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Diapycnal diffusivities in homogeneous stratified turbulence
D. D. Stretch
1
and S. K. Venayagamoorthy
2
Received 28 October 2009; revised 10 December 2009; accepted 16 December 2009; published 28 January 2010.
[1] Quantifying diapycnal mixing in stably stratified
turbulence is fundamental to the understanding and
modeling of geophysical flows. Data of diapycnal mixing
from direct numerical simulations of homogeneous
stratified turbulence and from grid turbulence experiments,
are analyzed to investigate the scaling of the diapycnal
diffusivity. In these homogeneous flows the instantaneous
diapycnal diffusivity is given exactly by K
d
=
r
/(@r/@z)
2
where
r
is the dissipation rate of density fluctuations, and
@r/@zis the mean density gradient. The diffusivity K
d
may be
expressed in terms of the large scale properties of the
turbulence as K
d
=gL
E
2
/T
L
,whereL
E
is the Ellison
overturning length-scale, T
L
is the turbulence decay time-
scale, and gis half the mechanical to scalar time-scale ratio.
Our results show that L
E
and T
L
can explain most of the
variations in K
d
over a wide range of shear and
stratification strengths while gremains approximately
constant. Citation: Stretch, D. D., and S. K. Venayagamoorthy
(2010), Diapycnal diffusivities in homogeneous stratified
turbulence, Geophys. Res. Lett.,37, L02602, doi:10.1029/
2009GL041514.
1. Introduction
[2] Many geophysical flows such as in the oceans,
atmosphere, lakes, or estuaries, are influenced by the
presence of stable density stratification. Quantifying dia-
pycnal mixing in these flows is important for estimating the
effects on the overall mass/energy balance [see, e.g., Gregg,
1987]. Methods of inferring diapycnal fluxes from micro-
structure measurements have therefore been developed and
are widely used in oceanography. These methods, princi-
pally due to Osborn and Cox [1972] and Osborn [1980], are
based on the notion that for homogeneous stationary turbu-
lence, the irreversible mixing at small scales is balanced by
the average advective transport by large scales. The appli-
cability of these methods has been widely questioned [e.g.,
Davis, 1994] because the basic assumptions are seldom
fully valid in practice. However, even in cases where the
simplifying assumptions are valid (e.g., in some lab experi-
ments and/or numerical simulations) a unifying scaling
framework for diapycnal mixing that relates the diffusivity
to the large scale properties of the flow has not emerged.
This is required for numerical models of geophysical
flows where small scale mixing cannot be directly resolved
and must be parameterized as a sub-grid scale process.
[3] The non-dimensional parameter /nN
2
(where is the
dissipation of turbulent kinetic energy, nis the kinematic
viscosity, and Nis the buoyancy frequency) is widely used
in oceanography to characterize turbulence ‘‘intensity’’ or
‘‘activity’’ and therefore mixing rates. Barry et al. [2001]
and Shih et al. [2005] used it to parameterize the turbulent
fluxes and diffusivities obtained from lab experiments and
direct numerical simulations (DNS) respectively [see also
Ivey et al., 2008]. Their results indicate that there are three
mixing regimes that may be identified in terms of /nN
2
:
(1) a ‘‘diffusive regime’’ with /nN
2
1 where the diffusivity
tends towards molecular values; (2) an ‘‘intermediate
regime’’ with 10 ]/nN
2
]100 where the diffusivity
increases linearly with /N
2
, consistent with the Osborn
[1980] model; (3) an ‘‘energetic regime’’ with /nN
2
^100
where the diffusivity increases more slowly with /nN
2
, e.g.,
diffusivity /(/nN
2
)
1/2
.
[4] There are two issues raised by the diffusivity scaling
presented by Barry et al. [2001] and Shih et al. [2005].
Firstly, their scaling, if generally applicable, seems to imply
that the diapycnal diffusivity depends on the viscosity, even
at very high Reynolds numbers, which is inconsistent with
theoretical concepts and observations. Pope [1998] and
Donzis et al. [2005] provide in depth discussions of this
issue. Secondly, their scaling does not yield any clear
‘‘passive’’ scalar limit for the diffusivity; i.e., how do
diascalar diffusivities in weakly stratified flows where
/nN
2
!1fit into the suggested scaling?
