Employing heisenberg’s turbulent spectral transfer theory to

Atmospheric Environment 41 (2007) 7059–7068

Employing Heisenberg’s turbulent spectral transfer theory to

parameterize sub-filter scales in LES models

Gervasio A. Degraziaa,�, Andre B. Nunesb, Prakki Satyamurtyb,Otavio C. Acevedoa, Haroldo F. de Campos Velhoc,

Umberto Rizzad, Jonas C. Carvalhoe

aDepartamento de Fısica, Universidade Federal de Santa Maria, Santa Maria, RS, BrazilbCPTEC, Instituto Nacional de Pesquisas Espaciais, Sao Jose dos Campos, SP, BrazilcLAC, Instituto Nacional de Pesquisas Espaciais, Sao Jose dos Campos, SP, Brazil

dIstituto di Scienze dellAtmosfera e del Clima, CNR, Lecce, ItalyeFaculdade 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

Large eddy simulation (LES) models represent a

well-established technique to study the physical

behavior of the planetary boundary layer (PBL)

(Deardorff, 1973; Moeng, 1984; Moeng and Wyn-

gaard, 1988; Schmidt and Schumann, 1989; Schu-

mann, 1991; Mason, 1994). In LES, only the energy-

containing eddies of the turbulent motion are explicitly

resolved and the effect of the smaller, more isotropic

eddies, needs to be parameterized. Modelling these

residual turbulent motions, which are also referred as

subgrid-scales or sub-filter motions (Pope, 2004) is in

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�Corresponding author.

E-mail address: [email protected] (G.A. Degrazia).

large part a phenomenological procedure based on

heuristic arguments (Sullivan et al., 1994).

In general, the governing equations in the LES

models are the incompressible Navier–Stokes equa-

tions described for a horizontally homogeneous

boundary layer. The resolved turbulent flow quan-

tities (wind components, pressure, temperature, etc.)

are obtained by the application of a low-pass spatial

filter of characteristic width, the turbulence resolu-

tion length scale (Pope, 2004), smaller than the

scales of the resolved turbulent motions. This low-

pass spatial filter width has the same order of

magnitude as the numerical grid dimensions and,

based on Kolmogorov spectral characteristics, can

be expressed in terms of the inertial subrange scales.

According to Wyngaard (1982), ‘‘the parameter-

ization of the residual stress term in the large eddy

equation is dynamically essential; it causes the

transference of kinetic energy to smaller scales.

Thus, its parameterization is a key step in develop-

ing a large eddy model’’. Interpreting the turbulence

field as a superposition of myriads of energy modes,

the parameterization of residual stress means to

model the physical effects of a large number of

degrees of freedom (turbulent scales), which, by

virtue of the filtering application are not explicitly

resolved in a numerical simulation.

The purpose of the present study is to show that

Heisenberg’s classical theory for inertial transfer of

turbulent energy (Heisenberg, 1948) can be used to

parameterize the residual stress tensor in LES

models applied to a convective boundary layer

(CBL). The idea is motivated by the fact that in the

inertial subrange the Heisenberg’s theory establishes

that the transfer of energy from wave numbers

smaller than a given particular value to those larger

can be represented as the effect of a turbulent

viscosity. This introduces a division between scales

at any arbitrary wave number in the inertial

subrange. This scale separation is indeed naturally

relevant in the LES frame, where a cutoff wave

number is arbitrarily chosen in the inertial range,

introducing then an artificial sharp division, to which

Heisenberg’s approach seems well suited to be

applied. Therefore, in this paper, we employ

Heisenberg’s theory to obtain a kinematic turbu-

lence viscosity (KTV), which represents the effects

of the filtered eddies of the inertial subrange.

From a physical point of view, the expression for

KTV provides a cutoff or limiting wave number,

separating the resolved scales from those filtered. It

allows to establish a relationship between the

horizontal subgrid characteristic length scales and

the filtered wavelengths of the inertial subrange

from the observed turbulent spectrum in the CBL.

