Fluxprofile relation with roughness sublayer correction_anrqvist_bergstrom

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Quarterly Journal of the Royal Meteorological Society Q. J. R. Meteorol. Soc. 141: 1191–1197, April 2015 B DOI:10.1002/qj.2426
Flux-profile relation with roughness sublayer correction
J. Arnqvist* and H. Bergström
Department of Earth Sciences, Uppsala University, Sweden
*Correspondence to: J. Arnqvist, Department of Earth Sciences, Uppsala University, Villavägen 16, 75236 Uppsala, Sweden.
E-mail: johan.arnqvist@geo.uu.se
Calculation of momentum flux using Monin–Obukhov similarity theory over forested
areas is well known to underestimate the flux. Several suggestions of corrections to the
standard flux-profile expression have been proposed in order to increase the magnitude
of turbulent flux. The aim of this article is to find a simple, analytical representation for
the characteristics of the flow within the canopy layer and the surface layer, including the
roughness sublayer. A new form of the roughness sublayer correction is proposed, based on
the desire to connect the shape of the roughness sublayer correction to forest characteristics.
The new flux-profile relation can be used to find the flux or the wind profile whenever
simple and fast estimations are needed, as for mesoscale modelling, scalar transport models,
or sound propagation models.
Key Words: roughness sublayer; wind profile; dimensionless gradient; stability expressions
Received 24 February 2014; Revised 13 June 2014; Accepted 7 July 2014; Published online in Wiley Online Library 26
August 2014
1. Introduction
When estimating wind profiles and momentum fluxes over any
surface Monin–Obukhov scaling (hereafter MO scaling) provides
a robust relation between surface characteristics and vertical
profiles. However it is a well-documented fact that MO scaling
underpredicts the momentum flux over forests (Thom et al., 1975;
Garratt, 1980; Högström et al., 1989; Finnigan, 2000). The part of
the surface layer where the flow deviates from the surface-layer
approximations by the effect of the roughness elements is defined
as the roughness sublayer. It is still an open question whether
this is due to a direct influence of the roughness elements or by
flow regimes created by the high roughness. The most prevalent
idea for the cause of the roughness sublayer is the mixing-layer
analogy, first put forward in detail in Raupach et al. (1996).
The mixing-layer analogy ascribes flow characteristics of the
roughness sublayer to vortices which are created at the canopy
top, as a result of the inflection point instability in the wind
profile. Reports in the literature gives height of the roughness
sublayer to be approximately 3h∗ where h is the height of the
trees (Raupach et al., 1991; Kaimal and Finnigan, 1994; Wenzel
et al., 1997). MO scaling underestimates the flux in the roughness
sublayer and, with large areas of the Earth covered by forests,
this is problematic on all ranges of atmospheric modelling that
use first-order closure for the surface layer. Physick and Garratt
(1995) developed corrections to increase the flux in a mesoscale
model and improved the diurnal variation of scalar profiles.
However, as pointed out by Harman and Finnigan (2007) their
correction did not produce a continuous profile and lacked a
wind profile within the canopy. For scalar flux estimations in
∗All variables are defined in the Appendix.
various models and noise-level estimations in sound propagation
models, the wind profile within the forest needs to be known.
The need for simple and fast calculation will exclude the use of
more sophisticated higher-order models for many purposes. An
analytical profile that is continuous from the ground to the top of
the surface layer is therefore important. The profile of Harman and
Finnigan (2007) was successful in predicting the mean flow but
there is no analytical solution to their flux-gradient expression.
De Ridder (2010) proposed an approximation to enable analytical
integration which resulted in an error of maximum 4% compared
to numerical integration. However the profile of De Ridder (2010)
does not include an expression for the wind within the forest.
Without a roughness sublayer, assuming a constant-flux layer,
the wind gradient can be described by
∂U
∂z
= u∗
κz
φm(z/L), (1)
where u∗ is the scaling velocity, defined as (u′w′2 + v′w′2)−1/4.
The Obukhov length L is defined as
L = −u2∗T0
κgθ∗
. (2)
Stability expressions for φm have been thoroughly examined,
and we shall adopt the expressions of Högström (1996), repeated
here for convenience:
φm =
{
(1 − 19z/L)−1/4 for z/Lfound a slight
variation of d with stability. The mean value of d was found to
be 22 m for Site 2 in the selected sector and is reported to be
15 m for Site 1 (Högström et al., 1989). This corresponds to 0.75h
for Site 1 and 0.88h for Site 2. In canopy flows, h is a natural
choice of length-scale central to the flow dynamics. However
at some distance above the canopy d becomes more important
than h, and thus it is convenient to follow Harman and Finnigan
(2007) and define the variable dt = h − d as it takes into account
both heights. In the following derivation of the mathematical
expressions dt will be a central length-scale.
