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despite being coupled by vertical viscosity, are thin enough
to yield Fr
v
∼
these scalings for Re
b
∼
O(
1
)
). Neither
L
b
nor
L
visc
is a
simple function of Re
b
alone, since
O(
1
)
and, at least intermittently, Kelvin-
Helmholtz instabilities at small scales. For the larger val-
ues of Re
b
≈
Re
b
Re
,
L
visc
≡
2, these instabilities appear to break down
into even smaller-scale three-dimensional turbulence, as
foreseen by
Lilly
[1983].
The turbulent statistics in these simulations are quite
sensitive to changes in Fr
h
and Re, and as
Brethouwer
et al.
[2007] have argued, some of this sensitivity appears
as a dependence on Re
b
. In particular, the kinetic energy
spectrum has a power law range with a slope that steep-
ens with decreasing Re
b
. For the range of parameters
considered here, the slope goes from around
1
√
Re
.
L
b
≡
2
πL
h
2
πL
h
(8.30)
Both depend on Re in addition to Re
b
, and both become
smaller as Re is increased at fixed Re
b
.
Waite
[2011]
described spectral bumps with similar scaling in simu-
lations that employed hyperviscosity to reduce viscous
effects at large scales. In those simulations, a direct, non-
local transfer of kinetic energy from large horizontal
scales to
L
b
was found to be responsible for the bumps
in the spectra. As in
Waite
[2011], the spectral bumps
in the DNS in this chapter are consistent with Kelvin-
Helmholtz instabilities transferring energy directly from
large scales to the billow scale. For our largest Re
b
≈
5
as Re
b
goes from 2 down to 0.2. These spectra are all
steeper than
Lindborg's
[2006] proposed
−
3to
−
5
3
,whichisto
be expected at
O(
1
)
values of Re
b
, since vertical dis-
sipation is non-negligible across all
k
h
. The very steep
−
−
2,
the billows break down into smaller-scale turbulence and
the peaks are broad; for smaller Re
b
, the instabilities
become more intermittent and seem to be directly damped
by viscosity, and the peaks are narrower. It is interest-
ing that these small-scale transitions occur even for small
Re
b
≈
5 spectrum at small Re
b
has been observed in other
simulations where the buoyancy scale is inside the ver-
tical dissipation range (viscous or ad hoc) and appears
to be the asymptotic behavior for Re
b
→
0[
Laval et al.
,
2003;
Waite and Bartello
, 2004;
Brethouwer et al.
, 2007;
Kurien and Smith
, 2014]. At small Re
b
, the entire range
of
k
h
is essentially a dissipation range for the verti-
cal part of the viscous term; energy injected at large
scales is dissipated at large scales, and no energy cas-
cade to small horizontal scales occurs. Furthermore,
Waite
[2013] recently showed that the potential enstro-
phy is approximately quadratic in this limit, by contrast
with the larger Re
b
regime where higher order terms are
significant.
The dependence of the turbulent statistics on Re
b
does
not explain all of the sensitivity to Fr
h
and Re observed
in these simulations. In fact, there does seem to be some
significant variability in the slope of the power law part
of the spectrum, even at constant Re
b
. For our largest
Re
b
values, the kinetic energy spectrum steepens as Fr
h
decreases (with a corresponding increase in Re to keep
Re
b
≈
0.2, at least for sufficiently large Re and small Fr
h
.
Even though the vertical Froude number based on mean
quantities is smaller than 1, it is likely that there are inter-
mittent patches of locally small Richardson number and
subsequent instability, as seen in Figure 8.8c. Presumably,
there is a threshold Re
b
below which no instabilities occur
for any Re and Fr
h
, but these simulations show that such
a cutoff value is less than 0.2.
Given the similarity between the simulations presented
in this chapter and those of
Waite
[2011], it seems reason-
able to speculate that the location of the spectral bumps
seen here will scale with the buoyancy scale
L
b
in DNS
with Re
b
1. This scaling would suggest the existence of a
distinct spectral range between the buoyancy scale, where
kinetic energy in injected by instability of the large-scale
flow, and the Ozmidov scale, where these instabilities ulti-
mately break down into three-dimensional isotropic tur-
bulence. However, reproducing this range requires large
Re
b
along with small Fr
h
and thus Reynolds numbers
and numerical resolutions that are beyond the laboratory
regime;
BartelloandTobias
[2013] suggest Re
3, and
the steepening of these spectra has not quite converged,
even for Fr
h
= 0.017 and Re = 9400; further steepening
may therefore occur as Fr
h
is decreased more. The size
of the vertical dissipation, as approximated by Re
−
b
in
(8.22), is therefore not the only parameter that determines
the shape of the spectra, even at small Fr
h
and large Re. It
would be interesting to see if this behavior occurs at higher
values of Re
b
and whether it yields spectra steeper than
−
2). The spectral slopes range from
−
2to
−
10
5
, which
is an order of magnitude larger than the highest Re val-
ues here. This range of scales between
L
b
and
L
O
is a key
feature of geophysical stratified turbulence that is miss-
ing in laboratory-scale experiments and most numerical
simulations because of the extremely high computational
cost of resolving it. Indeed, typical values of
L
b
and
L
O
in the atmosphere are
O(
1
)
km and
O(
10
)
m, and so this
scale range is not normally resolved in mesoscale simu-
lations with
x
5
1.
Furthermore, many of these simulations exhibit spec-
tral bumps at small horizontal scales, and the shape and
position of these bumps are not solely determined by Re
b
.
These bumps appear to result from the injection of kinetic
energy by nonlinear interactions, and their position scales
like
L
b
and
L
visc
(it is not possible to distinguish between
3
for Re
b
O(
1
)
km. Interestingly, bumps in the
atmospheric kinetic energy spectrum have been observed
at this scale [e.g.
Vinnichenko
, 1970].
∼