As a phenomenological consequence, the magnitude

of Dx, Dy and Dz are fixed in terms of these residual

turbulent wavelengths and described as a small

fraction of the dominant convective length scale.

In order to test the grid mesh spacing proposed,

the LES model developed by Moeng (1984) is

employed, where the subgrid parameterization is

based on Sullivan et al. (1994).

2. Subgrid scale model and spatial scales in a CBL

In LES studies, one is trying to model eddies of

inertial-range scales and to resolve explicitly the

largest energy-containing eddies. For a well-devel-

oped CBL, the typical Reynolds number is greater

than 107 and the inertial subrange is defined by the

interval Lblbm, where L is the integral scale, l

characterizes the size of the inertial subrange eddies,

and m is Kolmogorov’s microscale. The integral

scale characterizes the energy-containing eddies,

which perform most of the turbulent transport,

while Kolmogorov’s microscale defines the size of

the viscous eddies that dissipate the turbulent

energy. For a fully developed CBL, LE102 and

mE10ÿ3m, and the eddy size range in the CBL

covers roughly five decades (Sorbjan, 1989).

Furthermore, since the Reynolds number is very

high for the CBL, the effect of molecular diffusivity

is irrelevant compared to the turbulent diffusivity,

and hence molecular viscosity will have negligible

direct influence on the resolved-scale motions

(Mason, 1994). Therefore, the convergence of the

statistics of the studied quantities with LES is not

expected to be affected by the dissipative scales.

A frequently used parameterization for the residual

stress tensor tij in the large eddy models is expressed

by (Smagorinsky, 1963; Deardorff, 1973; Moeng,

1984; Sullivan et al., 1994; Lesieur and Metais, 1996)

tij ¼ ÿnTqmiqxj

þqmj

qxi

� �

, (1)

where mi and mj are resolved velocity components, i,

j ¼ 1,2,3, corresponding to the x, y, z directions,

respectively, and nT is the turbulent eddy viscosity

which is expressed as

nT ¼ ckl0e1=2, (2)

where ck ¼ 0.1 is a constant, l0 is a mixing-length

scale related to the filter operation (Mason, 1994),

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G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–70687060

and e is the turbulent kinetic energy (TKE) of

subfilter scales (SFSs). For neutral (zero surface

turbulent heat flux) or unstable (positive turbulent

surface heat flux) conditions, l0 is equal to the low-

pass LES filter width D.

Following Weil et al. (2004), the low-pass LES

filter width is given by

D ¼3

2

� �2

DxDyDz

" #1=3

, (3)

where Dx, Dy and Dz are the computational mesh

sizes in the three coordinate directions x, y, z and

the constant (3/2)2 accounts for the dealiasing.

Therefore, the filter width D is of the order of the

numerical grid dimension and the choice of their

value may be guided by the typical scales of the

CBL flow. In fact, this choice will depend on the

available computer power and the range of scales

that needs to be actually resolved for the pertinent

description of the investigated problem. For a LES

resolving the energy-containing eddies in a CBL the

filter width D can be chosen in the inertial subrange,

closer to the integral scale, and far from Kolmogor-

ov’s dissipative scale.

Furthermore, the turbulence dissipation rate e is

parameterized as (Moeng, 1984)

� ¼ c�e3=2

l, (4)

where ce is a constant (0.93).

3. Heisenberg model for the KTV

In his classical work, based on intuitive argu-

ments, Heisenberg (1948) assumed that the process

of the energy transfer from the small to the large

wave numbers in a Kolmogorov turbulent spectrum

is similar to the conversion of mechanical energy

into thermal energy through the agency of mole-

cular viscosity. In other words, the physical picture

that forms the basis of Heisenberg’s theory is that,

in the energy cascade process within the kinetic

turbulent spectrum, the mechanism of inertial

exchange of energy from large to small eddies is

controlled by a KTV. Thus, the effect of the inertia

term can be regarded as equivalent to a virtual

turbulent friction, nT, produced by the small-scale

turbulence (inertial subrange eddies), acting on the

large-scale turbulence (energy-containing eddies). nTrepresents the KTV, which as a consequence of the