3. Theory
3.1. Mixing-layer theory
The mixing-layer analogy as responsible for the enhanced
momentum transport is an attractive idea, as it offers an
explanation to the origin of the large organized structures found
above canopies. In a plane mixing layer, large horizontal rolls
are formed from a flow instability due to the different wind
speeds in the two layers. Even a thick ideal forest does not
resemble a plane mixing layer, but the inflection point in
the wind profile is present and that is a necessary (but not
sufficient) condition for the instability to form (Drazin, 2002).
Drag forces and diabatic stability might stabilize the flow so
that in stable diabatic conditions evanescent waves are formed
instead. However, in a fully developed flow, with moderate
to high wind speed, coherent structures resembling those in
a mixing layer are likely to form. To these structures much
of the momentum transport can be ascribed (Bergström and
Högström, 1989; Raupach et al., 1996; Finnigan, 2000; Thomas
and Foken, 2007). The concept of a flux-profile relationship relies
on the existence of eddies and their diffusing property. However,
with a pure mixing-layer, all the eddies originating from the
inflection instability would be centred at the inflection point.
This means that the eddies superimposed on a shear flow from
an inflection instability would enhance the exchange coefficient
at some distance above and below the inflection point. There are
results that indicate that a second instability leads to the formation
of rollers which are aligned with the wind speed. Between rolls of
opposite vorticity, areas of downward or upward moving air will
be created (Raupach et al., 1996). Finnigan et al. (2009) found
evidence of this from studying large-eddy simulation of a canopy
flow. They found that the rolls form pairs of horseshoe vortices,
with one upward and one downward transporting component.
This is a conceptual model that fits with both the organized
motion and the exchange coefficients observed. Still, with the
arguments presented above, it is not evident that the correct form
of the roughness sublayer correction is an exponential decrease
from the treetops and upwards as assumed by Mölder et al.
(1999), Harman and Finnigan (2007) and De Ridder (2010).
We propose a slightly different shape of the roughness sublayer
correction.
3.2. Formulation of the roughness sublayer correction
To compensate for the roughness sublayer effects on the flux-
profile relationship, a correction function, ϕ, is added to the φ
functions (Physick and Garratt, 1995; Mölder et al., 1999; Harman
and Finnigan, 2007; De Ridder, 2010). The form of this function
determines how the correction changes with height:
∂U
∂z
= u∗
κz
φm(z/L)ϕ(z, zr), (6)
where zr is a length-scale connected to the height of the roughness
sublayer. While earlier studies have assumed exponential decay of
the roughness sublayer effects with height (Physick and Garratt,
1995; Mölder et al., 1999; Harman and Finnigan, 2007; De Ridder,
2010), to the authors knowledge there is no physical reasoning
behind this assumption. We propose, in addition, to include a
non-dimensional length-scale as an amplitude to the exponential
function. This expression has the benefit of being easy to integrate,
and has the possibility of controlling the decay rate of the
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Flux-Profile Relation with Roughness Sublayer Correction 1193
0.2 0.4 0.6 0.8 1
1
1.5
2
2.5
3
φm
z 
/ h
Figure 1. Vertical shape of the ϕ function; the solid line is from Eq. (7), the
dashed line from Harman and Finnigan (2007), and the dash-dotted line from
De Ridder (2010). ϕm at the treetops was set to be 0.5 and the roughness sublayer
height was set to 2.5h in the Ridder expression.
roughness sublayer effect close to the canopy top. The form
of the ϕ function will then be
ϕ = 1 − z
η
e−z/dt , (7)
where η is a length scale that controls the magnitude of the
enhanced flux. The choice of dt as the appropriate length-scale in
the exponential expression in Eq. (7) comes from the desire to
have the maximum of ϕm at the treetops, as we expect that the
mixing-layer eddies are centred there. It also means that as the
ratio dt/h shrinks (increasing density of the forest), the depth of
the roughness sublayer will shrink. This is consistent with basic
scaling laws as a rough measure of the eddy size is U/(dU/dz),
and dU/dz is expected to grow with increasing forest density. If
the flux is magnified by a factor 1/χ at the treetops compared to
MO scaling the value of η becomes
η = dt
(1 − χ)
e−1. (8)
The shape of ϕ can be seen in Figure 1 together with the
expressions given by Harman and Finnigan (2007) and De Ridder
(2010).
