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∂w
∂z
length scales on Fr
h
,Re,andRe
b
is discussed. Conclusions
are given in Section 8.4.
∇
h
·
u
h
+Fr
v
= 0,
(8.15)
∇
h
+
b
,
∂b
∂t
∂b
∂z
∂
2
∂z
2
1
Re
1
α
2
+
u
h
·
∇
h
b
+Fr
v
w
+
w
=
8.2. BACKGROUND
(8.16)
8.2.1. Equations and Scale Analysis
where primes denote dimensionless variables and sub-
script
h
denotes the horizontal component. The aspect
ratio of the flow is
The equations of motion for an incompressible strati-
fied fluid subject to the Boussinesq approximation are
∂
u
∂t
+
u
1
ρ
0
∇
L
v
L
h
≡
Fr
h
Fr
v
2
u
,
·
∇
u
=
−
p
+
b
z
+
ν
ˆ
∇
(8.9)
α
≡
.
(8.17)
∇
·
u
= 0,
(8.10)
Consider first the inviscid dynamics of (8.13)-(8.16)
(viscous effects will be reviewed in Section 8.2.3). By defi-
nition, strong stratification means Fr
h
∂b
∂t
+
u
b
+
N
2
w
=
κ
2
b
,
·
∇
∇
(8.11)
1. But the limit-
ing behavior of (8.13)-(8.16) is largely controlled by the
size of Fr
v
, which has so far not been specified. Early
work on stratified turbulence was based on the premise of
small Fr
v
[
Rileyetal.
, 1981;
Lilly
, 1983], which leads to the
neglect of several terms in (8.13)-(8.16). This assumption
implies that vertical advection is small, the flow is nearly
in hydrostatic balance, and the horizontal velocity field
is approximately nondivergent. The equations of motion
reduce to vertically decoupled layers of two-dimensional
turbulence in this limit, which is commonly referred to
as quasi-two-dimensional, layerwise two-dimensional, or
pancake turbulence. More rigorous approaches, based
on averaging over high-frequency internal gravity waves,
yield the same limiting equations (with the possible inclu-
sion of advection by a vertically sheared horizontal mean
flow) [
Babin et al.
, 1997;
Embid and Majda
, 1998].
There is a self-destructive paradox built into the lay-
erwise two-dimensional turbulence picture, which was
anticipated by
Lilly
[1983]. Vertical decoupling implies
a collapse of vertical scale. Depending on the ultimate
size of
L
v
after this collapse, the vertical Froude num-
ber may no longer be small and the scaling may break
down. In the inviscid case,
Lilly
[1983] predicted that
Kelvin-Helmholtz instabilities would ultimately develop,
halting the vertical collapse and three-dimensionalizing
the flow at small scales. Such instabilities were subse-
quently observed in the numerical simulations of
Laval
et al.
[2003] and
Riley and deBruynKops
[2003], along with
a number of subsequent studies.
Billant and Chomaz
[2001] revisited the scale analysis of
Riley et al.
[1981] and
Lilly
[1983] and argued that their
assumption of Fr
v
where
u
=
u
z
is the velocity,
b
is the buoyancy,
and
p
is the dynamic pressure. The definition of
b
depends
on the fluid in question: It is
gθ/θ
0
in a dry atmosphere and
−
x
+
v
ˆ
ˆ
y
+
w
ˆ
gρ/ρ
0
in the ocean, where
θ
and
ρ
are potential temper-
ature and density perturbations,
θ
0
and
ρ
0
are constant
reference values, and
g
is gravity. The mass diffusivity
κ
is
assumed to be equal to
ν
(i.e., unit Schmidt number
Sc
),
but much of the following discussion is also valid for the
oceanic regime of
Sc
1. The Brunt-Väisälä frequency
N
is defined for the atmosphere as
g
θ
0
dθ
dz
,
N
2
≡
(8.12)
where
θ(z)
is the basic state potential temperature. Here,
N
is assumed to be constant.
The equations of motion can be nondimensionalized
in different ways depending on whether the underlying
flow is vortical or wavelike. The main difference in these
approaches is in the time scale, which is characterized by
the advection time scale
L
h
/U
for vortical flows and the
buoyancy time scale 1
/N
for waves [
Drazin
, 1961;
Riley
et al.
, 1981;
Lilly
, 1983]. Using the vortical time scale, the
dimensionless equations are [following
Riley and Lelong
,
2000]
∂
u
h
∂t
∂
u
h
∂z
+
u
h
·
∇
h
u
h
+Fr
v
w
u
h
,
∂
2
∂z
2
1
Re
1
α
2
−
∇
h
p
+
∇
h
+
=
(8.13)
1 is inappropriate for stratified tur-
bulence. They claimed that the vertical scale would adjust
naturally to keep Fr
v
∼
Fr
h
∂w
∂w
∂z
+
u
h
·
∇
h
w
+Fr
v
w
O(
1
)
, implying that
L
v
would be
set by the buoyancy scale
∂t
w
,
+
b
+
Fr
h
Re
∂p
∂z
∂
2
∂z
2
1
α
2
∇
h
+
2
π
U
=
−
(8.14)
L
b
≡
N
.
(8.18)
