1. ARTICLE IN PRESS
Atmospheric Environment 41 (2007) 7059–7068
www.elsevier.com/locate/atmosenv
Employing Heisenberg’s turbulent spectral transfer theory to
parameterize sub-filter scales in LES models
Gervasio A. Degraziaa,Ã, Andre B. Nunesb, Prakki Satyamurtyb,
´ ´
Otavio C. Acevedo , Haroldo F. de Campos Velhoc,
´ a
Umberto Rizzad, Jonas C. Carvalhoe
a
´
Departamento de Fısica, Universidade Federal de Santa Maria, Santa Maria, RS, Brazil
b
´
CPTEC, Instituto Nacional de Pesquisas Espaciais, Sa Jose dos Campos, SP, Brazil
˜o
c
´
LAC, Instituto Nacional de Pesquisas Espaciais, Sa Jose dos Campos, SP, Brazil
˜o
d ´
Istituto di Scienze dellAtmosfera e del Clima, CNR, Lecce, Italy
e
Faculdade de Meteorologia, Universidade Federal de Pelotas, Pelotas, RS, Brazil
Received 11 October 2006; received in revised form 23 April 2007; accepted 2 May 2007
Abstract
A turbulent subfilter viscosity for large eddy simulation (LES) models is proposed, based on Heisenberg’s mechanism of
energy transfer. Such viscosity is described in terms of a cutoff wave number, leading to relationships for the grid mesh
spacing, for a convective boundary layer (CBL). The limiting wave number represents a sharp filter separating large and
small scales of a turbulent flow and, henceforth, Heisenberg’s model agrees with the physical foundation of LES models.
The comparison between Heisenberg’s turbulent viscosity and the classical ones, based on Smagorinsky’s parameteriza-
tion, shows that both procedures lead to similar subgrid exchange coefficients. With this result, the turbulence resolution
length scale and the vertical mesh spacing are expressed only in terms of the longitudinal mesh spacing. Through the
employment of spectral observational data in the CBL, the mesh spacings, the filter width and the subfilter eddy viscosity
are described in terms of the CBL height. The present development shows that Heisenberg’s theory naturally establishes a
physical criterium that connects the subgrid terms to the large-scale dimensions of the system. The proposed constrain is
tested employing a LES code and the results show that it leads to a good representation of the boundary layer variables,
without an excessive refinement of the grid mesh.
r 2007 Elsevier Ltd. All rights reserved.
Keywords: LES subfilter; Heisenberg model; Convective boundary layer
1. Introduction (Deardorff, 1973; Moeng, 1984; Moeng and Wyn-
gaard, 1988; Schmidt and Schumann, 1989; Schu-
Large eddy simulation (LES) models represent a mann, 1991; Mason, 1994). In LES, only the energy-
well-established technique to study the physical containing eddies of the turbulent motion are explicitly
behavior of the planetary boundary layer (PBL) resolved and the effect of the smaller, more isotropic
eddies, needs to be parameterized. Modelling these
ÃCorresponding author. residual turbulent motions, which are also referred as
E-mail address: degrazia@ccne.ufsm.br (G.A. Degrazia). subgrid-scales or sub-filter motions (Pope, 2004) is in
1352-2310/$ - see front matter r 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.atmosenv.2007.05.004

2. ARTICLE IN PRESS
7060 G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068
large part a phenomenological procedure based on horizontal subgrid characteristic length scales and
heuristic arguments (Sullivan et al., 1994). the filtered wavelengths of the inertial subrange
In general, the governing equations in the LES from the observed turbulent spectrum in the CBL.
models are the incompressible Navier–Stokes equa- As a phenomenological consequence, the magnitude
tions described for a horizontally homogeneous of Dx, Dy and Dz are fixed in terms of these residual
boundary layer. The resolved turbulent flow quan- turbulent wavelengths and described as a small
tities (wind components, pressure, temperature, etc.) fraction of the dominant convective length scale.
are obtained by the application of a low-pass spatial In order to test the grid mesh spacing proposed,
filter of characteristic width, the turbulence resolu- the LES model developed by Moeng (1984) is
tion length scale (Pope, 2004), smaller than the employed, where the subgrid parameterization is
scales of the resolved turbulent motions. This low- based on Sullivan et al. (1994).
