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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 Kd = $\epsilon$rho/(∂${\overline{\rho}/∂z)2 where $\epsilon$rho is the dissipation rate of density fluctuations, and ∂${\overline{\rho}/∂z is the mean density gradient. The diffusivity Kd may be expressed in terms of the large scale properties of the turbulence as Kd = gammaLE2/TL, where LE is the Ellison overturning length-scale, TL is the turbulence decay time-scale, and gamma is half the mechanical to scalar time-scale ratio. Our results show that LE and TL can explain most of the variations in Kd over a wide range of shear and stratification strengths while gamma remains approximately constant.
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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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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 hizdenotes 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., g0.7) 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 gconstant, 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 gconstant.
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 g0.7.
[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 g0.7 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
4of5
[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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... As such, many different combinations of characteristic length scale and velocity scale are used to define K q in a stratified fluid. 36,38,40,45,46 This demands more study to better understand the physics and dynamics of the stratified flows and the associated dominant scales of irreversible diapycnal mixing. ...
... Thus, for stationary state, using Eq. (1), Eq. (6) can be written by as 46,48 It should be noted that the concept of introducing an eddy diffusivity in a turbulence closure model is to represent the density flux in a closed form, but even though hq 0 w 0 i is available/measurable, K q defined with hq 0 w 0 i would be erroneous in a strongly stratified flow. This is because in a stratified flow, both w 0 and q 0 are generally contaminated by internal wave induced reversible components such that hq 0 w 0 i might provide an overestimation of irreversible diapycnal mixing, especially in the strongly stratified regime. ...
... It is plausible to assume that in a turbulent flow, T q should scale with the turbulent kinetic energy decay timescale T L ¼ k=e. It is worth noting here that this assumption has been deemed to be reasonable in earlier works 46,53 irrespective of the strength of stratification. The density variance and turbulent kinetic energy dissipate at the same rate such that the scalar timescale ðT q Þ and the turbulent timescale (T L ) are of same order. ...
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New insights for inferring diapycnal diffusivity in stably stratified turbulent flows are obtained based on physical scaling arguments and tested using high-resolution direct numerical simulation (DNS) data. It is shown that the irreversible diapycnal diffusivity can be decomposed into a diapycnal length scale that represents an inner scale of turbulence and a diapycnal velocity scale. Furthermore, it is shown that the diapycnal length scale and velocity scale can be related to the measurable Ellison length scale ($L_E$) that represents outer scale of turbulence and vertical turbulent velocity scale ($w^\prime$) through a turbulent Froude number scaling analysis. The turbulent Froude number is defined as $Fr=\epsilon/Nk$, where $\epsilon$ is the rate of dissipation of turbulent kinetic energy, $N$ is the buoyancy frequency, and $k$ is the turbulent kinetic energy. The scaling analysis suggests that the diapycnal diffusivity $K_\rho \sim w^\prime L_E$ in the weakly stratified regime ($Fr>1$) and $K_\rho \sim (w^\prime L_E )\times Fr $ for the strongly stratified regime ($Fr<1$).
... The global ocean, like most geophysical flows, is stably stratified yet subject to complex flow energetics such as overturning circulations and eddy generation [16,53,61]. It is well known that small scale mixing processes are required to balance global ocean energetics (e.g. ...
... [8,22,61]). However, there is currently disagreement between the results of laboratory experiments, observational data and numerical simulations on how much this mixing needs to be to balance global ocean energetics [1,8,22,53]. This has implications for models of global ocean circulation. ...
... However, [20,30] showed that equivalence between L O and L T only occurs when the ratio between buoyancy and turbulent timescales is O (1). The nature of the problem arises due to the complexity in reproducing, simulating or observing episodic and transient mixing events that occur at a range of temporal and spatial scales within geophysical flows [53]. Direct measurements of both temperature and velocity microstructure (and subsequently ) in energetic regions of the ocean is a means to address this issue (e.g. ...
