BoundaryLayer Meteorol (2009) 131:277–292DOI 10.1007/s105460099360zARTICLE
A Variable Mesh Spacing for LargeEddy SimulationModels in the Convective Boundary Layer
Gervásio Annes Degrazia
·
Umberto Rizza
·
Franciano Scremin Puhales
·
Antônio Gledson Goulart
·
Jonas Carvalho
·
Guilherme Sausen Welter
·
Edson Pereira Marques Filho
Received: 27 March 2008 / Accepted: 5 February 2009 / Published online: 24 February 2009© Springer Science+Business Media B.V. 2009
Abstract
A variable vertical mesh spacing for largeeddy simulation (LES) models in aconvective boundary layer (CBL) is proposed. The argument is based on the fact that in thevertical direction the turbulence near the surface in a CBL is inhomogeneous and thereforethe subﬁlterscale effects depend on the relative location between the spectral peak of thevertical velocity and the ﬁlter cutoff wavelength. From the physical point of view, this lack of homogeneity makes the vertical mesh spacing the principal length scale and, as a consequence, the LES ﬁlter cutoff wavenumber is expressed in terms of this characteristic lengthscale. Assuming that the inertial subrange initial frequency is equal to the LES ﬁlter cutoff frequency and employing ﬁtting expressions that describe the observed convective turbulentenergy onedimensional spectra, it is feasible to derive a relation to calculate the variablevertical mesh spacing. The incorporation of this variable vertical grid within a LES modelshowsthatboththemeanquantities(andtheirgradients)andtheturbulentstatisticsquantitiesare well described near to the ground level, where the LES predictions are known to be achallenging task.
Keywords
Convective boundary layer
·
Largeeddy simulation
·
Variable mesh spacing
G. A. Degrazia (
B
)
·
U. Rizza
·
F. S. Puhales
·
G. S. WelterLaboratório de Fisica da Atmosfera, Universidade Federal de Santa Maria, Santa Maria, RS, Brazilemail: Degrazia@ccne.ufsm.brU. RizzaIstituto di Scienze dell’Atmosfera e del Clima, Consiglio Nazionale delle Ricerche, Lecce, Italyemail: u.rizza@isac.cnr.itA. G. GoulartCentro de Tecnologia de Alegrete, Universidade Federal do Pampa, Alegrete, RS, BrazilJ. CarvalhoUniversidade Federal de Pelotas, Pelotas, RS, BrazilE. P. Marques FilhoUniversidade Federal do Rio de Janeiro, Rio de Janeiro, RJ, Brazil
1 3
278 G. A. Degrazia et al.
1 Introduction
In largeeddy simulation (LES), the energycontaining eddies of the turbulent motion areexplicitly resolved and the effect of the smaller, more isotropic, eddies (typical eddies in theinertial subrange) needs to be parameterized (Muschinski 1996). Modelling these residual
turbulent motions, which are also termed subﬁlterscale (SFS) motions (Pope 2004), is in
largepartaphenomenologicalprocedurebasedonheuristicarguments(Sullivanetal.1994).The resolved turbulent ﬂow timedependent random variables are obtained by the application of a lowpass spatial ﬁlter of characteristic width
, the turbulent resolution length scale(Pope 2004), smaller than the scales of the resolved turbulent motions. According to Mason
(1994) “The term largeeddy simulation has indeed been usually reserved for applicationsin which the scale of the ﬁlter operation is within or at least close to an inertial subrangeof turbulence”. Therefore, this lowpass spatial ﬁlter width that plays a decisive role in theconstruction of the resolved turbulence variables can be expressed in terms of the inertialsubrange parameters. Thus, there is a clear expectation that an adequate physical derivation of the parameter
provides robust ﬂuxes of heat and momentum from resolvedscalemotions and that the contributions of the unresolved subﬁlter motions are responsible for thedissipation of the resolved turbulence. However, most geophysical ﬂows present boundaryregions where the turbulence scale reduces and becomes smaller than the spatial ﬁlter scalewidth. Indeed, in the atmospheric convective boundary layer (CBL), the wavelength of thepeak in the vertical velocity spectrum
(λ
m
)
w
decreases in proximity to the ground (Kaimalet al. 1976; Caughey and Palmer 1979) and this surface closeness constrains our capacity
to accomplish highReynoldsnumber LES (Sullivan et al. 2003). Near the ground, the ﬁl
ter cutoff wavelength
λ
f
employed in LES modelling is of the same order as
(λ
m
)
w
andconsequently the SFS ﬂuxes in LES become important and their contribution to the totalﬂux (resolved and ﬁltered eddies) grows in proximity to the surface. Recent testing of SFSmodels using observations show that the contribution of the SFS turbulent energy modes tothe total turbulence is dominant when
(λ
m
)
w
/λ
f
≤
1, while the resolved motions play apreponderant role for
(λ
m
)
w
/λ
f
≫
1 (Sullivan et al. 2003).