To get the height dependence of U , Eq. (6) is integrated
according to Physick and Garratt (1995):
{U(z)−U(z0)}κ/u∗
= ln(z/z0)−{�m(z/L)−�m(z0/L)}
−
∫ z
z0
φm(1 − ϕm)z−1dz.
(9)
To remove the wind speed at z0 from the left-hand side of Eq. (9),
we recognize that the roughness sublayer correction vanishes as z
approaches infinity. Applying Eq. (9) at z → ∞ and subtracting
Eq. (4) at z → ∞ gives
U(z0)κ/u∗ =
∫ ∞
z0
φm(1 − ϕm)z−1dz. (10)
Substracting Eq. (10) from Eq. (9) gives the correct expression
for U(z):
U(z)κ/u∗ = ln(z/z0) − {�m(z/L) + �m(z0/L)}
+
∫ ∞
z
φm(1 − ϕm)z−1dz.
(11)
With the current form of the correction function (Eq. (7)) the
integrated correction from Eq. (11) reduces to
�̂m = η−1
∫ ∞
z
φm e−z/dt dz, (12)
which has an analytical solution for both stability expressions
(Eq. (3)) of the φm function. For Lat h must be equal for both the profile within and above the
canopy, and that∂U/∂z is equal at h. Combining Eqs (11) and (16)
gives the first condition while derivation of Eq. (16) and combina-
tion with Eq. (6) gives the second. This way the constant α can be
set in a way that ensures the profile is continuous at the canopy top:
α = −h φm(dt/L) ϕm(dt/L)
dt
[
ln(z/z0)−{�m(dt/L)−�m(z0/L)}−�̂m(dt)
] . (17)
This expression means that the only parameters controlling the
wind profile within the forest are the wind and the wind shear
at the canopy top. Naturally this is a vast simplification, since
drag distribution and thermal stability within the forest is sure to
affect the flow. However, it is convenient from a computational
perspective and will provide a first approximation of the wind
within the canopy.
4. Comparisons with measurements
4.1. Model parameters
The model has a few parameters that needs to be determined to
close the set of equations. The length scales, h, dt and η need to
be known a priori. For h and dt, mean values from calculations
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1194 J. Arnqvist and H. Bergström
10
−3
10
−2
10
−1
10
0
0
0.5
1
c
χ
1.5
−d
t
 / L
d
t
 / L
10
−6
10
−4
10
−2
10
0
0
0.5
1
1.5
(a)
(b)
Figure 2. The value of χ as a function of dt/L for (a) Site 1 and (b) Site 2.
Individual measurements are indicated In (a) by x marks (unstable conditions)
and In (b) by dots (stable conditions). The black line is the mean value of
logarithmically spaced bins with standard deviation shown by bars. The dashed
line is from Eq. (18).
according to section 2 will be used. The value of η is determined
using Eq. (8),but the value of χ needs to be known. χ is equal to
how much MO similarity underestimates u∗ at the tree height. For
Site 1, Högström et al. (1989) give a value of χ = 0.5. A review of
different values for χ can be found in Cellier and Brunet (1992),
where values range from 0.4 to 1. It is not clear from the literature
whether χ is a function of stability. The estimated u∗ value from
MO theory has a stability dependence from the φm function. If
the real value of u∗ does not follow the same stability dependence,
χ will be a function of stability. In Figure 2, χ is plotted against
dt/L. Note that in the figure the data from Site 1 are strictly
unstable and the data from Site 2 are strictly stable. Data from
the two sites were split in two parts by the data reference number
being odd or even. One part was used in Figure 2 and the other
part to compare the theory against the wind profile in section 4.2.
Both sites show a stability dependence of χ . To account for this,
a stability function is proposed where the maximum effect of the
roughness sublayer is for neutral conditions and approaching 1 as
the diabatic forcing grows. The stability expression has the form
χdt/L = χn +
(
1
1 − χn
+ 0.2
1
|dt/L|
)−1
, (18)
where χn is the value of χ in neutral conditions.
z / L
ϕ
m
−1 −0.5 0 0.5 1 1.5 2
0.1
1
10
Figure 3. The value of φm as a function of z/L from 38 m at Site 2. The thin line
is the mean value with standard deviations shown by bars. The bold line is from
Högström (1996) (Eq. (3)).