lack of correlation between large and small sepa-

rated Fourier elements of the spectrum (statistical

independence between turbulent scales), is a prop-

erty of the inertial subrange turbulent eddies, whose

scales range from an arbitrary but fixed wave

number to infinity (since dissipative scales and

molecular viscosity are irrelevant, as mentioned

before). Therefore, the magnitude of KTV will

depend on the inertial subrange eddies that cause

the viscosity, and hence must depend on eddies with

wave numbers in the range (kc,N), where kc is a

cutoff or limiting wave number for the inertial

subrange, which can be determinated from the

experimental TKE spectra.

An eddy viscosity is the product of a character-

istic turbulent length scale and a velocity, and thus

dimensional analysis yields (Heisenberg, 1948—see

also Hinze, 1975)

nT ¼

Z 1

kc

CH

ffiffiffiffiffiffiffiffiffiffi

EðkÞ

k3

s

dk, (5)

where k is the wave number, CH is Heisenberg’s

dimensionless spectral transfer constant and E(k) is

the three-dimensional (3-D) turbulence energy

spectrum in the inertial subrange, with the following

form (Kolmogorov, 1941):

EðkÞ ¼ aK�2=3kÿ5=3, (6)

where aK is the Kolmogorov constant. Assuming

that the small-scale turbulence (inertial subrange)

should act on the large-scale turbulence like an

additional eddy viscosity we substitute Eq. (6) into

Eq. (5), where CHE0.47 and aKE1.52 (Muschinski

and Roth, 1993; Corrsin, 1963), to obtain

nT ¼ 0:44�1=3kÿ4=3c . (7)

It is important to note that Eq. (7) was firstly

derived by Muschinski and Roth (1993).

Following the philosophy of Kraichnan’s eddy

viscosity in spectral space, Lesieur and Metais

(1996) present an eddy viscosity expressed by

nT ¼2

3aÿ3=2K

EðkcÞ

k

� �1=2

, (8)

where E(kc) is the kinetic energy spectrum at the

cutoff kc.

Now, substituting Eq. (6) with k ¼ kc in Eq. (8)

yields

nT ¼ 0:44�1=3kÿ4=3c (9)

suggesting that Heisenberg’s eddy viscosity is

identical to the one proposed by Lesieur and Metais

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G.A. Degrazia et al. / Atmospheric Environment 41 (2007) 7059–7068 7061

(1996, Eqs. (3.5) and (4.1)). A remarkable point here

is the consistency of the theoretical model (8) with

the experimental value for the Heisenberg’s con-

stant.

At this point, spectral turbulent viscosities, such

as those given by Eqs. (5) and (8) can be compared

with those given by Eq. (2) (Smagorinsky, 1963;

Moeng, 1984; Sullivan et al., 1994; Weil et al., 2004).

For establishing this comparison we rewrite

Eq. (4) as

e1=2 ¼1

C1=3�

�1=3l1=30 (10)

and substitute it into Eq. (2)

nT

�1=3¼ 0:102l

4=30 . (11)

Next, one assumes that the cutoff wave number in

Fourier space

kc ¼p

Dx. (12)

In expression (12), Dx is the longitudinal mesh

spacing, and lies within the kÿ5/3 Kolmogorov

spectrum (Eq. (6), inertial subrange). Furthermore,

all degrees of freedom (turbulent energy modes)

with k4kc. have been filtered and, in this case, the

filter is sharp in Fourier space (Lesieur and Metais,

1996; Armenio et al., 1999).

The argument for the choice of Dx in Eq. (12)

comes from the physical properties associated to the

CBL. Observations show that a CBL presents a

homogeneous character in the horizontal directions.