[5]Venayagamoorthy and Stretch [2006] (hereafter VS),
using DNS of shear-free decaying homogeneous stably-
stratified turbulence, showed that the diapycnal diffusiv-
ity K
d
in these temporally developing flows is propor-
tional to L
E
2
/T
L
where L
E
=(r02)1
2/j@r/@zjis the Ellison
overturning length-scale and T
L
=k/is the turbulence
decay time-scale. This was found to hold over a wide
range of stratification strengths, including the passive
scalar limit. However the generality of this result could
not be tested since the simulations did not include shear
effects, nor did they include variations in Prandtl number.
Furthermore, as with most DNS studies, they were limited
to low Reynolds numbers that in the strongly stratified
cases were within the ‘‘diffusive’’ regime (/nN
2
1)
mentioned above. The applicability of the suggested scaling
to a wider class of flows therefore remains an unresolved
issue.
[6] The objective of this paper is to address the above-
mentioned issues by investigating the generality of the
scaling suggested by VS using data from both DNS
and laboratory experiments that cover a range of stratifi-
cation strengths, shear rates, Reynolds numbers, and
Prandtl (or Schmidt) numbers. In sections 2 and 3 we
discuss the theoretical background to the scaling of the
diascalar diffusivity. In section 4 we describe the DNS
and experimental datasets used for the present analysis.
GEOPHYSICAL RESEARCH LETTERS, VOL. 37, L02602, doi:10.1029/2009GL041514, 2010
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A
rticle
1
School of Civil Engineering, Surveying and Construction, University
of KwaZulu-Natal, Durban, South Africa.
2
Department of Civil and Environmental Engineering, Colorado State
University, Fort Collins, Colorado, USA.
Copyright 2010 by the American Geophysical Union.
0094-8276/10/2009GL041514$05.00
L02602 1of5
The results are discussed in section 5 and conclusions in
section 6.
2. Diascalar Flux in Homogeneous Turbulence
[7]Winters and D’Asaro [1996] used a geometric
approach to derive an exact expression relating the instan-
taneous diascalar flux of an arbitrary scalar qto its mean-
square gradients averaged over iso-scalar surfaces, namely
fd¼k
rq
jj
2
DE
z*
dq=dz*
ð1Þ
where kis the molecular diffusivity of q,z
*
is an iso-scalar
co-ordinate and hizdenotes an average over the iso-scalar
surface corresponding to a given z
*
value. Derivation of this
result assumes only that the evolution of the scalar field qis
described by an advection– diffusion equation with a
solenoidal velocity field. In their formulation, Winters and
D’Asaro [1996] defined a reference state for the scalar field
where all fluid particles are adiabatically re-arranged to
form a profile q(z
*
) that is monotonic in z
*
. When the scalar
concerned is the density field in a stably stratified flow, this
reference state corresponds to a state of minimum potential
energy. VS argued that in the case of homogeneous
turbulence with a uniform background mean scalar gradient
dq/dz (where q(z)=hqi
z
), the gradient of the reference state
q(z
*
) is equal to the gradient of the mean field q(z). In this
case the diascalar flux given by (1), averaged over all iso-
scalar surfaces, simplifies to
fd¼q
dq=dz
ð2Þ
with an associated diascalar diffusivity given by
Kd¼q
dq=dz
2ð3Þ
where
q
=khjrq
0
j
2
iis the volume averaged dissipation
rate of the scalar variance. This result for the diascalar
diffusivity is exact for the specified conditions. It is also the
same as that inferred by Osborn and Cox [1972] but it is
more generally applicable than suggested by the Osborn-
Cox argument. Although it requires that the turbulent fields
are statistically homogeneous, it does not require stationar-
ity or a balance between large and small scale processes
represented in the scalar transport budget.
[8] VS derived (3) using a Lagrangian analysis of fluid
particle displacements which were decomposed into
(reversible) isopycnal and (irreversible) diapycnal compo-
nents. The growth rate of the mean square diapycnal displace-
ments was shown to be proportional to K
d
for high Reynolds
and Peclet numbers where there is large separation between
advective and diffusive scales.
