Geoscience Reference
In-Depth Information
2
R low
P up
0.075
1. 5
R low
1
K
0.05
0.5
R up
0.025
0
R up
P low
0
-0.5
0.03
RR
RK
0.4
0.02
RP
0.01
0.2
RK
0 0123456789
0 0
5
10
15
20
k
k
Figure 6.5. Dispersion diagram (upper panel) and growth rate
(lower panel) of the eigenmodes of the superrotating lid con-
figuration for Ro = 0.1 and Bu = 0.25 (see Figure 6.2a). Thick
gray line in the upper panel corresponds to the RR resonance
and respective unstable modes. Here and sunsequent figures
the subscripts “low” and “up” indicate a wave in the lower and
upper layer, respectively.
Figure 6.6. Dispersion diagram (upper panel) and growth rate
(lower panel) of the eigenmodes of the superrotating lid con-
figuration for Ro = 1.5 and Bu = 3.5 (see Figure 6.2b). Thick
gray lines on the upper panel correspond to the RK and RP
resonances and respective unstable modes.
P up
R low
9
the Froude number, and
7
g H
2 2 r 0
R = V
C R
5
K
=
,
3
a new Rossby number. With these definitions one finds
the baroclinic instability at small R and KH instabilities
at large F , which matches the traditional view of these
instabilities. However, to keep a closer link with exper-
imental results of Williams et al. [2005] and Flór et al.
[2011], the discussion below is based on Bu and Ro. (For
more details, see Gula et al. [2009b].)
Thus, as for the ageostrophic Phillips model in a straight
channel [cf. Sakai , 1989; Gula et al. , 2009a], several types
of instabilities are present, namely, (a) the baroclinic insta-
bility for small values of Bu and Ro (RR resonance), (b)
the Rossby-Kelvin instability (RK or RP resonance) for
intermediate values of Bu and Ro, and (c) the Kelvin-
Helmholtz instability (KK or KP resonance) for high
values of Bu and Ro. As usual, the KH instability is
characterized by highest growth rates and shortest wave-
lengths, the baroclinic instability is long-wave and low
growth-rate, and RK instability is intermediate, although
spanning a wide range of wave numbers.
In Figures 6.5-6.7 we give the dispersion diagrams cor-
responding to different values of (Ro, Bu) referring to typ-
ical cases (a), (b), (c), respectively, in Figures 6.2 and 6.3.
1
R up
P low
10
8
6
4
2
0 0
KP
RP
RK
KK
RP
5
10
15
k
Figure 6.7. Dispersion diagram (upper panel) and growth rate
(lower panel) of the eigenmodes of the superrotating lid config-
uration for Ro = 10 and Bu = 90 (see Figure 6.2c). Thick gray
lines on the upper panel correspond to the RK, KK, RP, and KP
resonances and respective unstable modes.
We present also in Figure 6.8 the structure of unstable
modes in both layers and the corresponding maps of the
interface deviation, which is an often measured quantity
in experiments.
 
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