Analyzing the turbulent velocity spectra, the vertical

component shows the largest height dependence

among the three velocity components (Kaimal et al.,

1976; Caughey, 1982). In fact, throughout the CBL

depth, both u (longitudinal velocity) and v (lateral

velocity) spectra display a well-established z-less

limiting wave number lI for the inertial subrange.

This inhomogeneity of the turbulence in the vertical

direction in a CBL makes the inertial subrange

limiting wavelength associated to the vertical

turbulent velocity vary with height. As a conse-

quence, for the turbulent vertical velocity, an

observational value for lI valid throughout the

depth of the CBL cannot be clearly established.

Therefore, a relationship such as kc ¼ p/Dz is ill

defined since lI for the vertical velocity is strongly

height dependent in the lower layers of the CBL.

This discussion shows that the choice of the

horizontal length scale Dx as a characteristic scale

of the flow is not an arbitrary assumption, but it is

imposed by the observed horizontal homogeneity.

This homogeneous behavior for a CBL sets an

approximately constant value for lI for the u and v

components, valid for all heights in a CBL.

Horizontal homogeneity, therefore, naturally im-

poses Dx as the characteristic length scale of the

flow and makes kc ¼ p/Dx a well-defined cutoff

wave number. This controlling length scale Dx can

be derived from observational data for the CBL and

related to a characteristic scale of the physical

system, such as the mixed layer depth. This is done

in the next session.

Now, inserting relation (12) into Eq. (7), Heisen-

berg’s eddy viscosity can be written as

nT

�1=3¼ 0:096ðDxÞ4=3. (13)

For convective conditions, where l0 ¼ D, the

comparison of Eqs. (11)–(13) leads to

D ¼ l0 ffi 0:96Dx � Dx. (14)

Taking into account relation (14), we can choose

kc ¼ p/D, then one would find ðnTÞHeis=ðnTÞSmag ¼

0:96 � 1. It means that the models are actually

equivalent, probably within the error bars of the

experimental determination of the phenomenological

constants appearing in each model (CH and Ck).

However, for deriving (14) an a priori assumption is

made, which consists in assuming the equivalence

between Smagorinsky and Heisenberg models. This is

expected since both models arise from the same

simple dimensional analysis arguments.

Considering that observational studies show the

existence of horizontal homogeneity of the turbu-

lent properties of the CBL, one can assume that

Dx ¼ Dy. This means that the spectral peak for the u

and v velocity components occurs in the same

frequency, and that this frequency is almost con-

stant with height (Kaimal et al., 1976; Caughey,

1982). Employing Dx ¼ Dy in Eq. (3) and substitut-

ing in Eq. (14), the following relation between Dz

and Dx results:

Dz ffi0:4Dx for D ¼ 0:96Dx;

0:44Dx for D ¼ Dx:

�

(15a, b)

Expressions (14) and (15) directly relate D and Dz

to the longitudinal mesh spacing Dx. This result

warrants that only one spatial scale is necessary to

determine all the mesh sizes. From Eq. (14): DEDx,

we can choose kc ¼ p/D. Therefore, this choice will

reduce the inaccuracies for the velocity field in the

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horizontal direction (since D is slightly smaller than

Dx), and it will preserve the relation DzE0.4Dx.

4. An expression for Dx and nT in terms of the CBL

height

Since the parameterization for the residual stress

tensor must be based on inertial subrange properties

(small eddies) and that modeling the effects of the

residual turbulent motions on the resolved motions

is a somewhat empirical method (Sullivan et al.,

1994), we can now utilize spectral observed data for

a CBL to estimate Dx and consequently D and Dz in

terms of a limiting or cutoff wavelength for the

inertial subrange. Accepting the observational

evidence that lI is approximately constant for the

u and v components, as discussed in the previous

section, kc can be written as kc ¼ 2p/lI and there-

fore a comparison of this relation with kc ¼ p/Dx

leads to

Dx ¼ Dy ¼lI

2, (16)

which establishes a direct relationship between mesh

spacing and inertial subrange scales. According to

atmospheric measurements (Kaimal et al., 1976;

Kaimal and Finnigan, 1994; Caughey and Palmer,

1979; Caughey, 1982), the spectra of the velocity

components in the bulk of the CBL can be

generalized within the framework of mixed layer

similarity. Therefore, experimental observations in

the CBL allow the determination of lI. Following

Kaimal et al. (1976), the onset of the inertial

subrange for the u and v spectra in the CBL, occurs

at limiting wavelength lIffi0.1zi, where zi is defined

as the height of the lowest inversion base in a CBL.