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Direct observations of turbulent mixing in energetic ocean currents are fundamentally important to improving the understanding of stratified turbulence and the predictive capability of ocean models. In this study, we have selected an energetic region of the Western Indian Ocean to make simultaneous measurements of temperature and velocity to directly infer the geography of turbulent mixing on the east coast of South Africa. Direct measurements of the rate of turbulent kinetic energy dissipation, 𝜖, were recorded by an autonomous underwater vehicle glider platform and vertical microstructure profiler (VMP). The glider was deployed in the Agulhas current in 2015 over a 200 km transect and measured approximately 250 vertical profiles of turbulent microstructure. Additional vertical profiles near the shelf were obtained from approximately 85 VMP casts during 2018 in a region where the powerful Agulhas current is formed and interacts with the shelf. VMP and glider data collected at Sodwana Bay and the Agulhas current showed evidence of elevated dissipation rates 10−8 < 𝜖 < 10−6 Wkg−1. Our results indicate that in general turbulent mixing in the Agulhas is concentrated spatially in pancake-shaped patches. Ordinary Kriging was used to interpolate between consecutive profiles and showed that the horizontal scale of the turbulent patches was typically O(1 km) while the vertical scales of the patches ranged between 0.01 and 10 m.
... The temporal as well as spatial burstiness of the flow renders difficult an evaluation of transport coefficients on longer time scales as occur in climate. When it is not possible to perform long averages, as in decaying turbulence, other means of measuring small-scale properties have been constructed, for example following the Lagrangian dynamics of the flow [30,47,48] (see also [49] for convectively unstable boundary-layer flows). These properties are associated with kinetic and potential energy exchanges, globally and among scales, and between the underlying waves and nonlinear eddies, rendering interpretation difficult as the small-scale dynamics may depend on both the global and local parameters of a given flow at a given time. ...
... [53]). In [47], variations of C κ with parameters are attributed to the variation of T V and L E with the level of stratification, and with γ itself staying relatively constant in the range of parameters studied, for 1 < R B < ∞, but with Taylor Reynolds numbers of at most 90, whereas in our study they are roughly up to ten times larger. ...
... The conclusion in [47] of the constancy of γ throughout a large range of parameters may be pointing to a selfadjustment of stratified flows whereby the difference in (large-scale) energy distribution is compensated by smallscale activity such that the time scale ratio (γ) remains constant, a conclusion which is at odds, however, with (quasi-) equipartition ideas (E V ≈ 2E P in some range of scales), as emphasized on a phenomenological basis for example in [54] (see also [55]). The idea of equipartition of kinetic and potential energy at small-scale has been backed up recently, in the context of the shallow water model, by theoretical arguments based on statistical mechanics: at statistical equilibrium, the kinetic and potential energy of the small-scale fluctuations are equal [56], in fact as expected for a linear wave. ...
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We study rotating stratified turbulence (RST) making use of numerical data stemming from a large parametric study varying the Reynolds, Froude and Rossby numbers, Re, Fr and Ro in a broad range of values. The computations are performed using periodic boundary conditions on grids of 10243 points, with no modeling of the small scales, no forcing and with large-scale random initial conditions for the velocity field only, and there are altogether 65 runs analyzed in this paper. The buoyancy Reynolds number defined as RB = ReFr2 varies from negligible values to ≈ 105, approaching atmospheric or oceanic regimes. This preliminary analysis deals with the variation of characteristic time scales of RST with dimensionless parameters, focusing on the role played by the partition of energy between the kinetic and potential modes, as a key ingredient for modeling the dynamics of such flows. We find that neither rotation nor the ratio of the Brunt-Väisälä frequency to the inertial frequency seem to play a major role in the absence of forcing in the global dynamics of the small-scale kinetic and potential modes. Specifically, in these computations, mostly in regimes of wave turbulence, characteristic times based on the ratio of energy to dissipation of the velocity and temperature fluctuations, TV and TP, vary substantially with parameters. Their ratio (Formula presented.) follows roughly a bell-shaped curve in terms of Richardson number Ri. It reaches a plateau --on which time scales become comparable, (Formula presented.) -- when the turbulence has significantly strengthened, leading to numerous destabilization events together with a tendency towards an isotropization of the flow. Graphical abstract: [Figure not available: see fulltext.]
