Geoscience Reference
In-Depth Information
Note that
L
b
is sometimes called the overturning scale
because of its association with the appearance of small-
scale density overturning [e.g.,
Munk
, 1981;
Waite and
Bartello
, 2006], and it is distinct from the smaller Ozmi-
dov scale (which, confusingly, is sometimes also called the
buoyancy scale). The argument of
Billant and Chomaz
[2001] was based on the self-similarity of the equations of
motion when
L
v
∼
by numerical simulations.
Herring and Métais
[1989]
attempted to obtain an inverse cascade by applying
small-scale forcing in simulations over a range of Froude
numbers, but they were unsuccessful.
Lilly et al.
[1998]
considered rotating stratified turbulence with small-scale
forcing; they found an inverse cascade when strong rota-
tion was present, but not for purely stratified turbulence.
Smith and Waleffe
[2002] found a direct transfer of energy
from small-scale forcing into a vertically sheared mean
flow, but there was no cascade through intermediate
scales. A theoretical explanation for the lack of an inverse
cascade was given by
Waite and Bartello
[2004]: The
two-dimensional inverse cascade is a result of the conser-
vation of enstrophy (along with energy) by wave number
triads; the analogous quantity in stratified turbulence
is the potential enstrophy, which is approximately con-
served by triads for Fr
v
L
b
as well as the finding that
L
b
is the
dominant vertical scale of the zigzag instability [
Billant
and Chomaz
, 2000]. However, this scaling is also implied
by
Lilly
[1983], since
L
b
is the vertical scale at which the
Richardson number of layerwise two-dimensional turbu-
lence becomes
O(
1
)
and, presumably, Kelvin-Helmholtz
instabilities develop.
Waite and Bartello
[2004] measured
L
v
in numerical simulations of stratified turbulence and
confirmed that
L
v
∼
L
b
, and subsequent numerical studies
have been consistent with this finding. The inviscid limit-
ing dynamics of (8.13)-(8.16) at Fr
h
1. But the relationship between
energy and (potential) enstrophy is weaker in stratified
turbulence than in two-dimensional turbulence, because
gravity waves carry some of the energy but no potential
enstrophy. Even when Fr
v
1are
very different from the classical picture of layerwise two-
dimensional turbulence. Such turbulence is three dimen-
sional in the sense that horizontal and vertical advection
have the same order of magnitude, but it is anisotropic
because
α
1andFr
v
∼
1, layerwise two-dimensional
turbulence eventually leaks energy into gravity waves,
which cascade downscale and destroy the possibility of an
inverse cascade. More recent data analysis after
Nastrom
and Gage
[1985] has also contributed to the rejection
of the inverse cascade theory, as it shows a downscale
flux of mesoscale energy below scales of around 100 km
[
Lindborg and Cho
, 2001].
The lack of an inverse cascade seemed to mark the end
of the stratified turbulence hypothesis for the atmospheric
mesoscale, but it was revived in a very different form by
Lindborg
[2006]. The idea that stratified turbulence natu-
rally develops a vertical scale of
L
b
, and hence Fr
v
∼
∼
Fr
h
1.
8.2.2. Cascade Theories and Application
to Atmospheric Mesoscale
Observations of the atmospheric kinetic energy spec-
trum suggest that it has a double power law form in
horizontal wave number: At synoptic scales, it has a spec-
tral slope of
−
3, in agreement with QG turbulence theory
[
Charney
, 1971], but in the mesoscale the slope shallows to
something resembling
O(
1
)
,
suggests that it should be anisotropic but three dimen-
sional.
Lindborg
[2006] proposed a theory for stratified
turbulence with a direct cascade of energy from large
to small horizontal scales. Proceeding on dimensional
grounds along the lines of the
Kolmogorov
[1941] theory
for three-dimensional turbulence,
Lindborg
[2006] argued
that the kinetic energy spectrum should have the form
5
3
[
Nastrom and Gage
, 1985;
Cho
etal.
, 1999]. A number of different theories were proposed
in the late 1970s and early 1980s to explain the observed
form of the mesoscale spectrum.
Lilly
[1983] and
Gageand
Nastrom
[1986] advanced a stratified turbulence hypoth-
esis: They argued that the layerwise two-dimensional
nature of stratified turbulence with small Fr
v
might sup-
port an inverse cascade of energy through the mesoscale,
in analogy with two-dimensional turbulence [
Kraichnan
,
1967]. Such a cascade would require a small-scale source
of kinetic energy, which
Lilly
[1983] speculated could be
due to moist convection at the
O(
1
)
km scale. Around the
same time, a different explanation based on a direct cas-
cade of gravity wave energy was proposed [
Dewan
, 1979;
VanZandt
, 1982]. More recent alternatives to the strati-
fied turbulence hypothesis have included theories based on
QG [
Tung and Orlando
, 2003] and surface-QG turbulence
[
Tulloch and Smith
, 2009].
Waite and Snyder
[2013] have
considered the effect of direct mesoscale forcing by latent
heating.
Although it was an intriguing idea, the inverse cas-
cade theory for stratified turbulence was not supported
−
2
/
3
k
−
5
/
3
h
N
2
k
−
3
v
E
K
(k
h
)
∼
,
E
K
(k
v
)
∼
,
(8.19)
in horizontal and vertical wave numbers
k
h
and
k
v
. He sug-
gested that the mesoscale kinetic energy spectrum could be
understood as a stratified turbulence direct cascade from
synoptic scales to the microscale. The same claim has also
been advanced by
Riley and Lindborg
[2008] to explain the
energy spectrum of the ocean sub-mesoscale.
Lindborg
[2006] presented numerical simulations that
showed good agreement with (8.19), at least in the hori-
zontal. To minimize viscous effects and capture the strong
anisotropy at small Fr
h
, he used thin numerical grids
with
z
x
, along with separate horizontal and verti-
cal hyperviscosity to keep dissipation focused around the