At this point, the substitution of lI ¼ 0.1zi in Eq.

(16) yields Dx ¼ Dyffi0.05zi and, as a consequence

of Eqs. (14) and (15), Dffi0.048zi and Dzffi0.02zi.

Therefore, Eq. (16) along with the relationship

lI ¼ 0.1zi, imparts a physical spatial constraint,

which helps the correct choice of the dimension of

the numerical grid in LES models.

There is an approximate nature in the arguments

of Kaimal et al. (1976). Although they conclude that

lIffi0.1zi in the mixed layer (0.1pz/zip1.0), they

state that the limiting wavelength (upper value) of

the inertial subrange is lIp0.1zi. Furthermore, there

are at least two approximations in these results: (1)

n/U is used as an approximation to the wavelength,

where n is the frequency and U is the horizontal

mean wind speed; (2) Kaimal et al. (1976) spectral

curves are analytical functions approximating the

data. These curves should be, therefore, viewed as

reasonable approximations to the observed spectra.

On the other hand, considering the spectral plots

provided by Kaimal et al. (1976) for the u and v

velocity components the lower frequency limit (nzi/

U) of the inertial subrange could be as large as 30,

which would result in lIffi0.03zi. If one uses this

result in the Dx expression, Dx could be 0.015zirather than 0.05zi. Although there is considerable

uncertainty in the lI estimate, which translates into

uncertainty in the required grid size, our relations

for Dx, D and Dz obtained from Heisenberg eddy

viscosity and constructed from the experimental

data of the observed turbulent spectra in a CBL, can

be now compared with those found in the literature.

With this purpose, we choose zi ¼ 1000m to obtain

Dx ¼ Dyffi50, Dffi48 and Dzffi20m. These mesh

spacing sizes are in agreement with those selected by

Moeng and Sullivan (1994) and Weil et al. (2004),

which were employed in a LES model to numeri-

cally simulate the CBL. Indeed, the grid sizes

selected by these authors support the conclusion

established by Kaimal et al. (1976), meaning that for

the horizontal velocity components in the mixed

layer lIffi0.1zi. This observed limiting wavelength

has also been assumed in another recent study

(Elperin et al., 2006).

Finally, we can introduce l0 ¼ D ¼ 0.048zi and

Dx ¼ 0.05zi, respectively, in Eqs. (11) and (13) to

obtain an unique expression for the subfilter eddy

diffusivity. In terms of the CBL height zi, this

turbulent viscosity can be written as

nT

�1=3¼ 0:0018z

4=3i . (17)

The fact that zi is present in the expressions for

Dx, D, Dz and nT is a consequence imposed by the

observations accomplished in the CBL. In spite of

the inertial subrange convective length scale being

small, it can still be described in terms of the

controlling convective scale zi, and this can be

understood as a direct consequence of the fact that

all turbulent energy modes must scale as a function

of the energy-containing eddies.

The present analysis considers only a highly CBL,

where ÿzi/L is quite large. Indeed, in a CBL with

significant wind shear effects (ÿzi/Lo10), the

horizontal spectra near the surface (z/zio0.1), will

not look as similar to those above that height as do

spectra in the highly CBL. For example, the

horizontal velocity spectra computed from LES by

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Khanna and Brasseur (1998), for ÿzi/L ¼ 8, when

compared with those from Kaimal et al. (1976),

present a different pattern of spectra curves. Close

to the ground, there is a disparity in length scale

between the u and w spectra (Khanna and Brasseur,

1998). In fact, in the surface CBL, the horizontal,

unstable spectra appear to have a somewhat

different structure with a tendency toward having

two peaks (Panofsky and Dutton, 1984). Hojstrup

(1981), describes these horizontal spectra by sums of

low-frequency (convective part) and high-frequency

(mechanical part) portions. The methodology pre-

sented in this study does not apply to a moderately

unstable CBL in which shear is an important

turbulence production mechanism.