The value of z0 is another crucial parameter for the estimation
of the wind profile. In the model z0 is determined from Eq. (15).
Use of Eq. (15) requires L, which has been taken from the
measurements. We can gain some insight to the validity of
Eq. (15) by estimating z0 from the measurements higher above
the forest where the roughness sublayer effects are negligible.
From Site 2, measurements from the highest boom in the mast
(38 m), displayed in Figure 3, show that φm follow very closely
the theoretical curve from Eq. (3). However, a seemingly good
agreement in Figure 3 could be an effect of self-correlation since
both x- and y-axes are functions of u∗. To determine this, the
data were also examined by the method proposed by Baas et al.
(2006) who explored self-correlation in φ function plots and
suggested isolating u∗ to only one axis. Plotting the data in
this way leads to an increased scatter and the result suggests
that d = 22 m is a little too large for Site 2, but the data still
follow the classical surface layer φm expression with a correlation
coefficient of 0.65. That result allows us to use Eq. (4) to determine
another estimate of z0 based on the measurements of the wind
profile and the momentum flux. The results of that comparison
is shown in Figure 4. Both the new theory and the values of z0
determined from measurements show a decrease in roughness
with increasing stability. Zilitinkevich et al. (2008) argued that
z0 should in fact decrease with increasing stable stratification
and developed a model for the stability dependence based on
arguments of relevant length-scales. Their model is given below
for reference and is also shown in Figure 4.
z0 =
{
z0n
{
1 + 1.24(−h/L)1/3
}
for L 0.055 0.055 to 0.031 0.031 to 0.01 0.01 to −0.014 −0.014 to −0.055for Site 2 but there is a large negative
bias for Site 1. The lack of data in unstable conditions makes the
form of the stability dependence of χ an uncertain point, and it is
evident that, although Eq. (18) leads to a decreasing z0 in unstable
conditions, the wind profiles from Site 1 suggest that the roughness
should decrease even more. By examining the wind profiles from
Site 1, the roughness length falls from 1.9 in neutral conditions to
0.7 in unstable conditions. The model predicts the corresponding
values z0n = 2.5 and z0u = 1.6 which are obviously too high. It
is possible that more data would lead to altered form of Eq. (18)
in unstable conditions. Furthermore, it is evident that for dense
forests with almost all the leaf area situated in the tree crowns, like
Site 2, the modelled profile cannot predict the increase in wind
speed near the ground. Consequently the lowest measurement
point is underestimated. Since the theory of Inoue (1963) assumes
a rod-like appearance of the roughness elements, there are no
physics in the model to predict such behaviour. Thus it should
not be regarded as a model fault, but rather a discrepancy of
the model.
In Figure 7 the new wind profile expression is compared to
the expressions of Harman and Finnigan (2007) and De Ridder
(2010) as well as the MO profile. Constants relating to the
roughness sublayer were taken as recommended in the original
articles. The stability was defined by an Obukhov length of 100,
1000, and −100 m for the stable, neutral and unstable profiles
respectively. In the MO profile and the De Ridder (2010) profile,
(e)
(c) (d)
(a) (b)
0.5
1
1.5
2
2.5
0 2.5 5 7.5 10
0.5
1
1.5
2
2.5
U / u*
0 2.5 5 7.5 10
U / u*
0 2.5 5 7.5 10
U / u*
0 2.5 5 7.5 10
U / u*
0 2.5 5 7.5 10
U / u*
z 
/ h
0.5
1
1.5
2
2.5
z 
/ h
0.5
1
1.5
2
2.5
z 
/ h
0.5
1
1.5
2
2.5
z 
/ h
z 
/ h
Figure 5. Site 1: evaluation of the new flux-profile expression (solid line) against
measured averages (x-marks) of U/u∗. The horizontal bars indicate one standard
deviation from the measured averages. The dashed line is the Monin–Obukhov
expression with z0 from Eq. (15). (a) stable, (b) stable close to neutral, (c) neutral,
(d) unstable close to neutral, and (e) unstable.
the roughness length z0 = 1.8 m was used. For the new profile χn
was set to 0.65 and then allowed to vary according to Eq. (18).