pass spatial filter width has the same order of
magnitude as the numerical grid dimensions and, 2. Subgrid scale model and spatial scales in a CBL
based on Kolmogorov spectral characteristics, can
be expressed in terms of the inertial subrange scales. In LES studies, one is trying to model eddies of
According to Wyngaard (1982), ‘‘the parameter- inertial-range scales and to resolve explicitly the
ization of the residual stress term in the large eddy largest energy-containing eddies. For a well-devel-
equation is dynamically essential; it causes the oped CBL, the typical Reynolds number is greater
transference of kinetic energy to smaller scales. than 107 and the inertial subrange is defined by the
Thus, its parameterization is a key step in develop- interval Lblbm, where L is the integral scale, l
ing a large eddy model’’. Interpreting the turbulence characterizes the size of the inertial subrange eddies,
field as a superposition of myriads of energy modes, and m is Kolmogorov’s microscale. The integral
the parameterization of residual stress means to scale characterizes the energy-containing eddies,
model the physical effects of a large number of which perform most of the turbulent transport,
degrees of freedom (turbulent scales), which, by while Kolmogorov’s microscale defines the size of
virtue of the filtering application are not explicitly the viscous eddies that dissipate the turbulent
resolved in a numerical simulation. energy. For a fully developed CBL, LE102 and
The purpose of the present study is to show that mE10À3 m, and the eddy size range in the CBL
Heisenberg’s classical theory for inertial transfer of covers roughly five decades (Sorbjan, 1989).
turbulent energy (Heisenberg, 1948) can be used to Furthermore, since the Reynolds number is very
parameterize the residual stress tensor in LES high for the CBL, the effect of molecular diffusivity
models applied to a convective boundary layer is irrelevant compared to the turbulent diffusivity,
(CBL). The idea is motivated by the fact that in the and hence molecular viscosity will have negligible
inertial subrange the Heisenberg’s theory establishes direct influence on the resolved-scale motions
that the transfer of energy from wave numbers (Mason, 1994). Therefore, the convergence of the
smaller than a given particular value to those larger statistics of the studied quantities with LES is not
can be represented as the effect of a turbulent expected to be affected by the dissipative scales.
viscosity. This introduces a division between scales A frequently used parameterization for the residual
at any arbitrary wave number in the inertial stress tensor tij in the large eddy models is expressed
subrange. This scale separation is indeed naturally by (Smagorinsky, 1963; Deardorff, 1973; Moeng,
relevant in the LES frame, where a cutoff wave ´
1984; Sullivan et al., 1994; Lesieur and Metais, 1996)
number is arbitrarily chosen in the inertial range,
qmi qmj
introducing then an artificial sharp division, to which tij ¼ ÀnT þ , (1)
Heisenberg’s approach seems well suited to be qxj qxi
applied. Therefore, in this paper, we employ where mi and mj are resolved velocity components, i,
Heisenberg’s theory to obtain a kinematic turbu- j ¼ 1,2,3, corresponding to the x, y, z directions,
lence viscosity (KTV), which represents the effects respectively, and nT is the turbulent eddy viscosity
of the filtered eddies of the inertial subrange. which is expressed as
From a physical point of view, the expression for
KTV provides a cutoff or limiting wave number, nT ¼ ck l 0 e1=2 , (2)
separating the resolved scales from those filtered. It where ck ¼ 0.1 is a constant, l0 is a mixing-length
allows to establish a relationship between the scale related to the filter operation (Mason, 1994),

3. ARTICLE IN PRESS
G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068 7061
and e is the turbulent kinetic energy (TKE) of rated Fourier elements of the spectrum (statistical
subfilter scales (SFSs). For neutral (zero surface independence between turbulent scales), is a prop-
turbulent heat flux) or unstable (positive turbulent erty of the inertial subrange turbulent eddies, whose
surface heat flux) conditions, l0 is equal to the low- scales range from an arbitrary but fixed wave
pass LES filter width D. number to infinity (since dissipative scales and
Following Weil et al. (2004), the low-pass LES molecular viscosity are irrelevant, as mentioned
filter width is given by before). Therefore, the magnitude of KTV will
#1=3 depend on the inertial subrange eddies that cause
3 2 the viscosity, and hence must depend on eddies with
D¼ DxDyDz , (3)
2 wave numbers in the range (kc,N), where kc is a
cutoff or limiting wave number for the inertial
where Dx, Dy and Dz are the computational mesh subrange, which can be determinated from the
sizes in the three coordinate directions x, y, z and experimental TKE spectra.
