Geoscience Reference
In-Depth Information
(a)
Shallow
1
(b)
0.9
Deep
Shallow
1
0.8
0.9
0.8
0.7
0.7
0.6
0.6
0.5
0.5
0.4
0.4
0.3
0.2
0.3
0.1
0.2
0
0
0.2
0.4
0.6
0.8
1
x
0.1
0
0
0.1
0.2
Deep
0.3
0.4
Figure 7.6.
Numerical simulations of self-organizing turbulent flows in the presence of a topographic step dividing the (dimen-
sionless) domain in shallow and deep regions. The plots represent relative vorticity distributions after several rotation periods of
the system. (a) Rectangular domain: relative vorticity contours; solid (dashed) contours represent positive (negative) values. The
step runs along the
x
direction (the step height is 0.05). The long-term state of the flow is a large vortex at each region. The sign
of these structures can change with slight variations of the initial conditions, so there is no preferential final state in this system.
Adapted from
Tenreiro et al.
[2010]. (b) Square domain: relative vorticity surfaces calculated from an ensemble of 12 simulations;
dark (light) colors indicate positive (negative) vorticity. The step runs along the
y
direction. In contrast with the previous case, the
long-term evolution is systematically given by the four-vortex arrangement shown in the figure, regardless of the initial conditions.
Adapted from
Tenreiro et al.
[2013].
(see Figure 7.6a). As the self-organization process occurs,
the final state is characterized by one or two dominant
vortices in each region. One could naively expect to find
an anticyclone over the shallow region and a cyclone over
the deep region, according to the results of
Bretherton and
Haidvogel
[1976]. However, the experiments showed that
it was impossible to predict the sign of the resulting vor-
tices: small variations in the initial conditions led the flow
toward a completely different final state. This result was
reinforced with a statistical analysis. On the other hand,
when using a square container, the results were radically
different: The long term evolution of the flow was charac-
terized by the four vortices shown in Figure 7.6b. It was
found that this arrangement was systematically obtained
in numerical simulations with very different initial condi-
tions, and this preferential vorticity distribution was asso-
ciated with the topographic step and the aspect ratio of
the domain. The analysis included numerical simulations
using rectangular domains with larger aspect ratios (up to
δ
C
= 5), and it was concluded that a preferential final state
is always found as long as the length of the step
L
S
is equal
to the longest side of the container
L
C
. When
L
S
/L
C
<
1,
the flow organization after long times remains uncertain.
7.4.3. Bottom Friction With Rotation
Up to now we have focused the discussion on the effects
of variable topography, that is, on inviscid topography
effects. In a rotating system, the viscous effects associated
with the bottom topography are related with the Ekman
boundary layer, as described in Section 7.2. Ekman fric-
tion is typically incorporated in the equations of motion
by adding a dominant, linear damping term, and this is
also the case in the vorticity formulation. However, the
vorticity equation (7.17) also includes nonlinear Ekman
terms. Perhaps the most important consequence of these