... A more recent study argues that scaling K q with Re b is inconsistent when the effects of viscosity or stratification become negligible. 11 More importantly, this study finds that the small-scale mixing represented by K q correlates well with a turbulent Péclet number Pe T , defined later, over the whole regime 1 < Re b < 1, i.e., there is no need to identify the scaling of K q with Re b in various mixing regimes. 11 This study highlights the importance of dividing diapycnal mixing into reversible and irreversible parts and models the latter with an interaction by exchange with the mean mixing model 12 by taking as the dominant mixing time scale the ratio of turbulent energy and viscous dissipation. ...
... 11 More importantly, this study finds that the small-scale mixing represented by K q correlates well with a turbulent Péclet number Pe T , defined later, over the whole regime 1 < Re b < 1, i.e., there is no need to identify the scaling of K q with Re b in various mixing regimes. 11 This study highlights the importance of dividing diapycnal mixing into reversible and irreversible parts and models the latter with an interaction by exchange with the mean mixing model 12 by taking as the dominant mixing time scale the ratio of turbulent energy and viscous dissipation. 11,13,14 More details on this approach are given later. ...
... 11 This study highlights the importance of dividing diapycnal mixing into reversible and irreversible parts and models the latter with an interaction by exchange with the mean mixing model 12 by taking as the dominant mixing time scale the ratio of turbulent energy and viscous dissipation. 11,13,14 More details on this approach are given later. Although the relationship obtained between K q and Pe T correlates existing data well, further support for it is needed, especially at various values of Sc, and for conditions where the stratification is strong. ...
Article
We conduct a parametric study of diapycnal mixing using one-dimensional-turbulence (ODT) simulations. Homogeneous sheared stratified turbulence is considered. ODT simulations reproduce the intermediate and energetic regimes of mixing, in agreement with previous work, but do not capture important physics of the diffusive regime. ODT indicates K&rgr;~&eh;/N2 for the intermediate regime, and K&rgr;~(&eh;h4)1/3 for the energetic regime and limit of near-zero stratification. Here K&rgr; is the turbulent diffusivity for mass, &eh; the dissipation rate, N the buoyancy frequency, and h the computational domain height, where h is relevant mainly in simulations with jump-periodic vertical boundary conditions. These scaling relationships suggest that K&rgr; is independent of the molecular diffusivity. ODT results for a wide range of parameters show that K&rgr; cannot be parametrized solely with the turbulent intensity parameter &eh;/(νN2), in contrast with the previous studies, but it is well predicted by correlations using the Ellison length scale.
... Although they did not take into account the Lagrangian dispersion model, their work has shed some light on a more general relation in nonstationary, inhomogeneous, and even unstably stratified turbulence. Such a relation may not be pursued in the traditional depth coordinate, as the advective vertical fluxes due to (reversible) internal wave motion can change sign and thus do not contribute significantly to irreversible mixing (e.g., Arthur et al. 2017;Stretch and Venayagamoorthy 2010;Taylor et al. 2019). ...
... Previous studies (e.g., Salehipour and Peltier 2015;Stretch and Venayagamoorthy 2010) have reported that the normalized diffusivity would be expected to reach high values of O(10 1 -10 4 ). Here the overall magnitude of normalized K WD is O(10 1 ) (Fig. 4a), corresponding to a small enhancing effect of mixing relative to the small-scale (molecular) k m . ...
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The present study provides a theoretical linkage between the Lagrangian dispersion diffusivity and the diapycnal diffusivity in the context of vertical mixing, although previous studies have demonstrated their equivalence under the assumptions of stationary, homogeneous, and stratified turbulence. This is achieved in a new coordinate in which the fluid density is adiabatically sorted in the vertical direction. In the density-sorted coordinate, 1) the vertical motion of Lagrangian particles is solely subjected to irreversible diffusion process; 2) relations between Lagrangian dispersion diffusivity, diapycnal diffusivity, and the generalized Osborn diffusivity are exact; and 3) a generalization of the classical Munk balance between vertical advection and diffusion is also illustrated in an exact sense. Since the adiabatic sorting of the fluid does not require the turbulence to be statistically stationary, homogeneous, and stably stratified, the present solution eliminated these requirements and is thus more general than previous studies. Upon this, a new Lagrangian diagnostic is proposed to quantify the local, instantaneous, and irreversible mixing. Applications are demonstrated in a turbulent scenario of internal wave breaking induced by current–topography interaction, in which the turbulence is intermittent, nonstationary, and inhomogeneous.