Based on the universal form of the energy spectrum for fully developed turbulence therelative proportion of resolved and SFS motions is established by the location of the ﬁltercutoff wavenumber
k
f
in the turbulent energy spectrum. If the energycontaining eddiesare described by the peak wavenumber
k
p
of the turbulent spectrum, for cutoff and peak wavenumbers, then the proportion of the SFS turbulent motions to the total turbulence islarge when
k
f
≤
k
p
, while the resolved turbulent motions are dominant for
k
f
≫
k
p
. As aconsequence,thedeterminationofaphysicalcriteriontoobtain
k
f
indistinctverticalregionsof a CBL is a fundamental procedure in LES methodology. A commonly used ﬁlter in LES isthe sharp Fourier cutoff (Moeng and Wyngaard 1988). The sharp spectral ﬁlter annihilates
all Fourier modes of wavenumber

k

greater than the cutoff wavenumber
k
f
≡
π
(1)whereas it has no effect upon lower wavenumber modes. In practical applications of LES,the parameter
is expressed in terms of
x
, and normally it is replaced by this horizontalmesh spacing (Degrazia et al. 2007). Following Moeng and Wyngaard (1988), the lowpass
LES ﬁlter width is given by
=
32
2
x
y
z
1
/
3
(2)
1 3
A Variable Mesh Spacing 279
where
x
,
y
and
z
are the computational mesh sizes in the coordinate directions
x
,
y
,
z
and the constant
(
3
/
2
)
2
accounts for the dealiasing. Furthermore, for a LES model thatresolves the most energetic turbulent eddies in a CBL, the ﬁlter characteristic width canbe chosen in the inertial subrange, closer to the integral scale, and far from Kolmogorov’sdissipative scales (Mason 1994; Degrazia et al. 2007).
Observations show that a CBL presents a homogeneous structure in the horizontal directions. Therefore, throughout the CBL, both
u
(longitudinal velocity) and
v
(lateral velocity)spectradisplayawellestablishedﬁlterzlesslimitingwavelength
λ
f
fortheinertialsubrange(Degrazia et al. 2007). This horizontal homogeneity imposes large values of
x
=
y
≈
0
.
05
z
i
(Degrazia et al. 2007, where
z
i
is deﬁned as the height of the lowest inversion base atthetopoftheCBL)inthesurfaceregionsofaCBL,andasaconsequencedecreasesthemagnitude of
k
f
and favours the condition
k
f
≤
k
p
. This experimental evidence indicates that aﬁlter wavenumber, such as
k
f
=
π/
x
, is physically not adequate for LES applications toreproduceturbulentﬂuxesinproximitytothesurface.Thus,toobtainapossiblesolutionthatavoids this condition and ensures that
k
f
≫
k
p
, it is necessary to deﬁne the sharp spectralﬁlter as
k
f
=
π
z
.
(3)However, in comparison to
u
and
v
, the vertical velocity component
w
exhibits the largest height dependence among the three velocity components. This vertical inhomogeneitymakes the inertial subrange limiting wavelength vary strongly with height in the lower CBL.Therefore, for the turbulent vertical velocity, observational values of
λ
f
or
k
f
valid nearthe surface in the CBL cannot be clearly determined. This lack of definition of
λ
f
for thevertical turbulent velocity does not allow the magnitudes of
z
and consequently of
to beestablished for the vertical region near the ground in a CBL.MotivatedbytheselectionofaﬁltercutoffwavenumberprovidedbyEq.3thepurposeof thepresentinvestigationistodevelopamethodologythatprovidesvaluesfor
z
inthelowerCBL.Theapproachisbasedonﬁttedmathematicalexpressionsdescribingtheobservedconvective turbulent energy spectra. Therefore, the idea is to determine, for distinct levels in thelower CBL, a limiting or ﬁlter wavelength associated with the vertical turbulent velocityspectrum. This procedure that allows derivation of a variable and nonuniform grid spacing
z
is based on the physical criterion that constrains
(λ
m
)
w
to
λ
f
and requires the condition
(λ
m
)
w
/λ
f
≫
1 in the proximity of the surface in a CBL.