3. Scaling of the Diapycnal Diffusivity in
Homogeneous Stratified Turbulence
[9] VS noted that the diapycnal diffusivity in homoge-
neous stably stratified flows can be expressed in terms of
the large scale properties of the turbulence as
Kd¼r
@r=@zðÞ
2¼gLE2
TL
;ð4Þ
where g=T
L
r
/r02is half the mechanical to scalar time-
scale ratio. Physically, the length scale L
E
describes the
generation of density fluctuations by vertical displacement
(or stirring) of fluid particles within the background mean
gradient, while T
L
is the time-scale over which fluid particles
exchange their density with their surrounding fluid by small-
scale mixing (refer VS for details). Note that although there
is no explicit buoyancy parameter in (4), the effects of stable
stratification on K
d
are implicit in L
E
and T
L
.
[10] An important reason to express K
d
in the form of
(4) is that the time-scale ratio is widely used as a parameter in
second moment closure models [e.g., Pope, 2000]. More-
over, VS found that their DNS results suggested that the time-
scale ratio was approximately independent of stratification.
The value they obtained was about 1.4 (i.e., g’0.7) which is
similar to values measured by Sirivat and Warhaft [1983] and
Yoon and Warhaft [1990] in thermally stratified decaying grid
turbulence. If the time-scale ratio remains independent of
stratification for a wider class of flows, including shear flows,
this has important simplifying implications for modeling.
Since both time scales are linked to the turbulence cascade
from large to small scales, it seems reasonable that they
maintain a constant ratio, provided there is no fundamental
change to these processes.
[11] Equation (4) may also be expressed in terms of
length scales as
Kd¼g1=3L4=3ð5Þ
where L=L
E
3/2
L
1/2
is a mixture of the displacement length-
scale L
E
and the dissipation length-scale L
=k
3/2
/. Note
that (5), with g’constant, is qualitatively consistent with
classical inertial range scaling for homogeneous turbulence
[e.g., Richardson, 1926].
[12] In the remainder of this paper we focus on investi-
gating the generality of the scaling given by (4) (or equiv-
alently (5)) with g’constant.
4. Data Sources
[13] Several previous DNS studies of homogeneous tur-
bulence, both with and without shear and stable stratifica-
tion, as well as laboratory grid turbulence studies, have
provided data that can be used to test the ideas presented in
section 3. We have selected several of these studies based
on accessibility of the data and the values of key parame-
ters, i.e., Reynolds numbers, Richardson numbers, and
Prandtl (or Schmidt) numbers. DNS studies that have used
artificially forcing to produce stationary turbulence were not
considered since the type of forcing scheme can affect the
mechanical to scalar time-scale ratio [see, e.g., Donzis et al.,
2005]. The selected data sources and typical parameter
values are summarized in Table 1. Further details are as
follows:
[14] 1. The DNS study of VS comprised freely decaying
homogeneous stably stratified turbulence. Each simulation
was initialized as isotropic turbulence with no scalar fluc-
tuations. Initial transients in the development of the scalar
fields (times less than one turnover time) were ignored.
Note that since these are temporally developing flows,
each simulation yields time histories for the instantaneous
L02602 STRETCH AND VENAYAGAMOORTHY: DIAPYCNAL DIFFUSIVITIES L02602
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diapycnal diffusivity K
d
,L
E
and T
L
, which can be used for
testing (4).
[15] 2. The DNS study by Shih et al. [2005] concerned
initially isotropic turbulent fields that were subjected to both
a uniform mean shear rate (S=@u/@z) and a uniform stable
stratification (buoyancy frequency N=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g=r0
ðÞ@r=@z
p).
A measure of the buoyancy effects is the gradient
Richardson number Ri
g
=N
2
/S
2
. Values used in the simu-
lations were 0.05 Ri
g
0.6. The flows are temporally
developing: for small Ri
g
the turbulent energy grows in time
while for large Ri
g
the energy decays. A stationary state
occurred when Ri
g
’0.17 for the data series used here. Initial
transients in the development of the scalar fields were omitted
from our analysis.
[16] 3. The DNS study of Rogers et al. [1986] provides a
reference case for passive scalars in homogeneous shear
flows, i.e., Ri
g
= 0. Available data comprised statistics at
discrete non-dimensional times St =2,4,6,... 12.
[17] 4. The experiments of Sirivat and Warhaft [1983]
and Yoon and Warhaft [1990] investigated the mixing of
temperature in spatially decaying grid turbulence. The data
used here was extracted from tables in the publications and
were measured at fixed downstream positions x/M= 100
and x/M= 76 respectively, where Mis the grid mesh size.
