Geoscience Reference
In-Depth Information
Hart
[1972] considered the top, bottom, and interfacial
friction layers and found that the rotation rates are
α
1
=
(
2+
χ)
Ro
/
2
(
1+
χ)
and
α
2
=Ro
/
2
(
1+
χ)
,where
χ
=
(ν
2
/ν
1
)
1
/
2
is the viscosity ratio between the two layers. If
the two layers have close viscosities
χ
=1,itleadsto
(α
1
,
α
2
)
=
(
0.75Ro,0.25Ro
)
.
A calculation based on a layerwise balance of the
torques in
Williams et al.
[2004] gives values for
(α
1
,
α
2
)
of the same order but depending on the turntable angular
velocity. The direct measurements of the radial veloc-
ity profiles by
Flór
[2007] are closer to
(α
1
,
α
2
)
and lower layers (the baroclinic instability), the resonance
between Rossby and Kelvin or Poincaré waves in respec-
tive layers (Rossby-Kelvin instability), and the resonances
between two Poincaré, or Kelvin and Poincaré, or two
Kelvin modes (Kelvin-Helmholtz shear instability). We
should recall at this point the physical nature of differ-
ent waves in the two-layer shallow-water system: Rossby
waves propagate due to potential vorticity gradients,
whatever their origin, Kelvin waves propagate due to
(and along) the boundaries in the rotating systems, and
Poincaré waves are inertia-gravity waves propagating due
to the density jump at the interface or at the free surface.
Although each instability occupies its proper domain in
the parameter space, we will see that there exist crossover
regions where two different instabilities coexist and may
compete.
≈
(
0.9Ro,0.1Ro
)
. We will therefore keep these last values
throughout the chapter, but this particular choice means
no loss of generality, as changing the relative rotation rate
just means rescaling the Rossby number.
Supposing a harmonic form of the solution in the
azimuthal direction,
6.2.2. Instabilities and Growth Rates
(u
j
(r
,
θ)
,
v
j
(r
,
θ)
,
π
j
(r
,
θ))
=
(
˜
u
j
(r)
,
˜
v
j
(r)
,
π
j
(r))
exp
˜
[
ik(θ
−
ct)
]
+ c.c., (6.7)
We irst present the overall stability diagram in the space
of parameters of the model and then illustrate different
parts of this diagram by displaying the corresponding
unstable modes and dispersion curves. The stability dia-
gram was obtained by calculating the eigenmodes and the
eigenvalues of the problem (6.5), (6.8) for about 50,000
points in the space of parameters (there are typically 200-
300 points along each axis in the figures below) and then
interpolating. Only discrete azimuthal wave numbers cor-
respond to realizable modes. We nevertheless present the
results as if the spectrum of wave numbers were con-
tinuous for better visualization. They are synthesized in
Figures 6.2 and 6.3 displaying the growth rates and the
wave numbers, respectively, of the most unstable modes.
Both figures represent the plane of parameters Ro-Bu
(Figures 6.2 and 6.3). We also show in Figure 6.4 how
the dispersion diagrams evolve while changing parame-
ters and approaching the instability band spreading from
low left to upper right in Figures 6.2 and 6.3. One clearly
sees how the initially stable flow without imaginary eigen-
values of
c
develops instabilities of various nature as
parameters change. Thus, as shown in the left column of
Figure 6.4, decreasing the Burger number leads to dis-
tortion of the dispersion curves of Rossby modes and
their reconnection leading to Rossby-Rossby (RR) reso-
nance, i.e., the baroclinic instability. Different distortion
of dispersion curves of Rossby modes takes place if Ro
increases at constant Bu, leading to reconnection with
(a) a Kelvin-mode curve and Rossby-Kelvin (RK) reso-
nance with corresponding instability and (b) a Poincaré-
mode curve and Rossby-Poincaré (RP) resonance and
corresponding instability. Further increase in Ro leads to
reconnection of Kelvin-mode curves and Kelvin-Kelvin
(KK) resonance and related shear instability with features
where
k
is the azimuthal wave number (
k
∈
N
), and
omitting tildes we get, from (6.2),
k(V
j
−
rc)iu
j
−
(r
+2
V
j
)v
j
+
r∂
r
π
j
=0,
−
(r
+
V
j
+
r∂
r
(V
j
))iu
j
+
k(V
j
−
rc)v
j
+
kπ
j
=0,
1
)
j
η
=0,
−
∂
r
(rH
j
iu)
+
kH
j
v
+
k(V
j
−
rc)(
−
π
2
−
π
1
+
s(π
2
+
π
1
)
=Bu
η
.
(6.8)
The system (6.8) is an eigenproblem of order 6 that
can be solved by applying the spectral collocation method
[
Trefethen
, 2000].
The dispersion diagrams we thus obtain show that the
branches of dispersion relation corresponding to different
modes can intersect, leading to linear wave resonances and
thus creating instabilities of various nature [
Cairns
, 1979;
Sakai
, 1989].
Following
Cairns
[1979] and
Ripa
[1983], the flow with
velocity
U
0
is unstable if there exists a pair of waves
with intrinsic frequencies
ω
2
that satisfy the fol-
lowing conditions: The waves propagate in the opposite
directions with respect to the basic flow
ω
1
and
˜
˜
ω
2
<
0, mean-
ing that they have opposite energy anomalies, and have
almost the same Doppler-shifted (absolute) frequencies
(
ω
1
˜
˜
kU
0
) and thus can phase lock and
resonate. The interpretation of the unstable modes as
resonances between the neutral waves provides a classifi-
cation of different instabilities and corresponding regions
of parameter space.
Namely, we will display below the instabilities result-
ing from the resonance between Rossby waves in upper
ω
1
+
kU
0
˜
∼˜
ω
2
−