... As a result, the distribution (in space and time) of turbulent mixing and stratification processes, and their characterization in terms of parameters such as L T and K Z continue to be widely-studied [24][25][26][27]. A novel approach to this problem has been proposed by [28], who adopted a statistical physics perspective and characterised the efficiency (increase in potential energy to total energy input ratio) as a distinction between changes in the coarse-grained buoyancy profile, which represents the irreversible increase in potential energy, to the remaining energy, which is lost to fine scale fluctuations of velocity and buoyancy. ...
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Thermal microstructure profiling is an established technique for investigating turbulent mixing and stratification in lakes and oceans. However, it provides only quasi-instantaneous, 1-D snapshots. Other approaches to measuring these phenomena exist, but each has logistic and/or quality weaknesses. Hence, turbulent mixing and stratification processes remain greatly under-sampled. This paper contributes to addressing this problem by presenting a novel analysis of thermal microstructure profiles, focusing on their multi-scale stratification structure. Profiles taken in two small lakes using a Self-Contained Automated Micro-Profiler (SCAMP) were analysed. For each profile, buoyancy frequency (N), Thorpe scales (LT), and the coefficient of vertical turbulent diffusivity (KZ) were determined. To characterize the multi-scale stratification, profiles of d2T/dz2 at a spectrum of scales were calculated and the number of turning points in them counted. Plotting these counts against the scale gave pseudo-spectra, which were characterized by the index D of their power law regression lines. Scale-dependent correlations of D with N, LT and KZ were found, and suggest that this approach may be useful for providing alternative estimates of the efficiency of turbulent mixing and measures of longer-term averages of KZ than current methods provide. Testing these potential uses will require comparison of field measurements of D with time-integrated KZ values and numerical simulations.
... In the case of homogeneous turbulence subject to a uniform mean stratification, Stretch & Venayagamoorthy (2010) show χ and M to be equivalent. Indeed, if such homogeneous turbulence is maintained in a steady state by energy transfers from the velocity field, then χ is also equivalent to −J . ...
... The ES (L E ) only requires the temperature or density distribution profile to quantify the local overturning caused by turbulence. Because of these simple input parameters, this index has been widely used in the oceanographic literature (Itsweire 1984;Baumert & Peters 2000;Stretch & Venayagamoorthy 2010). L E is increasingly used in the field of atmospheric science and hydrodynamics, and has been applied to identify strong mixed events and to estimate various turbulence parameters. ...
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The concurrence of stable and unstable stratification in stratified flows leads to dramatically different features of turbulent mixing. This unique flow is experimentally studied by introducing a horizontal jet of dense fluid into a pool of light fluid. The buoyancy flux from simultaneous velocity–density measurements is an indicator for competition between a stabilising mechanism and another destabilising mechanism. The difference of mixing efficiency, quantified by flux Richardson number Rif, between the (un)stable stratification is mainly contributed by the large-scale mixing. The behaviour of Rif can be modelled by the gradient Richardson number Rig linearly in the low-Ri case and nonlinearly in the high-Ri case (especially in a region where the counter-gradient flux emerges). The turbulent diapycnal diffusivity, quantifying the combined effect of reversible and irreversible mixing processes, increases as the buoyancy Reynolds number Reb increases only when Reb is large. The irreversible mixing diffusivity, which quantifies the sole irreversible mixing process, increases linearly as the turbulent Péclet number with the data points from the (un)stable stratification overlapped. The turbulent Prandtl number approaches 0.75 as Rig approaches zero, but does not show clear dependence on Rig in the examined regime.
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Using direct numerical simulation (DNS), we investigate stationary and homogeneous driven turbulence in various stratifications, ranging from neutral to very stable. The Taylor Reynolds number is about 400, allowing an adequate separation of scales for the study of stratified turbulence dynamics. Analysis of the simulations is used to elucidate several aspects of stratified turbulence, including flux--gradient relations, length scales, spectra, the formation of layer, and criteria of turbulence collapse.