the constant (3/2)2 accounts for the dealiasing. An eddy viscosity is the product of a character-
Therefore, the filter width D is of the order of the istic turbulent length scale and a velocity, and thus
numerical grid dimension and the choice of their dimensional analysis yields (Heisenberg, 1948—see
value may be guided by the typical scales of the also Hinze, 1975)
CBL flow. In fact, this choice will depend on the sffiffiffiffiffiffiffiffiffiffi
Z 1
available computer power and the range of scales EðkÞ
that needs to be actually resolved for the pertinent nT ¼ CH dk, (5)
kc k3
description of the investigated problem. For a LES
resolving the energy-containing eddies in a CBL the where k is the wave number, CH is Heisenberg’s
filter width D can be chosen in the inertial subrange, dimensionless spectral transfer constant and E(k) is
closer to the integral scale, and far from Kolmogor- the three-dimensional (3-D) turbulence energy
ov’s dissipative scale. spectrum in the inertial subrange, with the following
Furthermore, the turbulence dissipation rate e is form (Kolmogorov, 1941):
parameterized as (Moeng, 1984)
EðkÞ ¼ aK 2=3 kÀ5=3 , (6)
3=2
e where aK is the Kolmogorov constant. Assuming
¼ c , (4)
l that the small-scale turbulence (inertial subrange)
where ce is a constant (0.93). should act on the large-scale turbulence like an
additional eddy viscosity we substitute Eq. (6) into
3. Heisenberg model for the KTV Eq. (5), where CHE0.47 and aKE1.52 (Muschinski
and Roth, 1993; Corrsin, 1963), to obtain
In his classical work, based on intuitive argu- À4=3
nT ¼ 0:441=3 kc . (7)
ments, Heisenberg (1948) assumed that the process
of the energy transfer from the small to the large It is important to note that Eq. (7) was firstly
wave numbers in a Kolmogorov turbulent spectrum derived by Muschinski and Roth (1993).
is similar to the conversion of mechanical energy Following the philosophy of Kraichnan’s eddy
into thermal energy through the agency of mole- ´
viscosity in spectral space, Lesieur and Metais
cular viscosity. In other words, the physical picture (1996) present an eddy viscosity expressed by
that forms the basis of Heisenberg’s theory is that, 2 À3=2 Eðkc Þ 1=2
in the energy cascade process within the kinetic n T ¼ aK , (8)
3 k
turbulent spectrum, the mechanism of inertial
exchange of energy from large to small eddies is where E(kc) is the kinetic energy spectrum at the
controlled by a KTV. Thus, the effect of the inertia cutoff kc.
term can be regarded as equivalent to a virtual Now, substituting Eq. (6) with k ¼ kc in Eq. (8)
turbulent friction, nT, produced by the small-scale yields
turbulence (inertial subrange eddies), acting on the À4=3
nT ¼ 0:441=3 kc (9)
large-scale turbulence (energy-containing eddies). nT
represents the KTV, which as a consequence of the suggesting that Heisenberg’s eddy viscosity is
lack of correlation between large and small sepa- ´
identical to the one proposed by Lesieur and Metais

4. ARTICLE IN PRESS
7062 G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068
(1996, Eqs. (3.5) and (4.1)). A remarkable point here imposed by the observed horizontal homogeneity.
is the consistency of the theoretical model (8) with This homogeneous behavior for a CBL sets an
the experimental value for the Heisenberg’s con- approximately constant value for lI for the u and v
stant. components, valid for all heights in a CBL.
At this point, spectral turbulent viscosities, such Horizontal homogeneity, therefore, naturally im-
as those given by Eqs. (5) and (8) can be compared poses Dx as the characteristic length scale of the
with those given by Eq. (2) (Smagorinsky, 1963; flow and makes kc ¼ p/Dx a well-defined cutoff
Moeng, 1984; Sullivan et al., 1994; Weil et al., 2004). wave number. This controlling length scale Dx can
For establishing this comparison we rewrite be derived from observational data for the CBL and
Eq. (4) as related to a characteristic scale of the physical
1 system, such as the mixed layer depth. This is done
1=3
e1=2 ¼ 1=3 l 0 (10) in the next session.
C 1=3
Now, inserting relation (12) into Eq. (7), Heisen-
and substitute it into Eq. (2) berg’s eddy viscosity can be written as
nT 4=3 nT
¼ 0:102l 0 . (11) ¼ 0:096ðDxÞ4=3 . (13)
1=3 1=3
Next, one assumes that the cutoff wave number in
For convective conditions, where l0 ¼ D, the
Fourier space
comparison of Eqs. (11)–(13) leads to
p
kc ¼ . (12)
Dx D ¼ l 0 ffi 0:96Dx % Dx. (14)
In expression (12), Dx is the longitudinal mesh Taking into account relation (14), we can choose
spacing, and lies within the kÀ5/3 Kolmogorov kc ¼ p/D, then one would find ðnT ÞHeis =ðnT ÞSmag ¼
spectrum (Eq. (6), inertial subrange). Furthermore, 0:96 % 1. It means that the models are actually
all degrees of freedom (turbulent energy modes) equivalent, probably within the error bars of the
with k4kc. have been filtered and, in this case, the experimental determination of the phenomenological
´
filter is sharp in Fourier space (Lesieur and Metais, constants appearing in each model (CH and Ck).
