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In-Depth Information
UL
h
ν
and the Kolmogorov scale
Re
≡
,
(8.2)
2
π
ν
3
1
/
4
where
ν
is the kinematic viscosity. Stratified turbulence
requires small Fr
h
(for “stratified”) and large Re (for
“turbulence”), but the definition of “large Re” depends
on the degree of stratification. It has been recognized for
some time that, at large but fixed Re, decreasing Fr
h
can
suppress turbulence [e.g.,
Laval et al.
, 2003;
Riley and
deBruynKops
, 2003;
Waite and Bartello
, 2004]. Stronger
stratifications therefore require larger Reynolds numbers.
Building on the work of
Smyth and Moum
[2000] and oth-
ers (
Riley and deBruynKops
[2003],
Hebert and de Bruyn
Kops
[2006a,b]),
Brethouwer et al.
[2007] argued that strat-
ified turbulence requires large buoyancy Reynolds number
L
d
≡
,
(8.6)
where
is the kinetic energy dissipation rate (length
scales are defined with the factor 2
π
for consistency
with the usual wave number definitions). The Ozmidov
scale is the scale below which isotropic three-dimensional
turbulence occurs [
Lumley
, 1964;
Ozmidov
, 1965], while
L
d
is the scale below which three-dimensional turbu-
lence is damped by viscosity. Using the Taylor relation
U
3
/L
h
, it can be shown that the ratios between these
scales are
∼
L
O
L
h
∼
L
d
L
O
∼
1
Re
3
/
4
b
Fr
3
/
2
h
U
3
νN
2
L
h
,
,
(8.7)
Re Fr
h
≡
Re
b
≡
,
(8.3)
so strong stratification at large Re
b
necessitates
which implies that the criterion for “large” Re increases
like Fr
−
2
h
as Fr
h
→
0. Flows with Re
b
1 are strongly
L
h
L
O
L
d
(8.8)
damped by vertical viscosity, even if Re
1 (see Section
8.2.3).
Mesoscale motions in the atmosphere lie well inside the
stratified turbulence parameter regime: Typical values of
U
= 1 m/s,
L
= 100 km,
N
=10
−
2
s
−
1
,and
ν
=10
−
5
m
2
/s
yield
[e.g.,
Brethouwer et al.
, 2007]. Since DNS requires spatial
resolutions of
x
L
d
[e.g.,
Moin and Mahesh
, 1998],
the computational challenge presented by (8.8) is quite
demanding. As a consequence of this difficulty, numer-
ical studies of stratified turbulence commonly employ
hyperviscosity or other ad hoc sub-grid-scale models to
avoid direct resolution of
L
d
[e.g.,
Herring and Métais
,
1989;
Waite and Bartello
, 2004;
Lindborg
, 2006;
Waite
,
2011]. Nevertheless, a number of recent DNS studies have
reached modest Re
b
values up to
O(
100
)
at small Fr
h
[
Kimura and Herring
, 2012;
Bartello and Tobias
, 2013;
Almalkie and deBruynKops
, 2012].
In this chapter, we will review and investigate the
dynamics of stratified turbulence with buoyancy Reynolds
numbers around unity. This is the regime of laboratory-
scale stratified turbulence, though it is often employed,
via experiments and simulations, as an idealization of
the atmospheric mesoscale and oceanic sub-mesoscale. In
keeping with both themes of this topic, that is, labora-
tory and numerical models for atmospheric and oceanic
flows, we will use DNSs to investigate the dynamics of
laboratory-scale turbulence. This chapter is about the lab-
oratory parameter regime, not a particular set of labora-
tory experiments; the focus will be on the use of theory
and idealized simulation to better understand turbulence
at these scales. We will begin in Section 8.2 with a brief
review of stratified turbulence theory, including the dif-
ferent parameter regimes, applications to the atmospheric
mesoscale, and how viscous effects become important
when Re
b
Fr
h
=10
−
3
,
e 10
10
,
e
b
=10
4
.
(8.4)
Geophysical values of Fr
h
can be readily obtained in the
laboratory and computationally, but realizable values of
Re, and hence Re
b
, are many orders of magnitude smaller.
Contemporary laboratory experiments and direct numeri-
cal simulations (DNSs) of stratified turbulence can obtain
Re as high as
O(
10
4
)
[e.g.,
Praud et al.
, 2005;
Bartello
and Tobias
, 2013]. While this is a large value, it is not
necessarily sufficient to yield large Re
b
at small Fr
h
,espe-
cially under decaying conditions. For example, consider
the decaying grid turbulence experiments of
Praud et al.
[2005]: A representative experiment has initial Froude and
Reynolds numbers of 0.086 and 9000, which give Re
b
= 67.
However, as the turbulence decays,
U
decreases and
L
h
increases while
N
and
ν
stay fixed. Since Re
b
∝
U
3
/L
h
,the
buoyancy Reynolds number decreases rapidly and falls to
O(
1
)
after only a few turnover times. Indeed,
Brethouwer
et al.
[2007] surveyed a number of experimental papers
and found them all to have Re
b
O(
1
)
, except for when
the stratification was very weak.
Direct numerical simulations face a similar challenge
in capturing stratified turbulence with large Re
b
, because
such flows have a wide scale separation between the
energy-containing scale
L
h
, the Ozmidov scale
O(
1
)
, even when the Reynolds number is
large. In Section 8.3, new DNSs of stratified turbulence
are presented, with Re
b
in the range of 0.2-2. The depen-
dence of energy spectra, transfer spectra, and related
∼
2
π
N
3
1
/
2
,
L
O
≡
(8.5)