The roughness sublayer height was set to 40 m in the De Ridder
(2010) expression. The values of z0 and χn was chosen so that
the profiles collapse above the roughness sublayer in neutral
conditions. The comparison shows that the new profile has a very
similar shape to the profile from Harman and Finnigan (2007) in
neutral conditions, but that in stable and unstable conditions the
solution for z0 causes the profiles to spread. The biggest difference
is in unstable conditions where the new profile suggests that z0
should decrease, which is in contradiction to both Harman and
Finnigan (2007) and Zilitinkevich et al. (2008). However the
decrease in roughness in unstable conditions is supported by the
measurements from both Sites 1 and 2. Both the new profile
expression and the Harman and Finnigan (2007) expression have
a small roughness sublayer correction compared to the profile of
De Ridder (2010), partly owing to the fact that the latter does not
include a forest wind profile.
5. Conclusions
A new expression giving the wind profile within a forest canopy
and in the roughness sublayer above a forest has been proposed. It
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1196 J. Arnqvist and H. Bergström
0 2.5 5 7.5 10
0
0.5
1
1.5
2
z 
/ h
z 
/ h
z 
/ h
z 
/ h
z 
/ h
(a) (b)
(c) (d)
(e)
0 2.5 5 7.51012.51517.520
0
0.5
1
1.5
2
0 2.5 5 7.5 10
0
0.5
1
1.5
2
0 2.5 5 7.5 10
0
0.5
1
1.5
2
0 2.5 5 7.5 10
0
0.5
1
1.5
2
U / u* U / u*
U / u*U / u*
U / u*
Figure 6. As Figure 5, but for Site 2. (a) very stable, (b) stable, (c) stable close to
neutral (d) neutral and (e) unstable close to neutral.
2 4 6
0
1
2
3
0 5 10 15
0
1
2
3
U / u* U / u*
U / u*
z 
/ h
z 
/ h
z 
/ h
(a)
(c)
(b)
0 5 10
0
1
2
3
Figure 7. Comparison of four different flux-profile expressions. The solid line is
the new expression, the dashed line is from Harman and Finnigan (2007) and
the dash-dotted line is from De Ridder (2010). The dotted line is the normal
surface-layer expression without a roughness sublayer correction. (a) is for stable
stratification, (b) for neutral, and (c) for unstable.
includes a stability function for the roughness sublayer effect and
it predicts stability dependence of the roughness length. For a full
wind profile in the surface layer, the required input parameters
are: d (or dt), h, χ , L and u∗. The value of χ = ϕ(dt) is a crucial
parameter for the model and is likely to change from forest to
forest. However, a classification could be made for different types
of forest as guidance when measurements are not available. The
model is fairly simple to use, but it requires an implicit equation
to be solved to find the value of z0. If only the profile over the
trees is of interest, the value of z0 can be specified a priori as
with standard surface-layer schemes. Zilitinkevich et al. (2008)
proposed a stability-dependent roughness model with increasing
z0 in unstable conditions and decreasing z0 in stable conditions.
Both the stable data from Site 2 and the measurements from Site 2
support the theory that z0 should decrease in stable stratification.
In unstable conditions, an increasing z0 cannot be supported with
the current dataset. In fact, the new model predicts a decrease
in roughness in unstable conditions but the magnitude of this
decrease is matched by observations from only one of the two
datasets.
Acknowledgements
This research has been part of the part of the Vindforsk III project
V-312, Wind Power in Forests. Data from Site 1 were provided
by the Department of Earth Sciences at Uppsala University. The
authors would like to acknowledge Ebba Dellwik for running the
measurements at Site 2 and providing background information
about the site.
Appendix
List of variables
d Displacement height from ground
dt Displacement height from treetops
g Gravitational acceleration
h Canopy height
L Obukhov length
T0 Background virtual temperature
U Mean wind speed
u∗ Friction velocity
u Zonal wind speed
v Meridional wind speed
w Vertical wind speed
z Height over d
zr Length-scale related to
roughness sublayer height
z0 Roughness length
z0n Roughness length in neutral conditions
z0u Roughness length in unstable conditions
α Constant for the wind profile
within the canopy
 Gamma function
η Length-scale in the
roughness sublayer correction
θ∗ Surface-layer temperature scale
κ von Kármán constant
φm Dimensionless wind gradient
ϕm Correction function for the
roughness sublayer
χ ϕ(dt)
�m Integrated stability function for momentum
�̂m Integrated roughness sublayer correction
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