1996; Armenio et al., 1999). However, for deriving (14) an a priori assumption is
The argument for the choice of Dx in Eq. (12) made, which consists in assuming the equivalence
comes from the physical properties associated to the between Smagorinsky and Heisenberg models. This is
CBL. Observations show that a CBL presents a expected since both models arise from the same
homogeneous character in the horizontal directions. simple dimensional analysis arguments.
Analyzing the turbulent velocity spectra, the vertical Considering that observational studies show the
component shows the largest height dependence existence of horizontal homogeneity of the turbu-
among the three velocity components (Kaimal et al., lent properties of the CBL, one can assume that
1976; Caughey, 1982). In fact, throughout the CBL Dx ¼ Dy. This means that the spectral peak for the u
depth, both u (longitudinal velocity) and v (lateral and v velocity components occurs in the same
velocity) spectra display a well-established z-less frequency, and that this frequency is almost con-
limiting wave number lI for the inertial subrange. stant with height (Kaimal et al., 1976; Caughey,
This inhomogeneity of the turbulence in the vertical 1982). Employing Dx ¼ Dy in Eq. (3) and substitut-
direction in a CBL makes the inertial subrange ing in Eq. (14), the following relation between Dz
limiting wavelength associated to the vertical and Dx results:
turbulent velocity vary with height. As a conse-
quence, for the turbulent vertical velocity, an 0:4Dx for D ¼ 0:96Dx;
observational value for lI valid throughout the Dz ffi (15a, b)
0:44Dx for D ¼ Dx:
depth of the CBL cannot be clearly established.
Therefore, a relationship such as kc ¼ p/Dz is ill Expressions (14) and (15) directly relate D and Dz
defined since lI for the vertical velocity is strongly to the longitudinal mesh spacing Dx. This result
height dependent in the lower layers of the CBL. warrants that only one spatial scale is necessary to
This discussion shows that the choice of the determine all the mesh sizes. From Eq. (14): DEDx,
horizontal length scale Dx as a characteristic scale we can choose kc ¼ p/D. Therefore, this choice will
of the flow is not an arbitrary assumption, but it is reduce the inaccuracies for the velocity field in the

5. ARTICLE IN PRESS
G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068 7063
horizontal direction (since D is slightly smaller than curves are analytical functions approximating the
Dx), and it will preserve the relation DzE0.4Dx. data. These curves should be, therefore, viewed as
reasonable approximations to the observed spectra.
4. An expression for Dx and nT in terms of the CBL On the other hand, considering the spectral plots
height provided by Kaimal et al. (1976) for the u and v
velocity components the lower frequency limit (nzi/
Since the parameterization for the residual stress U) of the inertial subrange could be as large as 30,
tensor must be based on inertial subrange properties which would result in lIffi0.03zi. If one uses this
(small eddies) and that modeling the effects of the result in the Dx expression, Dx could be 0.015zi
residual turbulent motions on the resolved motions rather than 0.05zi. Although there is considerable
is a somewhat empirical method (Sullivan et al., uncertainty in the lI estimate, which translates into
1994), we can now utilize spectral observed data for uncertainty in the required grid size, our relations
a CBL to estimate Dx and consequently D and Dz in for Dx, D and Dz obtained from Heisenberg eddy
terms of a limiting or cutoff wavelength for the viscosity and constructed from the experimental
inertial subrange. Accepting the observational data of the observed turbulent spectra in a CBL, can
evidence that lI is approximately constant for the be now compared with those found in the literature.
u and v components, as discussed in the previous With this purpose, we choose zi ¼ 1000 m to obtain
section, kc can be written as kc ¼ 2p/lI and there- Dx ¼ Dyffi50, Dffi48 and Dzffi20 m. These mesh
fore a comparison of this relation with kc ¼ p/Dx spacing sizes are in agreement with those selected by
leads to Moeng and Sullivan (1994) and Weil et al. (2004),
which were employed in a LES model to numeri-
lI cally simulate the CBL. Indeed, the grid sizes
Dx ¼ Dy ¼ , (16)
2 selected by these authors support the conclusion
which establishes a direct relationship between mesh established by Kaimal et al. (1976), meaning that for
spacing and inertial subrange scales. According to the horizontal velocity components in the mixed
atmospheric measurements (Kaimal et al., 1976; layer lIffi0.1zi. This observed limiting wavelength
Kaimal and Finnigan, 1994; Caughey and Palmer, has also been assumed in another recent study
1979; Caughey, 1982), the spectra of the velocity (Elperin et al., 2006).
components in the bulk of the CBL can be Finally, we can introduce l0 ¼ D ¼ 0.048zi and
generalized within the framework of mixed layer Dx ¼ 0.05zi, respectively, in Eqs. (11) and (13) to
similarity. Therefore, experimental observations in obtain an unique expression for the subfilter eddy
the CBL allow the determination of lI. Following diffusivity. In terms of the CBL height zi, this
Kaimal et al. (1976), the onset of the inertial turbulent viscosity can be written as
subrange for the u and v spectra in the CBL, occurs nT 4=3
at limiting wavelength lIffi0.1zi, where zi is defined ¼ 0:0018zi . (17)
1=3
as the height of the lowest inversion base in a CBL.