[18] 5. The experiments of Itsweire et al. [1986] investi-
gated decaying stably stratified grid turbulence in a water
channel using salinity as the active scalar. The Schmidt
number of salt in water is about 700. Note that the small-
scale salinity fluctuations could not be resolved by con-
ductivity probes due to the high Schmidt number. The
dissipation rate of the scalar variance was therefore inferred
indirectly from the variance transport equations.
[19] 6. The experiments of Mydlarski [2003] [see also
Mydlarski and Warhaft, 1998] investigated temperature
mixing in decaying grid turbulence without significant
buoyancy effects. An ‘‘active’’ grid was used to generate
intense turbulence at high microscale Reynolds numbers.
5. Results and Discussion
[20] Consolidated results from DNS and grid turbulence
experiments are shown plotted in Figure 1 in the form of the
non-dimensional diascalar diffusivity K
d
/k(or Cox number)
versus the parameter Pe
t
=L
E
2
/T
L
k. The Cox number is the
ratio of turbulent to molecular diffusivity and is thus a
measure of the intensity of turbulent mixing, while Pe
t
may
be interpreted as a turbulent Peclet number based on the
vertical overturning scale L
E
and velocity scale L
E
/T
L
.
Equation (4) with g= 0.7 and with 5th and 95th percentile
values (0.5 and 1.0 respectively) are also shown on the plot
for comparison. More detailed insight into values of the
parameter gcan be obtained from Figures 2 and 3 where it
is shown plotted versus the Cox number and local Richard-
son number Ri
t
=(NT
L
)
2
respectively. The Richardson
number indicates the ratio of buoyancy to inertial forces.
[21] From Figure 1 it is evident that the diapycnal
diffusivity remains proportional to L
E
2
/T
L
for all cases and
all times, and over at least five orders of magnitude in the
Table 1. Summary of DNS and Laboratory Experimental Data
Analyzed for This Study
a
References Re
l
Ri
t
Pr, Sc
VS (2006) 40 0 – 100 0.5
Shih et al. [2005] 90 0 – 10 0.7
Rogers et al. [1986] 40 – 90 0 1 – 2
Sirivat and Warhaft [1983] 40 0 0.7
Yoon and Warhaft [1990] 30 0.04 – 1 0.7
Itsweire et al. [1986] 40 0.25 – 100 700
Mydlarski [2003] 80 – 731 0 0.7
a
Typical values are shown for the micro-scale Reynolds number Re
l
,
turbulent Richardson number Ri
t
=(NT
L
)
2
and Prandtl or Schmidt number
Pr, Sc = n/k.
Figure 1. The non-dimensional diapycnal diffusivity K
d
/k(or Cox number) plotted as a function of L
E
2
/T
L
kfor DNS and
experimental data. The solid line is g= 0.7, and the dashed lines are g= 0.5, 1.0.
L02602 STRETCH AND VENAYAGAMOORTHY: DIAPYCNAL DIFFUSIVITIES L02602
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Cox number. Figures 2 and 3 further show that the coeffi-
cient gdoes not vary systematically with Cox number or
with Ri
t
.
[22] The stratified DNS of VS and Shih et al. [2005]
show only weak mixing with Cox numbers generally less
than 30 and reducing as buoyancy effects increase. The
passive scalar simulations of Rogers et al. [1986] with
Pr = 2 attained Cox numbers of up to 1000 but Reynolds
numbers were a factor of two lower than the Pr =1
simulations (Table 1). Diffusivities in these DNS are all
consistent with (4) and suggest that gis insensitive to both
shear and stratification.
[23] The data of Itsweire et al. [1986] have the highest Cox
numbers ranging from about 1000 to 30000. The Reynolds
numbers for these experiments were low (Table 1) and the
high Cox numbers are a consequence of the high Schmidt
number of salt in water (Sc ’700). The scatter in this data
is probably due to the indirect method of measuring the
scalar dissipation rate. Nevertheless the data are consistent
with the suggested scaling. Note that as the turbulence
decays downstream in these experiments, local Richardson
numbers increase while the Cox numbers decrease.
[24] The high Reynolds number passive scalar grid tur-
bulence experiments of Mydlarski [2003] attained Cox
numbers ranging from 120 to 4250. Diffusivities from these
experiments are again consistent with (4) with g’0.7.