2 Methodology to Obtain a Vertical Mesh Spacing Near the Ground in a CBL
A review of the literature from the past 20 years shows that many observed turbulent velocityspectra present a simple shape in the planetary boundary layer (Kaimal and Finningan 1994;Panofsky and Dutton 1984; Foken 2006). Thus, in the frequency range associated with the
turbulent energy, spectra can often be approximated by a smooth curve containing one maximum only (when plotted as log
(
nS
(
n
))
versus log
(
n
))
. This statement is valid for the
w
spectrum under convective conditions (Kaimal et al. 1976; Caughey 1982; Olesen et al.
1984).The equation for Eulerian velocity spectra under unstable conditions can be expressed asa function of convective scales as follows (Degrazia et al. 2001):
nS
i
w
2
∗
=
1
.
06
c
i
f
ψ
2
/
3
(
z
/
z
i
)
2
/
3
(
f
m
)
5
/
3
i
(
1
+
1
.
5
(
f
/
f
m
)
i
)
5
/
3
(4)
1 3
280 G. A. Degrazia et al.
where
i
=
u
,v,w
are the turbulent wind components,
c
i
=
α
i
(
0
.
5
±
0
.
05
)(
2
πκ)
−
2
/
3
,α
i
=
(
1
;
4
/
3
;
4
/
3
)
(Champagne et al. 1977),
κ
=
0
.
4 is the von Karman constant,
f
=
nz
/
U
isthenondimensionalfrequency,
n
isthefrequencyinHz,
U
isthehorizontalmeanwindspeed,
(
f
m
)
i
is the nondimensional frequency of the spectral peak,
w
∗
is the convective velocityscale,andthenondimensionalmoleculardissipationratefunctionisdeﬁnedby
ψ
=
ε
z
i
/w
3
∗
,where
ε
is the mean dissipation of turbulent kinetic energy per unit time per unit mass of ﬂuid, with the order of magnitude of
ε
determined only by those parameters that characterizethe energycontaining eddies. The magnitude of the above deﬁned
α
i
parameters is derivedfrom the turbulence isotropic properties in the inertial subrange.From Eq.4, the vertical turbulent velocity spectrum can be written as
nS
w
w
2
∗
ψ
2
/
3
=
1
.
06
c
w
nzU
ψ
2
/
3
(
z
/
z
i
)
2
/
3
(
f
m
)
5
/
3
w
1
+
1
.
5
(
nz
/
U
)(
f
m
)
w
5
/
3
(5)where
c
w
=
0
.
36.The peak frequency,
(
f
m
)
w
, is an important parameter for studies on turbulent dispersionin the CBL. The knowledge of this quantity in Eq.5 allows reproduction of the observedspectral curves and captures the temporal and spatial characteristics associated with theenergycontaining eddies in distinct vertical regions of a CBL. From the Ashchurch andMinnesota experiments (Caughey 1982), for
z
<
0
.
1
z
i
(surface layer, Weil et al. 2004), therelation for
(
f
m
)
w
is linear and can be approximated by
(
f
m
)
w
=
z
(λ
m
)
w
=
15
.
9
=
0
.
17
,
(6)which is precisely the free convection limit observed in the Kansas observations (Kaimalet al. 1972; Caughey 1982). Substituting the relation (6) into Eq.5, one obtains the following
vertical turbulent spectrum in the lower CBL:
nS
w
w
2
∗
ψ
2
/
3
=
7
.
32
f
i
(
z
/
z
i
)
5
/
3
1
+
8
.
85
f
i
(
z
/
z
i
)
5
/
3
.
(7)Furthermore, the asymptotic behaviour of Eq.7 for large values of
f
i
in the inertial subrangeof the spectrum, can be described as
nS
w
w
2
∗
ψ
2
/
3
=
0
.