At this point, the substitution of lI ¼ 0.1zi in Eq. The fact that zi is present in the expressions for
(16) yields Dx ¼ Dyffi0.05zi and, as a consequence Dx, D, Dz and nT is a consequence imposed by the
of Eqs. (14) and (15), Dffi0.048zi and Dzffi0.02zi. observations accomplished in the CBL. In spite of
Therefore, Eq. (16) along with the relationship the inertial subrange convective length scale being
lI ¼ 0.1zi, imparts a physical spatial constraint, small, it can still be described in terms of the
which helps the correct choice of the dimension of controlling convective scale zi, and this can be
the numerical grid in LES models. understood as a direct consequence of the fact that
There is an approximate nature in the arguments all turbulent energy modes must scale as a function
of Kaimal et al. (1976). Although they conclude that of the energy-containing eddies.
lIffi0.1zi in the mixed layer (0.1pz/zip1.0), they The present analysis considers only a highly CBL,
state that the limiting wavelength (upper value) of where Àzi/L is quite large. Indeed, in a CBL with
the inertial subrange is lIp0.1zi. Furthermore, there significant wind shear effects (Àzi/Lo10), the
are at least two approximations in these results: (1) horizontal spectra near the surface (z/zio0.1), will
n/U is used as an approximation to the wavelength, not look as similar to those above that height as do
where n is the frequency and U is the horizontal spectra in the highly CBL. For example, the
mean wind speed; (2) Kaimal et al. (1976) spectral horizontal velocity spectra computed from LES by

6. ARTICLE IN PRESS
7064 G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068
Khanna and Brasseur (1998), for Àzi/L ¼ 8, when S2 have very similar profiles for the average zonal
compared with those from Kaimal et al. (1976), wind speed, as expected. Simulations S3 and S4
present a different pattern of spectra curves. Close present a zonal wind vertical gradient inside the
to the ground, there is a disparity in length scale mixed boundary layer greater than in the previous
between the u and w spectra (Khanna and Brasseur, simulations, and a lower gradient in the entrainment
1998). In fact, in the surface CBL, the horizontal, region. This gradient is steeper in S4 than in S3. The
unstable spectra appear to have a somewhat average zonal wind speed in the mixed layer is
different structure with a tendency toward having greater when the computational grid is coarser.
two peaks (Panofsky and Dutton, 1984). Hojstrup However, these differences are not due to changes in
(1981), describes these horizontal spectra by sums of the geometry (boundary layer height, for example)
low-frequency (convective part) and high-frequency or the dynamic conditions in the system, since the
(mechanical part) portions. The methodology pre- energy provided to the system is the same. There-
sented in this study does not apply to a moderately fore, the differences shown in Fig. 1 are due to a
unstable CBL in which shear is an important wrong representation of the CBL induced by a bad
turbulence production mechanism. choice for the mesh discretization.
A comparison among the potential temperature
5. Numerical experiments profile is shown in Fig. 2. Again, it is noted that for
coarser the grid resolution, the simulated CBL
In order to provide numerical experimentation temperature decreases. There is a similarity between
for the approach developed in this paper, the LES the profiles S1 and S2, and between S3 and S4. It is
model developed by Moeng (1984) is used here, with also noted that for a lower r (finer grid) there is a
the subgrid turbulence parameterization based on steeper gradient in the entrainment region, i.e., the
Sullivan et al. (1994). boundary layer height is better identified for the
Some numerical experiments were carried out, finest grid resolution.
where different grid spacing were adopted (Table 1). The SFS TKE fraction varies little with height
The grid spacing is indicated by the ratio rDx/(zi)0, (Fig. 3), being approximately 0.11 over the bulk of
where (zi)0 is the initial height of the CBL, and Nj is the simulated CBL for cases S1 and S2 (those
the number of grid points for each direction (j ¼ x, presenting smaller r ratios). This behavior is similar
y, z). The same domain (5 km  5 km  2 km) was to those simulated and discussed by Weil et al.
considered for all simulations. As indicated, a value (2004). On the other hand, in the case of simulations
r ¼ 0.04 was employed in the simulation S1, while
r ¼ 0.05 was used in the simulation S2. Moeng and
Sullivan (1994, 2002) considered r ¼ 0.05, but in the
literature other values for r have been used, such as
rE0.156 (Moeng, 1984) or rE0.1 (Hadfield et al.,
1991; Brown, 1996; De Roode et al., 2004).
