Geoscience Reference
In-Depth Information
perturbation
ρ
assumed to be proportional to the salinity
˜
S
, i.e.
r)
. The diffusivity of this scalar quantity
S
is represented by
κ
.
Following
Hart
[1972] the equations are nondimension-
alized with length scale
L
=
R
2
−
ρ(
˜
r)
=
αS(
2
Ω
Δ
Ω
R
1
, velocity scale
U
=
L
, pressure
P
=
ρ
1
U
2
, time scale
T
=1
/
, and for
the density (salt concentration)
ρ
=
ρ
2
−
ρ
1
. The experi-
mental setup and notations are represented in Figure 11.1.
The scaled equations then become
∂
Fluorescein dye
H
1
H
2
ρ
1
ρ
1
H
=27cm
ρ
2
Laser sheet
ρ
2
L
=75cm
R
1
=25cm
u
∂t
+
(
1
B
u
R
o
H/L
S
1
R
e
∇
R
o
k
g
2
ˆ
u
·∇
)
u
=
−∇
p
−
×
u
−
z
+
u
+
g
/g
4
R
o
Figure 11.1.
Sketch of experimental setup, with
ρ
1
and
ρ
2
the
upper and lower layer fluid densities of depth
H
; the annular
disk at the surface is driven by three motors with rotation
.
Sr
ˆ
r
,
∂S
∂t
+
1
Pe
∇
2
S
,
∇·
(S
u)
=
(11.2)
We show the presence of interfacial Ekman layers and
the influence of the Schmidt number. These properties
are relevant for the small-scale instabilities discussed in
Section 11.3.3. When the internal fluid interface is not dis-
crete but continuous as in a salt-stratified two-layer fluid,
questions about instability such as Kelvin-Helmholtz and
Hölmböe instability, internal Ekman layers, as well as the
effect of stratification on the RK instability come into
play (Section 11.3.3). In the conclusions (Section 11.4)
we discuss the different results in a geophysical con-
text in considering realistic Burger and Rossby numbers
in the atmosphere and oceans and a range of different
experiments on baroclinic (Figure 11.2) instabilities, and
small-scale instabilities.
with the nondimensional parameters defined as
the Burger number (or Froude number Fr = 1
/
Bu)
g
H
4
2
L
2
,
Bu =
the Rossby number
Ro =
2
,
and the Ekman number
ν
2
H
2
,
Ek =
(a)
(b)
11.2. EQUATIONS AND SCALES FOR
LABORATORY EXPERIMENTS
The dynamics for a two-layer salt-stratified and rotating
fluid with viscous boundary effects is described by the
Navier-Stokes equations with boundary conditions. They
are composed of the continuity equation, the equation of
motion (including Coriolis force and centrifugal forces),
and the conservation of salt. In the Boussinesq approxi-
mation, the general equations for a salt-stratified fluid in
rotation are
(c)
(d)
∇·
u
=0,
∂
u
∂t
+
(
−
∇
p
ρ
o
−
g
ρ
−
ρ
o
ρ
o
2
2
u
·∇
)
u
=
×
u
−
z
+
ν
ˆ
∇
u
−
+
ρ
ρ
o
2
r
r
,
ρ
o
∂S
∂t
+
2
S
,
∇·
(S
u)
=
κ
∇
(11.1)
Figure 11.2.
Typical observations of the instabilities reported in
the diagram of Figure 11.3 with (a) the Hölmböe instability (H),
(b) and (c) respectively, mode 4 and mode 6 of the RK instability,
and (d) the baroclinic instability (BI) from
Flór et al.
[2011].
with
u
and
p
, the velocity and pressure field, respectively,
ρ
o
the mean density, i.e.,
ρ
o
=
(ρ
1
+
ρ
2
)/
2, and the density
field is assumed to vary as
ρ
=
ρ
o
+
˜
ρ
with the density
