Geoscience Reference
In-Depth Information
(b)
(c)
(a)
2
π
c
ϕ
>0
3
2
c
ϕ
<0
π
c
ϕ
<0
2
c
ϕ
>0
0
480
500
520
540
560
Time (s)
Figure 11.11.
Observation of Hölmböe instability, with (a) cusped-shaped wave crests and (b) space-time diagram correspond-
ing to the observation regime. The retrograde propagating Hölmböe waves are characterized by the oblique crests and troughs
(indicated by the white dashed line). The diagram in (c) indicates the sign of the phase speed that corresponds to the inclination
of the crests with respect to the mean flow (dashed line).
11.4. CONCLUSIONS AND DISCUSSION
14
t
We have focused our attention on experimental and
numerical investigations of a baroclinic front in an annu-
lar tank. We summarize the results below and end with the
consideration of some geophysical flows.
The numerical simulations confirm the presence of a
lower layer Kelvin wave and upper layer Rossby wave
for parameters in the domain where RK instability was
seen experimentally. Preliminary simulations suggest that
the change in interface thickness, such as due to different
Schmidt numbers, modifies the appearance of instabil-
ity modes. The growth rates of the higher RK modes
increase, and the higher baroclinic modes appear for much
lower Burger numbers. A possible mechanism is that the
Bu number locally decreases due to the amplified inclina-
tion of the interface at positions where Kelvin waves and
Rossby waves interact.
Quasi-geostrophic theory [
Hart
, 1972] predicts the
baroclinic instability at a threshold Burger number of
Bu
<
0.12 (see, e.g., Figure 11.3). Calculations of the
ageostrophic baroclinic instability [
Gula et al.
, 2009b] pre-
dict the first mode (mode 1) at Bu
<
0.3 and mode 2 at
Bu
12
10
8
6
4
2
3
4
R
Figure 11.12.
Numerical results for the evolution of the values
of
R
and
J
o
=
JR
as a function of time, with the initial time
around
J
o
≈
14 and
R >
4. Note that
R
decreases below 2 and
then increases again in time.
other waves than those due to Hölmböe instability (see
Figure 11.7d). The DNS simulations indeed confirmed
the presence of small-scale perturbations superimposed
on the RK mode (Figure 11.13 and 11.14) that, because
of the values of
R
and
J
, could not be due to stratified
shear instability.
The origin of these waves is not understood. Next to the
possibility of spontaneous emitted inertia-gravity waves
mentionedabove,ahypothesisisthatthevariationsininter-
face slope as induced by the RK instability modes radiate
inertia-gravity waves. The ambient presence of the large
vertical shear and the presence of horizontal shear are in
favor of the wave capture process mentioned above. Since
the front is in an ageostrophic state, small-scale adjust-
ment waves are not impossible. Further investigations are
necessary to conjecture the type of wave radiation.
0.2 being in excellent agreement with the experimen-
tal observations, noting that mode 2 is generally the first
mode being observed.
Amplitude vacillation of the baroclinic instability seems
a general feature in these flows. It appears also for the
continuous stratifications here considered, but for larger
Burger numbers than the threshold Bu-number value for
baroclinic instability. The amplitude of the oscillation
decreases in time most likely due to the mixing at the
interface and corresponding change in Burger number.
≥