Antonelli et al. (2003) employed r ¼ 0.04, and
Sullivan et al. (1996) used several values ranging
from r ¼ 0.01 up to 0.06.
The simulations were performed to represent
2.5 h of fully developed turbulence in the CBL.
The initial boundary layer height is the same for
all simulations. Fig. 1 shows that simulations S1 and
Table 1
Description of the simulations
Simulations r ¼ Dx/(zi) (Nx, Ny, Nz) (zi)0 (m)
S1 0.040 (128,128,128) 1000
S2 0.050 (96,96,96) 1000
S3 0.078 (64,64,64) 1000
S4 0.156 (32,32,32) 1000
Fig. 1. Average zonal wind speed for the simulations S1–S4.

7. ARTICLE IN PRESS
G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068 7065
Fig. 2. Same as in Fig. 1, but for potential temperature.
Fig. 4. SFS sensible heat flux (S) and resolved scale sensible heat
flux (LE) as a function of height for the different simulations.
with less TKE and sensible heat flux is relevant,
considering that in LES methodology one hopes that
inertial subrange eddies contribute much less than
the energy containing eddies.
Therefore, from Figs. 1 to 4, it is possible to
identify two distinct simulation groups: finer resolu-
tion (S1 and S2) and coarser resolution (S3 and S4).
Additionally, it is verified that for an enhanced
resolution (S1), the CBL properties are not much
better represented than in the S2 simulation, i.e., it
is not expected a better representation with a finer
resolution than that used in S2. Therefore, the
criterion developed in this study (based on a
physical constrain for the grid spacing) guides
towards an optimized mesh size, which depends
only on the CBL height.
Recent testing of SFS models using observational
data shows that the coefficients Ck and Cs (Smagor-
Fig. 3. SFS fraction of total TKE as a function of height for the inski coefficient) in LES SFS models depend on the
different simulations. s2 and s2 are, respectively, the subfilter
s LE ratio of lw/lI (Sullivan et al., 2003), where lw is the
and resolved scale velocity variances.
peak wavelength of the w-spectrum. For small lw/
lI, Ck and Cs approach zero and thus are not
S3 and S4, that employed a larger r ratio, the SFS constant, as assumed in models with a sharp filter
TKE fraction is larger, and, for S4, varies continually cutoff. Additionally, all of the SFS fluxes vary with
with height. The SFS sensible heat fluxes (Fig. 4) are lw/lI e.g., the SFS variances of the u and w
larger near the surface for simulations S3 an S4 than components are anisotropic for lw/lI near 1.0 and
for S1 and S2. In all cases, the linear sensible heat only tend toward the isotropic limit for lw/lI410 or
flux profile relationship is well simulated. The fact 20. However, field observation in the bulk of a CBL
that, in simulations S1 and S2, the SFSs contribute (Caughey, 1979; Caughey, 1982) show that there is a

8. ARTICLE IN PRESS
7066 G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068
models of a convective boundary layer (CBL). The
theoretical framework is classical Heisenberg’s
turbulent spectral transfer theory, which provides
a kinematic turbulence viscosity (KTV), in terms of
a cutoff or limiting wave number kc (Eq. (7)). In
Heisenberg’s model, the KTV is invoked to explain
the mechanism of inertial transfer of energy from
large to small eddies. From a physical point of view,
the KTV is assumed to represent the friction
produced by the smaller eddies and acting on the
larger eddies.
The presence of this cutoff or limiting
wave number in the subgrid turbulent viscosity
introduced by Heisenberg’s model establishes a
Fig. 5. CPU-time for computational mesh with different number sharp filter in the turbulent energy modes
of points in the grid. and, consequently, this theory is in good conformity
with the main idea contained in LES models,
in which energy-containing eddies (large scales)
strong tendency for the peak wavelengths of all are explicitly resolved, whereas inertial subrange
velocity components to be the same and roughly eddies (small scales) are parameterized. Expressing
equal to E1.5zi. Therefore, the substitution of the key physical quantity kc in terms of the
lwE1.5zi and lIE0.1zi in the quotient lw/lI yields longitudinal mesh spacing Dx (Eq. (12)), we
a value of approximately 15 for this ratio, meaning compare the Heisenberg’s turbulent viscosity
that SFS motions are dominated by small-scale (Eq. (13)) with the classical one based on Smagor-
turbulent eddies (Sullivan et al., 2003). As a insky’s model (Eq. (11)) (Moeng, 1984; Sullivan
consequence, in mid-CBL regions where the turbu- et al., 1994). The comparison showed that both
lence is well resolved (over the bulk of the simulated approaches (Eqs. (11) and (13)) provide a similar
CBL, 0.1pz/zip1, the SFS TKE fraction is 0.11 in value for this subfilter eddy viscosity, and it leads to
the simulations S1 and S2), the local isotropy relationships for the filter width D (Eq. (14)) and
assumption is consistent with most LES models. for the vertical mesh spacing Dz (Eq. (15)) only in
Another important question to mention is the terms of Dx.
