the nonstationary case.
Consider the flow from a strongly heated surface to a weakly heated surface,
as observed in SGP where air from senescent wheat and bare fields is advected
over green grasslands with large transpiration and weaker surface heat flux. The
surface heat flux decreases from about 0.15 to about 0.02 K m s−1. Here a well-
defined internal boundary layer is not observed in terms of mean variables such
as temperature and moisture. Temperature decreases by a few tenths of a kelvin,
mainly due to the absence of thermals over the grassland (Figure 7). That is, the
mean air temperature over the grassland is about the same as the air between the
thermals over the upstream region of senescent wheat. The weak surface heating
over the grassland is sufficient to maintain mixing between the air near the sur-
face and overlying boundary layer advected from upstream, preventing significant
vertical flux divergence and formation of an internal boundary layer.
However, the temperature variance decreases rapidly in the downstream direc-
tion due to the absence of strong thermals (Figure 7). Such thermals are absent
because the potential temperature of the advected air is not significantly different
from the surface radiation temperature of the grassland. Therefore, the adjusting
boundary layer does exhibit horizontal changes of some of the higher moments
but is missing significant horizontal change of the mean variables. Such boundary-
layer adjustments are probably common but have received little attention in the
literature.
8. Conclusions
Through interpretation of the existing literature and analysis of new data, this
study has examined the vertical extent of the influence of surface heterogeneity
on different scales. If the influence of mesoscale heterogeneity extends vertically
to levels where the spatially-averaged flux is significantly different from the surface
flux, then MO similarity theory and bulk methods cannot be used to estimate the
area-averaged surface flux. As a possible result, a consensus for parameterization
of effective roughness lengths and transfer coefficients for heterogeneous surfaces
has not emerged for these scales of surface heterogeneity. The analysis in Sec-
tion 4.1 indicates that the flux aggregation method (tile or mosaic approach) for
semi-explicit inclusion of subgrid surface heterogeneity in numerical models is
formally valid only with small-scale heterogeneity and not valid for larger scales
of heterogeneity.
The response of boundary-layer fluxes to surface heterogeneity increases with
the scale of the heterogeneity and decreases with observational height, wind speed,
boundary-layer depth and stability. In addition, ubiquitous background transient
motions influence the spatial variation of the measured flux. Actual surface het-
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 59
a)
b)
Figure 7. (a) The instantaneous spatial variation of potential temperature and (b) standard deviations
of temperature and vertical velocity computed over moving 100-m windows. An internal boundary
layer is not well-defined in terms of mean variables or change of σw.
erogeneity normally consists of simultaneous spatial variations on more than one
scale. Because of these multiple influences, simple scaling arguments can only
explain a limited fraction of the spatial variation of fluxes. Nonetheless, traditional
blending height formulations are able to roughly predict the degree of relationship
between the heat flux at the top of the surface layer and the surface radiation tem-
perature (Section 6.2). However, the blending height formulation cannot predict
the amplitude of the atmospheric response to the surface heterogeneity because:
(a) it does not include information on the amplitude of the surface heterogeneity,
and (b) the amplitude of the atmospheric response is reduced by horizontal mixing
due to large boundary-layer eddies that scale with the boundary-layer depth. Tra-
60 L. MAHRT
ditional blending height formulations do not include information on the height of
the boundary layer.
The scaling arguments of Raupach and Finnigan (1995) include information on
the boundary-layer depth and consequently successfully explain part of the spatial
variation of the heat flux when the flow is unstable. The scaling arguments of
Raupach and Finnigan (1995) are used to derive a new formulation which includes
the role of the amplitude of the surface heterogeneity. The resulting predictions
can be used to estimate the range of scales of surface heterogeneity that influence
the boundary layer at a given height (Section 6.2) and the required resolution of the
remotely-sensed surface heterogeneity for inferring the spatial variation of heat and
moisture fluxes. The spatial variation of the moisture flux is much less organized
than that for the heat flux and more data are required before a formulation can be
derived for the moisture flux. The above simple scaling arguments did not appear
to be as successful when applied to additional data sets with less passes per flight
(not discussed above). Data are required for near-neutral conditions to combine
with the present data sets. This would allow evaluation of combining the blending
height and convective scaling approaches.
With flows over surface discontinuities, local internal boundary layers often
fail to develop. When the change of surface properties is not sharp and/or of
small-amplitude, the boundary layer adjusts without formation of a new internal
boundary layer within the existing boundary layer. Suchadjusting boundary layers
exhibit significant horizontal variation of some of the higher moments, such as heat
flux and temperature variance, but do not show significant horizontal variation of
mean quantities. Even when local internal boundary layers do form, they normally
include a vertically thick diffuse transition layer without a sharp top. The top of
the internal boundary layer is only occasionally observed as a sharp interface from
instantaneous data. The top fluctuates in time so that the top of the time-averaged
internal boundary layer is diffuse. Over heated surfaces, the top of the internal
boundary layer represents the upper limit of the thermals over the new surface and
can rise quite steeply in weak wind conditions.
In contrast, internal boundary layers forming due to marked surface changes
affecting a large area, such as flow of marine air over a heated land surface (meso-
scale internal boundary layers), are characterized by well-defined tops associated
with capping inversions (Garratt, 1990). The literature has concentrated on flow
over sharp discontinuities. Little is known about flow over surface heterogeneity
where the boundary layer adjusts in a less dramatic fashion and traditional internal
boundary-layer concepts are of limited use.
Acknowledgements
I gratefully acknowledge Ian McPherson for the NRC Twin Otter data and Sesha
Sadagopan for computational assistance. The extensive comments of the reviewers,
SURFACE HETEROGENEITY AND VERTICAL STRUCTURE OF THE BOUNDARY LAYER 61
Dean Vickers, Nigel Wood and Jielun Sun are greatly appreciated. This material is
based upon work supported by Grants DAAH04-96-10037 and DAAD19-9910249
from the U.S. Army Research Office and Grant NAG5-8601 from the NASA
Hydrology Program.
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