Geoscience Reference
In-Depth Information
where
H
is the height for each layer,
g
the reduced gravity
g
=2
g(ρ
2
−
From the general equations (11.1), the two-layer quasi-
geostrophic model was derived to study the baroclinic
instability of a geostrophic flow analytically. This simpli-
fied model can be obtained from an asymptotic devel-
opment of the equations in the limit of small Rossby
numbers [
Pedlosky
, 1987]. At order zero, the flow sim-
ply verifies the geostrophic and hydrostatic balances. At
order one, the development gives the quasi-geostrophic
equations. This two-layer model gives major properties for
the baroclinic instability and allowed predictions for the
threshold Burger number for the baroclinic instability in
Hart's experiments [
Hart
, 1972]. For layers of the same
depth and same viscosity, the potential vorticity equations
for the two-layer quasi-geostrophic flow including Ekman
boundary layers [
Hart
, 1972;
Pedlosky
, 1970],
d
dt
1
ρ
1
)/(ρ
2
+
ρ
1
)
,
the background rotation
and
the differential rotation of the disk. Often the
dissipation number
d
=
√
ν
H
=
√
E
k
√
2Ro
is considered [
Hart
, 1972]. This number represents the dis-
sipation of the spinning disk motion at the surface due to
Ekman pumping. Alternatively, one can also consider the
Reynolds number
L
H
2
R
e
=
R
o
E
k
∇
ψ
1
)
=
d
3
2
,
as dissipation parameter. For a given geometric con-
figuration and stratification, the three other parameters
are the ratio of accelerations
=
g
/g
, the aspect ratio
γ
=
H/L
, and for the diffusivity of momentum compared
to salt, the Schmidt number Sc=
ν/κ
. Instead of the
Schmidt number, one can also consider the Peclet number
Pe =
L
2
/κ
=Re
1
Bu
(ψ
2
−
1
2
∇
2
ψ
1
+
2
ψ
1
−
2
ψ
2
−
−
2
∇
∇
ψ
2
)
=
d
3
2
ψ
1
,
(11.3)
d
dt
2
1
Bu
(ψ
1
−
1
2
∇
2
ψ
2
+
2
ψ
2
−
−
2
∇
Sc.
Given an experimental setup, the flow can be deter-
mined by three parameters,
d
, Bu, and the Rossby number
Ro. In experiments with immiscible fluids,
g
is set by the
available optically active fluids so that these experiments
are associated with a Bu-Ro diagram (for a given range of
rotation frequencies).A different reduced gravity
g
would
allow for the same Bu number but different Ro, Ek, or
d
numbers. The complete regime diagram is therefore set
by Bu, Ro, and
d
. When viscous effects are negligible,
this reduces to a two-dimensional diagram spanned by
Burger and Rossby
number. In order to compare different
experiments, in
Flóretal.
[2011], the parameters were cho-
sen equal to those in the work of
Williams et al.
[2005].
Because of the miscible fluid interface and larger setup,
we expect that threshold critical values may vary in this
regime diagram. Here, the nondimensional parameters are
the same as before, but for a cylindrical tank, the length
scale
L
is taken equal to the tank radius. Table 11.1 gives
the experimental values of these systems so far tested in
different experimental setups.
To examine the small-scale shear instabilities at the
interface, we use the definition of
Alexakis
[2005] for the
global Richardson number
J
o
, i.e. the gradient Richard-
son number Ri at the interface,
·
with
∂t
+
J(ψ
i
,
)
,
where
J
is the Jacobian operator and
ψ
i
the stream func-
tion in cylindrical coordinates for the layer
i
. The same
dimensional parameters Bu and
d
appear in the equations.
The coupling between the layers occurs via the Burger
number term and viscous effects in the form of Ekman
pumping. Topographic effects are neglected since the bot-
tom and disk are flat, and centrifugal effects on the shape
of the interface are neglected [
Hart
, 1973]. Because of
the quasi-geostrophic balance, inertia-gravity waves are
filtered out, and the equations apply to length scales that
are of the order of the deformation radius, for example,
the scale of the Rossby waves.
By contrast, in the two-layer shallow-water model stud-
ied by
Sakai
[1989] and more recently by
Gula and Zeitlin
[2010] and
Gula et al.
[2010], gravity waves are not fil-
tered out. Since this model contains no Rossby-number
approximation or asymptotic development, it includes
ageostrophic motions. It is an inviscid model and the
effects due to Ekman pumping are neglected.
The quasi-geostrophic and the shallow-water model
both rely on the shallow-water approximation, which
assumes thin layers with a small aspect ratio and small
vertical compared to horizontal gradients. The validity
of this approximation in experimental configurations is
achieved as long as
(H
j
/L)
2
Ro
j
in layer
j
is small. For
H
1
=
H
2
, it corresponds to the product of the square
of the aspect ratio,
γ
2
, with the Rossby number. This
number gives the ratio between the acceleration of the
vertical velocity and the vertical pressure gradient. For
=
∂
d
dt
i
g
δ
v
(U)
2
R
,
J
o
=Ri
(
0
)
=
JR
=
with the usual bulk Richardson number
J
, the total shear
thickness
δ
v
, and total density thickness
δ
ρ
of the inter-
face and their ratio defined as
R
=
δ
v
/δ
ρ
. In immiscible
fluid layers, the total density thickness is given by twice
the thickness of the Ekman layer.