5. Numerical experiments

In order to provide numerical experimentation

for the approach developed in this paper, the LES

model developed by Moeng (1984) is used here, with

the subgrid turbulence parameterization based on

Sullivan et al. (1994).

Some numerical experiments were carried out,

where different grid spacing were adopted (Table 1).

The grid spacing is indicated by the ratio r�Dx/(zi)0,

where (zi)0 is the initial height of the CBL, and Nj is

the number of grid points for each direction (j ¼ x,

y, z). The same domain (5 km� 5 km� 2 km) was

considered for all simulations. As indicated, a value

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

S2 have very similar profiles for the average zonal

wind speed, as expected. Simulations S3 and S4

present a zonal wind vertical gradient inside the

mixed boundary layer greater than in the previous

simulations, and a lower gradient in the entrainment

region. This gradient is steeper in S4 than in S3. The

average zonal wind speed in the mixed layer is

greater when the computational grid is coarser.

However, these differences are not due to changes in

the geometry (boundary layer height, for example)

or the dynamic conditions in the system, since the

energy provided to the system is the same. There-

fore, the differences shown in Fig. 1 are due to a

wrong representation of the CBL induced by a bad

choice for the mesh discretization.

A comparison among the potential temperature

profile is shown in Fig. 2. Again, it is noted that for

coarser the grid resolution, the simulated CBL

temperature decreases. There is a similarity between

the profiles S1 and S2, and between S3 and S4. It is

also noted that for a lower r (finer grid) there is a

steeper gradient in the entrainment region, i.e., the

boundary layer height is better identified for the

finest grid resolution.

The SFS TKE fraction varies little with height

(Fig. 3), being approximately 0.11 over the bulk of

the simulated CBL for cases S1 and S2 (those

presenting smaller r ratios). This behavior is similar

to those simulated and discussed by Weil et al.

(2004). On the other hand, in the case of simulations

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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) 1000Fig. 1. Average zonal wind speed for the simulations S1–S4.


S3 and S4, that employed a larger r ratio, the SFS

TKE fraction is larger, and, for S4, varies continually

with height. The SFS sensible heat fluxes (Fig. 4) are

larger near the surface for simulations S3 an S4 than

for S1 and S2. In all cases, the linear sensible heat

flux profile relationship is well simulated. The fact

that, in simulations S1 and S2, the SFSs contribute

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-

inski coefficient) in LES SFS models depend on the

ratio of lw/lI (Sullivan et al., 2003), where lw is the

peak wavelength of the w-spectrum. For small lw/

lI, Ck and Cs approach zero and thus are not

constant, as assumed in models with a sharp filter

cutoff. Additionally, all of the SFS fluxes vary with

lw/lI e.g., the SFS variances of the u and w

components are anisotropic for lw/lI near 1.0 and

only tend toward the isotropic limit for lw/lI410 or

20. However, field observation in the bulk of a CBL

(Caughey, 1979; Caughey, 1982) show that there is a

ARTICLE IN PRESS

Fig. 2. Same as in Fig. 1, but for potential temperature.

Fig. 3. SFS fraction of total TKE as a function of height for the

different simulations. ss2 and sLE

2 are, respectively, the subfilter

and resolved scale velocity variances.

