Geoscience Reference
In-Depth Information
The vertical scale of the layers seen in Figure 8.8 can be
measured from the first moment of the vertical wave num-
ber spectrum of kinetic energy
E
K
(k
v
)
(which is computed
in an analogous fashion to the horizontal spectrum). We
define the characteristic vertical scale to be
(a)
10
3
E
K
(k
v
)
d
k
v
k
v
E
K
(k
v
)
d
k
v
L
v
≡
2
π
(8.28)
(a similar definition based on the
2
moment was used by
Waite and Bartello
[2004];
Brethouwer et al.
[2007] used
(8.28)). As discussed in Section 8.2, the vertical scale of
stratified turbulence is expected to scale like
L
visc
when
the layer thickness is set by viscous coupling and
L
b
when the layers break down into shear instabilities and
small-scale three-dimensional turbulence. In Figure 8.9,
the vertical scale in all simulations is plotted against
L
b
and
L
visc
. The buoyancy and viscous scales are similar in
these simulations, since (8.18) and (8.21) imply that
L
b
L
visc
=
Re
b
, (8.29)
and our
√
Re
b
values range from 0.39 to 1.6. Nevertheless,
the
L
visc
scaling is more convincing than the
L
b
scaling.
The layer thickness in these simulations therefore seems
to be set by viscosity, which is consistent with our rela-
tively small values of Re
b
[as in
Brethouwer et al.
, 2007].
Note, however, that (8.29) can also be interpreted as a ver-
tical Froude number for
L
v
(b)
10
3
L
visc
, which is therefore
not much smaller than
O(
1
)
. Since
U
and
L
v
are based
on mean quantities, we expect our simulations to have
patches with locally larger
U
and smaller
L
v
, which would
therefore be susceptible to instabilities like those visible in
Figure 8.8.
In the horizontal, the spectral bumps described above
suggest a definition of a transition length scale. Follow-
ing
Waite
[2011], let
k
m
be the horizontal wave number
of the local minimum of buoyancy flux seen in many of
the transfer spectra in Figures 8.5-8.7, and define
L
m
≡
2
π/k
m
. This is the horizontal length scale where kinetic
energy is injected by nonlinear interactions and partially
converted to potential energy when Fr
h
is sufficiently small
and Re is sufficiently large. In Figures 8.10a and 8.10b,
L
m
is plotted against the buoyancy and viscous scales for the
simulations in which a clear bump in
B(k
h
)
could be iden-
tified. For the range of parameters considered here,
L
m
scales fairly well with both
L
b
and
L
visc
; a wider range
of Re
b
would possibly be able to distinguish between
these two scalings. By contrast,
L
m
does not scale partic-
ularly well with either the Ozmidov or Kolmogorov scales
(Figures 8.10c and 8.10d), which are both much smaller
than
L
m
.
Waite
[2011] found
L
m
∼
Figure 8.9.
Characteristic vertical length scale
L
v
plotted
against (a) the buoyancy scale
L
b
and (b) the viscous scale
L
visc
.
Symbols denote Re
b
≈
2(
·
), 0.6 (
+
), and 0.2 (
∗
) (simulation sets
A, B, and C).
therefore appear to be signatures of the vertical layer
thickness in the horizontal spectrum. This connection
between vertical and horizontal is consistent with the
roll-up of thin layers of horizontal vorticity into more
isotropic Kelvin-Helmholtz billows, as seen in Figure 8.8.
L
b
in simulations
with viscous effectsreduced by hyperviscosity, which mod-
els the regime of effectively large Re
b
. The small-scale
injection of kinetic energy and associated spectral bumps
∼
