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(a)
(b)
0.0080
0.0080
0.0040
0.0040
0.0000
0.0000
T K
T P
B
T K
T P
B
-0.0040
-0.0040
-0.0080
-0.0080
10 0
10 1
10 2
10 3
10 0
10 1
10 2
10 3
k h k
k h k
(c)
0.0080
0.0040
0.0000
T K
T P
B
-0.0040
-0.0080
10 0
10 1
10 2
10 3
k h
/
Δ k
Figure 8.6. Horizontal wave number spectra of kinetic and potential energy transfer and buoyancy flux in simulations with fixed
Re b
2 and different Fr h and Re, corresponding to Fr h = (a) 0.069, (b) 0.032, and (c) 0.017 (runs A2, A4, and A6). Spectra are
multiplied by k h to preserve area under the curves with loglinear axes.
Fr h with Re b
2 would yield a similar injection of kinetic
energy and negative buoyancy flux at even larger k h .
At the smallest values of Re b
8.3.4. Length Scales
In physical space, these simulations exhibit the familiar
structure seen in a number of recent studies of stratified
turbulence: thin layers of predominantly horizontal veloc-
ity at large horizontal scales along with, in some cases,
shear instabilities and small-scale turbulence [e.g., Laval
et al. , 2003; Riley and deBruynKops , 2003; Brethouwer
et al. , 2007; Waite , 2011; Bartello and Tobias , 2013].
Figure 8.8 shows representative snapshots of the y com-
ponent of vorticity, which is dominated by the vertical
shear ∂u/∂z , for three different Re b values (the simulation
with the largest Re is shown in each case). For Re b = 2.6,
there are several patches resembling Kelvin-Helmholtz
instabilities, some of which appear to have transitioned to
smaller-scale turbulence. However, much of the domain
remains quiet, with a smooth, layerwise structure. As Re b
decreases, the regions of instability become increasingly
intermittent and without the associated breakdown into
smaller-scale turbulence; at the time of the snapshots in
Figure 8.8, only a few patches of instability are visible for
Re b = 0.69 and one for Re b = 0.18.
0.2, Figure 8.5c sug-
gests that the transfer becomes very small just downscale
of the forcing, where the energy spectra have steep slopes
of around
5. This range is magnified with logarithmic
axes in Figure 8.7, which shows transfer and buoyancy
flux spectra for two simulations with Re b
0.2. The left
panel is representative of the cases when Fr h
0.007. For
all k h
k f , the total transfer dominates the buoyancy
flux by at least an order of magnitude and is balanced
by the vertical part of the dissipation. It has a spec-
tral slope of around -4 at large scales but falls off faster
than a power law at larger k h . As observed above for
larger Re b , a small-scale transition emerges in the energy
balance at sufficiently small Fr h and large Re (Figure
8.7b), where a small bump of positive transfer and neg-
ative buoyancy flux emerges at large k h . These bumps
are quite intermittent, appearing and disappearing over
the averaging interval 1200
2000 s. In the aver-
aged spectra in Figure 8.7b, they are centered around
k h /k
t
60.
 
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