CPU-time spent in the simulations. All simulations Finally, setting a relationship between Dx and the
were run in a sequential computer. The simulation approximately constant limiting wavelength for the
time could be reduced in a parallel machine, but the inertial subrange of a CBL (observational data,
goal here is just a comparison among the simula- Eq. (16)) we describe Dx, D, and Dz as a minute
tions. The comparison among the CPU-times is fraction of the CBL heigth zi. Therefore, with
shown in Fig. 5. Dx, D, and Dz expressed in terms of zi an unique
The simulation time is given in hours versus the formula for the turbulent viscosity, obtained from
number of the points in the computational grid. Eqs. (11) and (13), can be found. This KTV
Clearly, increasing the grid points more CPU-time is associated to the inertial subrange eddies can be
required (Fig. 5). There is a big difference among employed to parameterize the residual stress tensor
the CPU-time for the simulations, where S1 in LES models.
simulation spent approximately 123 h and S2 The semiempirical analysis developed in this
simulation spent 37 h. Therefore, it is extremely work, which leads to the relationships between Dx,
relevant to have a good representation for the D, Dz and nT in terms of zi shows that Heisenberg’s
turbulence phenomena employing as little grids theory allied to observational data (heuristic argu-
point as possible. ments) provides a physical basis to the choice of
numerical values on the different formulas that
6. Conclusion constitute the parameterization of the subfilter
scales in LES models.
In this study, we have developed relationships for From LES simulations, it was verified that if the
the subfilter scales in large eddy simulation (LES) physical constrain presented here is followed (S2

9. ARTICLE IN PRESS
G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068 7067
simulation), the result is similar to that obtained boundary layer. Part I: A small-scale circulation with zero
with a finer grid (S1 simulation), but with significant wind. Boundary-layer Meteorology 57, 79–114.
reduction of the CPU-time. However, for S3 and S4 Heisenberg, W., 1948. Zur Statistischen Theorie der Turbulenz.
Zeitschrift fur Physik 124, 628–657.
simulations, Figs. 1 and 2 show different results Hinze, J.O., 1975. Turbulence. McGraw-Hill, New York.
from those obtained with S1 simulation, indicating Hojstrup, J., 1981. A simple model for the adjustment of velocity
that both simulations are not a good representation spectra in unstable conditions downstream of an abrupt
for modeling a CBL. In addition, in simulation S2, change in roughness and heat flux. Boundary-layer Meteor-
with the parameters suggested by the present study, ology 21, 341–356.
Kaimal, J.C., Finnigan, J.J., 1994. Atmospheric Boundary Layer
the subfilter scale turbulent kinetic energy fraction Flows. Oxford University Press, New York.
and sensible heat fluxes are similar to those obtained ´
Kaimal, J.C., Wyngaard, J.C., Haugen, D.A., Cote, O.R., Izumi,
with a finer mesh (S1), but smaller than those Y., 1976. Turbulence structure in the convective boundary
obtained with a coarser grid (S3 and S4). layer. Journal of the Atmospheric Sciences 33, 2152–2169.
Khanna, S., Brasseur, J., 1998. Three-dimensional Buoyancy-and
Shear-Induced Local Structure of the Atmospheric Boundary
Acknowledgments Layer 55, 710–743.
Kolmogorov, A.N., 1941. The local Structure of turbulence in
incompressible viscous fluid for very large reynolds numbers.
This work has been supported by Brazilian Doklady Akademii Naukite SSSR 30, 301–305.
Research Agencies: Conselho Nacional de Desen- ´
Lesieur, M., Metais, O., 1996. New trends in large-eddy
´
volvimento Cientı´ fico e Tecnologico (CNPq) and simulations of turbulence. Annual Review of Fluid Mechanics
Coordenac ao de Aperfeic oamento de Pessoal de
-˜ - 28, 45–82.
Ensino Superior (CAPES). Mason, P.J., 1994. Large-eddy simulation: a critical review of
technique. Quarterly Journal of the Royal Meteorological
Society 120, 1–26.
Moeng, C.-H., 1984. A large-eddy-simulation model for the study
References of planetary boundary-layer turbulence. Journal of the
Atmospheric Sciences 41, 2052–2062.