19
f
−
2
/
3
i
.
(8)Theideainthisanalysisistodeterminealimitingorinitialcutoffwavelength(orfrequency)for the inertial subrange at heights
z
/
z
i
<
0
.
1 in the CBL, and consider it identical to theLES ﬁlter cutoff wavelength (or frequency). We are interested in obtaining the initial frequencies (or wavelengths) of the inertial subrange for distinct vertical levels in the proximityof the surface. From these initial frequencies the turbulent energy will be ﬁltered and theireffects parameterized in LES models. Therefore, to obtain mathematically these ﬁlter cutoff frequencies or limits, we equate Eq.7 with its asymptotic behaviour given by (8) and solve
this equality to obtain the fundamental frequencies,
f
f
, that represent the limiting or ﬁltercutoff initial frequency of the inertial subrange,0
.
026
f
f
5
/
3
=
(
z
/
z
i
)
5
/
3
1
+
8
.
85
f
f
(
z
/
z
i
)
5
/
3
.
(9)
1 3
A Variable Mesh Spacing 281
After some simple algebraic manipulations, we obtain
(
z
/
z
i
)
f
f
×
0
.
03
=
0
.
11
,
(10)and
f
f
=
3
.
66
z z
i
−
1
.
(11)Now employing the relation
f
f
=
z
i
λ
f
one obtains
λ
f
=
0
.
27
z z
i
z
i
. Furthermore,
k
f
can be written as
k
f
=
2
πλ
f
and a comparison of this ratio with
k
f
=
π/
z
leads to
z
=
λ
f
2
∼=
0
.
14
z z
i
z
i
.
(12)Equation12 imparts a physical spatial constraint (obtained from observations), that helps usdetermine the correct choice of the dimension of the vertical mesh spacing in LES modelsin the lower CBL; this discussion shows that the choice of the vertical length scale,
z
, asgiven by relation (12), is not an arbitrary assumption, but is imposed by observations in the
CBL.On the other hand, the determination of
z
, as given by Eq.12, makes the relation
k
f
=
π/
z
welldeﬁned since the height dependency of
λ
f
is now known in the lower CBL.Therefore, the inhomogeneous character of the turbulence in the vertical direction in a CBLis responsible for the distinct values of the ﬁlter cutoff wavenumber
k
f
that need to beselected in a LES model. For
z
/
z
i
<
0.1, the vertical inhomogeneity naturally imposes
z
as a characteristic length scale of the ﬂow and selects the ratio
k
f
=
π/
z
as having thequalities needed for a LES ﬁlter cutoff wavenumber.The above development enables calculation of the magnitude of the ratio between thewavelength of the peak in the vertical velocity spectrum and the LES ﬁlter cutoff wavelength. From our derivation the value for
(λ
m
)
w
/λ
f
is of the order of 20. As the magnitudefor this ratio is much larger than 1.0 one expects that the employment of
k
f
=
π/
z
in LESmodels, with
z
given by Eq.12 in the lower CBL, reproduces SFS ﬂuxes that are a small
fraction of the resolved turbulent ﬂuxes.
3 Numerical Experiment Using a Variable Vertical Grid Spacing
z
In order to provide numerical experiments using the approach previously presented in thisstudy, the LES model, ﬁrstly developed by Moeng (1984) and later modiﬁed by Sullivan
et al. (1994), is used with a variable vertical grid spacing
z
for
z
<
0
.
1
z
i
. Regarding thedependence of the SGS closure on the ﬁlter scale, we would like to point out that in order toaccomplish our simulations the ﬁlter has not been modiﬁed, as well as the
(λ
m
)
w
/λ
f
ratio.Thus, the subﬁlter turbulence parameterization aspects in this study are based exclusively onSullivan et al. (1994).
A (5, 5, 2) km box domain with 128 grid points in each direction
(
x
,
y
,
z
)
has beenused. In the simulation we held the kinematic turbulent heat ﬂux constant with a magnitude
w
′
θ
′
0
=
0
.
24K ms
−
1
, and the geostrophic wind was set to
U
g
=
10ms
−
1
. The initialvalues for the CBL height and the surface potential temperature were set respectively equalto
(
z
i
)
0
=
1
,
000m and
θ
=
300K. Furthermore, to establish a comparison, two simula
1 3