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We examine available data from experiment and recent numerical simulations to explore the supposition that the scalar dissipation rate in turbulence becomes independent of the fluid viscosity when the viscosity is small and of scalar diffusivity when the diffusivity is small. The data are interpreted in the context of semi-empirical spectral theory of Obukhov and Corrsin when the Schmidt number, $\hbox{\it Sc}$, is below unity, and of Batchelor's theory when $\hbox{\it Sc}$ is above unity. Practical limits in terms of the Taylor-microscale Reynolds number, $R_\lambda$, as well as $\hbox{\it Sc}$, are deduced for scalar dissipation to become sensibly independent of molecular properties. In particular, we show that such an asymptotic state is reached if $R_\lambda \hbox{\it Sc}^{1/2}\,{\gg}\,1$ for $\hbox{\it Sc} \,{}\,1$.
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The Osborn-Cox model is a simplified tracer variance budget that is a basis for direct estimates of the diapycnal diffusivity K V. When used to interpret temperature variance dissipation measurements, it indicates K V to be much smaller in the thermocline than the diffusivities found by matching large-scale observations to models or budgets. It is argued that, if the Osborn-Cox model is to describe fluxes in the general circulation, it must describe the variance budget of all fluctuations around the long-term average used to define the general circulation. Within this framework, the simplifications leading to the Osborn-Cox model are reexamined to find if they still hold and which is most likely to cause K V errors. -from Author
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The statistics of a turbulent passive scalar (temperature) and their Reynolds num-ber dependence are studied in decaying grid turbulence for the Taylor-microscale Reynolds number, R λ , varying from 30 to 731 (21 6 P e λ 6 512). A principal objective is, using a single (and simple) flow, to bridge the gap between the existing passive grid-generated low-Péclet-number laboratory experiments and those done at high Péclet number in the atmosphere and oceans. The turbulence is generated by means of an ac-tive grid and the passive temperature fluctuations are generated by a mean transverse temperature gradient, formed at the entrance to the wind tunnel plenum chamber by an array of differentially heated elements. A well-defined inertial–convective scaling range for the scalar with a slope, n θ , close to the Obukhov–Corrsin value of 5/3, is observed for all Reynolds numbers. This is in sharp contrast with the velocity field, in which a 5/3 slope is only approached at high R λ . The Obukhov–Corrsin constant, C θ , is estimated to be 0.45–0.55. Unlike the velocity spectrum, a bump occurs in the spectrum of the scalar at the dissipation scales, with increasing prominence as the Reynolds number is increased. A scaling range for the heat flux cospectrum was also observed, but with a slope around 2, less than the 7/3 expected from scaling theory. Transverse structure functions of temperature exist at the third and fifth orders, and, as for even-order structure functions, the width of their inertial subranges dilates with Reynolds number in a systematic way. As previously shown for shear flows, the existence of these odd-order structure functions is a violation of local isotropy for the scalar differences, as is the existence of non-zero values of the transverse temperature derivative skewness (of order unity) and hyperskewness (of order 100). The ratio of the temperature derivative standard deviation along and normal to the gradient is 1.2 ± 0.1, and is independent of Reynolds number. The refined similarity hypothesis for the passive scalar was found to hold for all R λ , which was not the case for the velocity field. The intermittency exponent for the scalar, µ θ , was found to be 0.25 ± 0.05 with a possible weak R λ dependence, unlike the velocity field, where µ was a strong function of Reynolds number. New, higher-Reynolds-number results for the velocity field, which smoothly follow the trends of Mydlarski & Warhaft (1996), are also presented.
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We examine observations of turbulence in the geophysical environment, primarily from oceans but also from lakes, in light of theory and experimental studies undertaken in the laboratory and with numerical simulation. Our focus is on turbulence in density-stratified environments and on the irreversible fluxes of tracers that actively contribute to the density field. Our understanding to date has come from focusing on physical problems characterized by high Reynolds number flows with no spatial or temporal variability, and we examine the applicability of these results to the natural or geophysical-scale problems. We conclude that our sampling and interpretation of the results remain a first-order issue, and despite decades of ship-based observations we do not begin to approach a reliable sampling of the overall turbulent structure of the ocean interior.