Fig. 4. SFS sensible heat flux (S) and resolved scale sensible heat

flux (LE) as a function of height for the different simulations.


strong tendency for the peak wavelengths of all

velocity components to be the same and roughly

equal to E1.5zi. Therefore, the substitution of

lwE1.5zi and lIE0.1zi in the quotient lw/lI yields

a value of approximately 15 for this ratio, meaning

that SFS motions are dominated by small-scale

turbulent eddies (Sullivan et al., 2003). As a

consequence, in mid-CBL regions where the turbu-

lence is well resolved (over the bulk of the simulated

CBL, 0.1pz/zip1, the SFS TKE fraction is 0.11 in

the simulations S1 and S2), the local isotropy

assumption is consistent with most LES models.

Another important question to mention is the

CPU-time spent in the simulations. All simulations

were run in a sequential computer. The simulation

time could be reduced in a parallel machine, but the

goal here is just a comparison among the simula-

tions. The comparison among the CPU-times is

shown in Fig. 5.

The simulation time is given in hours versus the

number of the points in the computational grid.

Clearly, increasing the grid points more CPU-time is

required (Fig. 5). There is a big difference among

the CPU-time for the simulations, where S1

simulation spent approximately 123 h and S2

simulation spent 37 h. Therefore, it is extremely

relevant to have a good representation for the

turbulence phenomena employing as little grids

point as possible.

6. Conclusion

In this study, we have developed relationships for

the subfilter scales in large eddy simulation (LES)

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

sharp filter in the turbulent energy modes

and, consequently, this theory is in good conformity

with the main idea contained in LES models,

in which energy-containing eddies (large scales)

are explicitly resolved, whereas inertial subrange

eddies (small scales) are parameterized. Expressing

the key physical quantity kc in terms of the

longitudinal mesh spacing Dx (Eq. (12)), we

compare the Heisenberg’s turbulent viscosity

(Eq. (13)) with the classical one based on Smagor-

insky’s model (Eq. (11)) (Moeng, 1984; Sullivan

et al., 1994). The comparison showed that both

approaches (Eqs. (11) and (13)) provide a similar

value for this subfilter eddy viscosity, and it leads to

relationships for the filter width D (Eq. (14)) and

for the vertical mesh spacing Dz (Eq. (15)) only in

terms of Dx.

Finally, setting a relationship between Dx and the

approximately constant limiting wavelength for the

inertial subrange of a CBL (observational data,

Eq. (16)) we describe Dx, D, and Dz as a minute

fraction of the CBL heigth zi. Therefore, with

Dx, D, and Dz expressed in terms of zi an unique

formula for the turbulent viscosity, obtained from

Eqs. (11) and (13), can be found. This KTV

associated to the inertial subrange eddies can be

employed to parameterize the residual stress tensor

in LES models.

The semiempirical analysis developed in this

work, which leads to the relationships between Dx,

D, Dz and nT in terms of zi shows that Heisenberg’s

theory allied to observational data (heuristic argu-

ments) provides a physical basis to the choice of

numerical values on the different formulas that

constitute the parameterization of the subfilter

scales in LES models.

From LES simulations, it was verified that if the

physical constrain presented here is followed (S2

ARTICLE IN PRESS

Fig. 5. CPU-time for computational mesh with different number

of points in the grid.


simulation), the result is similar to that obtained

with a finer grid (S1 simulation), but with significant

reduction of the CPU-time. However, for S3 and S4

simulations, Figs. 1 and 2 show different results

from those obtained with S1 simulation, indicating

that both simulations are not a good representation

for modeling a CBL. In addition, in simulation S2,

with the parameters suggested by the present study,

the subfilter scale turbulent kinetic energy fraction

and sensible heat fluxes are similar to those obtained

with a finer mesh (S1), but smaller than those

obtained with a coarser grid (S3 and S4).

Acknowledgments

This work has been supported by Brazilian

Research Agencies: Conselho Nacional de Desen-

volvimento Cientıfico e Tecnologico (CNPq) and

Coordenac- ao de Aperfeic-oamento de Pessoal de

Ensino Superior (CAPES).

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Employing heisenberg’s turbulent spectral transfer theory to