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grid scale. Most other numerical studies of stratified tur-
bulence have employed isotropic grids with
z
strong. Assuming a balance between the vertical viscos-
ity and advective terms in (8.13), they estimated that
the resulting vertical scale would be on the order of the
viscous scale
x
and
have obtained a wider range of spectral slopes. Several
DNS studies [
Riley and deBruynKops
, 2003;
Brethouwer
etal.
, 2007;
AlmalkieanddeBruynKops
, 2012;
Bartelloand
Tobias
, 2013;
Kimura and Herring
, 2012] are broadly con-
sistent with
Lindborg
[2006], despite having much larger
dissipation ranges. Steeper spectra, with slopes as low as
−
∼
2
π
L
h
ν
U
L
visc
≡
(8.21)
(note that
L
visc
is not the same as the Kolmogorov scale
L
d
;seealso
Godoy-Diana et al.
[2004]). Indeed, numerical
simulations have shown strongly layered flows with lam-
inar coupling in the vertical scale when the stratification
is increased at fixed Re (or, analogously, fixed resolution
with ad hoc grid-scale dissipation) [
Laval et al.
, 2003;
Waite and Bartello
, 2004].
Brethouwer et al.
[2007] showed
that the vertical scale is set by
L
visc
at small Re
b
and
L
b
at
large Re
b
, with a transition range over 1
5, have been found in some studies [
Laval et al.
, 2003;
Waite and Bartello
, 2004], but as
Brethouwer et al.
[2007]
pointed out, this steepening seems to result from excessive
vertical dissipation. Horizontal slopes of
3have
also been found, raising some questions about the univer-
sality of (8.19) [
Waite
, 2011;
Kimura and Herring
, 2012].
The vertical spectrum in (8.19) has been more difficult
to reproduce numerically, but
Kimura and Herring
[2012]
have reported a clear
−
2to
−
10. The
laboratory experiments of
Praud et al.
[2005] also exhibit
a viscous scaling of vertical scale, consistent with their
relatively small values of Re
b
.
Viscous effects can be important in strongly stratified
turbulence, even at large Reynolds number. When Re
Re
b
3 slope at small vertical scales. At
vertical scales larger than
L
b
, the vertical spectrum tends
to be flat, which is consistent with the layerwise decou-
pling that occurs for
L
v
−
L
b
[
Waite and Bartello
, 2004].
Brethouwer et al.
[2007] speculated that the strati-
fied turbulence inertial subrange extends down to the
Ozmidov scale, below which it transitions to isotropic
three-dimensional turbulence. However, the details of
this transition are not entirely clear. As envisioned by
Lilly
[1983], the layered structure of stratified turbulence
can lead to the development of shear instabilities at small
horizontal scales. Such instabilities have been observed in
numerical simulations and are associated with bumps in
the kinetic energy spectrum at large
k
h
[
Laval et al.
, 2003;
Brethouwer et al.
, 2007;
Waite
, 2011]. These bumps are
located at horizontal scales around the buoyancy scale and
appear to result from a nonlocal transfer of energy from
large, quasi-horizontal vortices to small-scale Kelvin-
Helmholtz billows [
Waite
, 2011;
Khani and Waite
, 2014].
These results suggest that
L
b
rather than
L
O
marks the
small-scale end of the
Lindborg
[2006] inertial subrange.
In practice, these scales are usually quite similar, as
1,
the horizontal part of the viscous term in (8.13) is small,
but the vertical part has magnitude
⎧
⎨
1
if e
b
1
(L
v
∼
L
visc
)
,
1
Re
α
2
∼
(8.22)
⎩
1
/
Re
b
if Re
b
1
(L
v
∼
L
b
)
,
[
Brethouwer et al.
, 2007]. So viscous effects due to verti-
cal gradients can be significant at large Reynolds number
if the stratification is strong enough to make Re
b
1.
The apparent paradox of strong vertical viscosity at large
Reynolds numbers is explained by the anisotropy of strat-
ified turbulence. As stratification increases, the character-
istic vertical scale decreases like
L
b
and the aspect ratio of
the flow decreases like Fr
h
. At large but fixed Re,
L
b
ulti-
mately falls inside the vertical dissipation range, at which
point the horizontal cascade is suppressed; this transi-
tion occurs around Re
b
∼
1[
RileyanddeBruynKops
, 2003;
Brethouwer et al.
, 2007]. The cascade theory of
Lindborg
[2006] assumes an inertial subrange in
k
h
over which vis-
cous effects are negligible and thus cannot be expected to
hold in the laboratory regime of Re
b
L
O
L
b
∼
Fr
1
/
2
h
,
(8.20)
1, where vertical
viscosity may not be restricted to large
k
h
. Instead, simu-
lations point to a steep -5 spectrum when vertical damping
(viscous or ad hoc) is strong [
Laval et al.
, 2003;
Waite and
Bartello
, 2004;
Brethouwer et al.
, 2007].
so very small Fr
h
are required to get a wide separation
between
L
b
and
L
O
[e.g.,
Brethouwer et al.
, 2007].
8.2.3. Viscous Effects
Our discussion so far has been focused on the invis-
cid dynamics of (8.13)-(8.16). However, it is viscosity that
ultimately distinguishes the laboratory and geophysical
regimes of stratified turbulence.
Riley and deBruynKops
[2003] pointed out that viscous coupling rather than
Kelvin-Helmholtz instabilities sets the vertical scale of
stratified turbulence when viscous effects are sufficiently
8.3. DIRECT NUMERICAL SIMULATION
8.3.1. Setup
In order to further explore the parameter regime of
laboratory-scale stratified turbulence, we have performed
a number of DNSs with modest buoyancy Reynolds