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Preface; Nomenclature; Part I. Fundamentals: 1. Introduction; 2. The equations of fluid motion; 3. Statistical description of turbulence; 4. Mean flow equations; 5. Free shear flows; 6. The scales of turbulent motion; 7. Wall flows; Part II. Modelling and Simulation: 8. Modelling and simulation; 9. Direct numerical simulation; 10. Turbulent viscosity models; 11. Reynolds-stress and related models; 12. PDF models; 13. Large-eddy simulation; Part III. Appendices; Bibliography.
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Linearly stratified salt solutions of different Prandtl number were subjected to turbulent stirring by a horizontally oscillating vertical grid in a closed laboratory system. The experimental set-up allowed the independent direct measurement of a root mean square turbulent lengthscale L t , turbulent diffusivity for mass K ρ , rate of dissipation of turbulent kinetic energy ε, buoyancy frequency N and viscosity v , as time and volume averaged quantities. The behaviour of both L t and K ρ was characterized over a wide range of the turbulence intensity measure, ε/ vN ² , and two regimes were identified. In the more energetic of these regimes (Regime E, where 300 < ε/ vN ² < 10 ⁵ ), L t was found to be a function of v , κ and N , whilst K ρ was a function of v , κ and (ε/ vN ² ) 1/3 . From these expressions for L t and K ρ , a scaling relation for the root mean square turbulent velocity scale U t was derived, and this relationship showed good agreement with direct measurements from other data sets. In the weaker turbulence regime (Regime W, where 10 < ε/ vN ² < 300) K ρ was a function of v , κ and ε/ vN ² . For 10 < ε/ vN ² < 1000, our directly measured diffusivities, K ρ , are approximately a factor of 2 different to the diffusivity predicted by the model of Osborn (1980). For ε/ vN ² > 1000, our measured diffusivities diverge from the model prediction. For example, at ε/ vN ² ≈ 10 ⁴ there is at least an order of magnitude difference between the measured and predicted diffusivities.
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The spatial decay and structural evolution of grid-generated turbulence under the effect of buoyancy was studied in a ten-layer, salt-stratified water channel. The various density gradients were chosen such that the initial overturning turbulent scale was slightly smaller than any of the respective buoyancy scales. The observed general evolution of the flow from homogeneous turbulence to a composite of fossil turbulence or quasi-two-dimensional turbulence and internal wavefield is in good agreement with the predictions of Gibson (1980) and the lengthscale model of Stillinger, Helland & Van Atta (1983). The effect of the initial size of the turbulent lengthscale compared with the buoyancy scale on the decay and evolution of the turbulence is investigated and the observed influence on the rate of decay of both longitudinal and vertical velocity fluctuations pointed out by Van Atta, Helland & Itsweire (1984) is shown to be related to the magnitude of the initial internal wavefield at the grid. An attempt is made to remove the wave-component kinetic energy from the vertical-velocity-fluctuation data of Stillinger, Helland & Van Atta (1983) in order to obtain the true decay of the turbulent fluctuations. The evolution of the resulting fluctuations is similar to that of the present large-grid data and several towed-grid experiments. The rate of destruction of the density fluctuations (active-scalar dissipation rate) is estimated from the evolution equation for the potential energy, and the deduced Cox numbers are compared with those obtained from oceanic microstructure measurements. The classical Kolmogorov and Batchelor scalings appear to collapse the velocity and density spectra better than the buoyancy scaling proposed by Gargett, Osborn & Nasmyth (1984). The rise of the velocity spectra at low wavenumbers found by Stillinger, Helland & Van Atta (1983) is shown to be related to internal waves.
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In 1921 G. I. Taylor introduced (with little discussion) the notion that the dispersion of a conserved passive scalar in a turbulent flow is determined by the motion of fluid particles (independent of the molecular diffusivity). Here, a hypothesis of diffusivity independence is introduced, which provides a sufficient condition for the validity of Taylor's approach. The hypothesis, which is supported by DNS data, is that, at high Reynolds number, the mean of the scalar conditional on the velocity is independent of the molecular diffusivity. From this hypothesis it is shown that (at high Reynolds number) the conditional Laplacian of the scalar is zero. This new result has several significant implications for models of turbulent mixing, and for the scalar flux. Primarily, a model of turbulent scalar mixing that is independent of velocity is inconsistent with the hypothesis, and gives rise to a spurious source or (more likely) sink of the scalar flux.