Antonelli, M., Mazzino, A., Rizza, U., 2003. Statistics of Moeng, C-H., Sullivan, P.P., 2002. Large-eddy simulation. In:
temperature fluctuations in a buoyancy-dominated boundary Holton, J.R., Pyle, J., e Cury, J.A. (Eds.), Encyclopedia of
layer flow simulated by a large eddy simulation model. Atmospheric Sciences. Academic Press, New York, pp.
Journal of the Atmospheric Sciences 60, 215–224. 1140–1150.
Armenio, V., Piomelli, U., Fiorotto, V., 1999. Effect of the Moeng, C-H., Sullivan, P.P., 1994. A comparison of shear-and
subgrid scales on particle motion. Physics of Fluids 11, buoyancy-driven planetary boundary layer flows. Journal of
3030–3042. the Atmospheric Sciences 51, 999–1022.
Brown, A.R., 1996. Evaluation of parameterization schemes for Moeng, C.-H., Wyngaard, J.C., 1988. Spectral analysis of large-
the convective boundary layer using large-eddy simulation eddy simulations of the convective boundary layer. Journal of
results. Boundary-layer Meteorology 81, 167–200. the Atmospheric Sciences 45, 3575–3587.
Caughey, S.J., 1982. Observed characteristics of the atmospheric Muschinski, A., Roth, R., 1993. A local interpretation of
boundary layer. In: Nieuwstadt, F.T.M., van Dop, H. (Eds.), Heisenberg’s transfer theory. Beitrage Physics Atmosphere
Atmospheric Turbulence and Air Pollution Modelling. 66, 335–346.
Reidel, Dordrecht, pp. 107–158. Panofsky, H.A., Dutton, J.A., 1984. Atmospheric Turbulence.
Caughey, S.J., Palmer, S.G., 1979. Some aspects of turbulence Wiley, New York.
structure through the depth of the convective boundary Pope, S.B., 2004. Ten questions concerning the large-eddy
layer Quart. Journal of Royal Meteorological Society 105, simulation of turbulent flows. New Journal of Physics 6,
811–827. 1–24.
Corrsin, S., 1963. Estimates of the relations between Eulerian and Schmidt, H., Schumann, U., 1989. Coherent structure of the
Lagrangian scales in large Reynolds number turbulence. convective boundary layer derived from large-eddy simula-
Journal of the Atmospheric Sciences 20, 115–119. tions. Journal of Fluid Mechanics 200, 511–562.
De Roode, S., Duynkerke, P.G., Jonker, H.J.J., 2004. Large-eddy Schumann, U., 1991. Subgrid length-scales for large-eddy
simulation: how large is large enough? Journal of the simulation of stratified turbulence. Theoretical and Computa-
Atmospheric Sciences 61, 403–421. tional Fluid Dynamic 2, 279–290.
Deardorff, J.W., 1973. The use of subgrid transport equations in Smagorinsky, J., 1963. General circulation experiments with the
a three-dimensional model of atmospheric turbulence. Jour- primitive equations. Monthly Weather Review 91, 99–164.
nal of Fluids Engineering 95, 418–429. Sorbjan, Z., 1989. Structure of the Atmospheric Boundary Layer.
Elperin, T., Kleeorin, N., Rogachevskii, I., Zilitinkevich, S.S., Prentice-Hall, Eaglewood Cliffs, NJ.
2006. Tangling turbulence and semi-organized structures in Sullivan, P.P., Mc Williams, J.C., Moeng, C.-H., 1994. A subgrid
convective boundary layers. Boundary-layer Meteorology model for large-eddy simulation of planetary boundary layer
119, 449–472. flows. Boundary-layer Meteorology 71, 247–276.
Hadfield, M.G., Cotton, W.R., Pilke, R.A., 1991. Large-eddy Sullivan, P.P., McWilliams, J.C., Moeng, C.-H., 1996. A grid
simulations of thermally forced circulations in the convective nesting method for large-eddy simulation of planetary

10. ARTICLE IN PRESS
7068 G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068
boundary-layer flows. Boundary-layer Meteorology 80, Weil, J.C., Sullivan, P.P., Moeng, C.-H., 2004. The use
167–202. of large-eddy simulations in Lagrangian particle dispe-
Sullivan, P.P., Horst, T.W., Lenschow, D.H., Moeng, C.-H., rsion models. Journal of the Atmospheric Sciences 61,
Weil, J.C., 2003. Structure of subfilter-scale fluxes in the 2877–2887.
atmospheric surface layer with application to large-eddy Wyngaard, J.C., 1982. Boundary-layer modeling. In: Nieuwstadt,
simulation modelling. Journal of Fluid Mechanics 482, F.T.M., van Dop, H. (Eds.), Atmospheric Turbulence and Air
101–139. Pollution Modelling. Reidel, Dordrecht, pp. 69–106.