[25] In summary the results shown in Figures 1, 2, and 3
support the scaling suggested in section 3 and the data are
well described by (4) with g’0.7 for all stratifications
(including passive cases) and both with/without shear. The
scaling also seems to be valid for Prandtl (Schmidt)
numbers 0.7 Pr 700 although simulations or experi-
ments at higher Reynolds numbers (and with shear and
stable stratification) are needed to fully address this issue.
6. Conclusions
[26] In homogeneous (but not necessarily stationary)
turbulent flows the instantaneous diapycnal diffusivity is
given exactly by K
d
=
r
/(@r/@z)
2
and may be formulated
in terms of the large scale properties of the turbulence as
K
d
=gL
E
2
/T
L
. Our analysis of DNS and grid turbulence
data shows that L
E
and T
L
can explain all the variations in
K
d
(over several orders of magnitude in the Cox number
K
d
/k) for a broad range of shear and stratification strengths
(including shear-free and neutrally stratified cases) while g
remains approximately constant.
Figure 2. The coefficient gplotted as a function of the Cox number K
d
/kfor DNS and experimental data. The solid line is
g= 0.7, and the dashed lines are g= 0.5, 1.0.
Figure 3. The coefficient gplotted as a function of the Richardson number Ri
t
=(NT
L
)
2
for DNS and experimental data.
The solid line is g= 0.7, and the dashed lines are g= 0.5, 1.0.
L02602 STRETCH AND VENAYAGAMOORTHY: DIAPYCNAL DIFFUSIVITIES L02602
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[27] This result suggests a unified scaling framework for
diascalar fluxes in homogeneous turbulence that seems to
have considerable generality and may therefore be useful for
turbulence models. It is worth noting that advective vertical
fluxes r0w0for the flows discussed in this paper show very
different characteristics from the diapycnal fluxes. For
example in developing stably stratified flows, the advective
fluxes can oscillate and change sign due to (reversible)
internal wave motions that do not contribute significantly to
irreversible diascalar mixing.
[28] An example of how these results may be applied to
turbulence modeling is given by Venayagamoorthy and
Stretch [2010], who used them to derive a new formulation
of the turbulent Prandtl number based on irreversible scalar
and momentum fluxes.
[29] Our results also clarify the issue of Reynolds number
effects that was discussed in section 1 regarding previously
suggested scalings, especially for the energetic regime
where /nN
2
^100. The new scaling we have presented
in this paper does not show any distinct regime(s) of
applicability, but remains valid over the whole range
1</nN
2
1represented in the data discussed here.
Furthermore the results do not show significant Prandtl (or
Schmidt) number effects on the diapycnal diffusivity. How-
ever, this issue requires further investigation due to the
limited range of Reynolds numbers in both the DNS and
laboratory experiments, particularly for stably stratified
flows. The DNS studies used here have relatively low
resolution compared to current state of the art so it is feasible
to extend the simulations to higher Reynolds numbers to
further check the validity of the suggested scaling for K
d
.
Another important extension of this study is to explore the
applicability of these scaling results to inhomogeneous
turbulent flows and to field-scale flows in the atmosphere
and oceans. Natural geophysical flows are more complex
than the idealized flows considered here, e.g., they are
typically intermittent in space and time and have both high
Reynolds and Richardson numbers, a regime which is not
represented in the data considered here. These flows can
develop a quasi two-dimensional layered structure in the
strongly stable limit [see, e.g., Riley and Lelong, 2000].
Whether these changes in structure influence the scaling
results obtained in our study requires further investigation.
The detailed measurements now becoming available from
field studies [e.g., Zaron and Moum, 2009] are making it
feasible to test the K
d
scaling in natural geophysical flows.
[30]Acknowledgments. We thank Lucinda Shih for providing post-
processed DNS results. We also thank the two anonymous reviewers for
their helpful comments.
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D. D. Stretch, School of Civil Engineering, Surveying and Construction,
University of KwaZulu-Natal, Durban 4001, South Africa. (stretchd@ukzn.
ac.za)
S. K. Venayagamoorthy, Department of Civil and Environmental
Engineering, Colorado State University, Fort Collins, CO 80523-1372,
USA. (vskaran@colostate.edu)
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