Geoscience Reference
In-Depth Information
(a)
(b)
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
(c)
1
0.5
0
-0.5
-1
-1
-0.5
0
0.5
1
Figure 11.5. Comparison between a high-resolution 3D numerical simulation (right) and linear stability results (left) for the same
parameters (Gula, personal communication): (a) interface height field; (b) perturbed velocity field in the lower layer showing a
Kelvin wave; (c) perturbed velocity field in the upper layer showing a Rossby wave. The linear stability results also show the
pressure field in shading. Numerical parameters: Bu = 0.62, Ro = 0.67, = 0.077 rad / s, = 0.104 rad / s, g = 0.0598 m 2 / s,
N θ ×
N r ×
N z = 1201
×
301
×
257.
number. For low Schmidt numbers, the first growing RK
modes are 5 and 6, whereas a baroclinic mode 2 started
to grow at a later instant with a much higher growth rate.
The type of instability was identified from the presence
of a Rossby or a Kelvin wave in the top or bottom layer,
respectively. With increasing Schmidt numbers, the mode
number of both instabilities increased, and RK modes
7 and 8 were observed, followed by a faster growing
baroclinic mode 4. These RK modes could well incline
locally the interface by an increase in the local shear and
thus decrease locally the Burger number. In this manner,
the presence of RK modes could act as a finite amplitude
perturbation and initiate the baroclinic instability for
lower Bu numbers.
0.0006
Mode 1
Mode 2
Mode 3
Mode 4
Mode 5
Mode 6
Mode 7
Mode 8
0.0004
0.0002
0.0000
60
65
70
75
80
85
90
Time
Figure 11.6. Evolution of
the energy amplitude of
the
11.3.2. Interface Conditions and Ekman Circulation
modes defined by E(k) = r z ˆ
u dr dz with
u(r , k , z)
the Fourier transform of the velocity vector u(r , θ , z) in
the azimuthal direction. Same simulation parameters as in
Figure 11.5.
u
· ˆ
ˆ
To leading order, the flow generated by the rotat-
ing disk at the surface is horizontal. In the vertical, a
second-order vertical Ekman circulation is generated by
the fluid expelled at the boundary of the rotating disk.
In the case of immiscible fluids, this circulation is asso-
ciated with thin Ekman layers at the internal interface
[e.g., van Heijst , 1984]. But when a tanh-like stratifica-
tion profile is present, such as between two miscible fluids
instability (see Figure 11.3). Different Schmidt numbers
result in different interface thicknesses, especially for large
times. The growth rate of the different amplitudes of the
modes show a shift in growth rate with increasing Schmidt
 
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