Geoscience Reference
In-Depth Information
Vortex attraction toward a topographic obstacle
:Itis
not necessary to use a topographic
β
-plane to study
vortex-topography interactions. The sole presence of a
topographic feature might be enough to set the vor-
tex motion, even if the vortex is relatively far from the
obstacle. The reason is the
β
-drift induced by the near-
est slope. Thus, on the Northern Hemisphere, cyclones
(anticyclones) are attracted toward topographic upslopes
(downslopes). The important point is that the trajectory
depends on the far-field structure of the vortex (given
a certain topography). This effect was used in the stud-
ies described above to attract cyclonic vortices toward a
ridge and a submarine mountain [
Zavala Sansón
, 2002;
Zavala Sansón etal.
, 2012]. It has been proposed by
Zehn-
der
[1993] that this mechanism, the effect of the vortex
far field, might play a role on the trajectory of tropical
cyclones under the influence of continental topography.
in the laboratory. One reason may be that bottom topog-
raphy variations in typical experiments using medium-size
tanks are usually not very small, which restricts the use
of the quasi-geostrophic model. One of the few exper-
imental studies on this topic was performed by
Zavala
Sansón
[2007], who examined the evolution of dipolar and
tripolar vortices over a sine-shaped topography in one of
the horizontal directions. The qualitative results showed
that in both cases the long-term evolution of the vortices
was characterized by the alignment of the flow along the
topographic contours. Cyclonic relative vorticity was dis-
tributed over the deep regions, while anticyclonic vorticity
spreads over shallow parts of the domain.
Another important reason for the scarcity of exper-
imental cases on rotating turbulence over topography
is that the generation of an initially turbulent flow in
a medium-scale facility might be difficult to reproduce.
Forcing methods are usually mechanical. For instance, a
disordered small-scale initial flow field was obtained by
Maassen et al.
[2002] by passing a grid of vertical bars
through the fluid in nonrotating experiments with strat-
ified fluids. In rotating tank experiments,
Morize et al.
[2005] generated a turbulent initial condition by lifting a
square grid placed at the flat bottom of the container.
Using the former technique, the studies of
Tenreiroetal.
[2010, 2013] provide recent examples of rotating turbu-
lence over a discontinuous topography. In their experi-
ments, the turbulent low was examined in rectangular and
square containers (1.5 m
7.4.2. Turbulence Over Variable Topography
In the absence of external forcing, strictly two-
dimensional turbulent flows are characterized by a self-
organization process in which energy is transferred from
small scales of motion toward larger scales. This is
the concept of the inverse energy cascade proposed by
Kraichnan
[1967], based on the pioneering ideas of L. F.
Richardson since 1922. A similar process is observed in
homogeneous, quasi-two-dimensional turbulence, where
rotation effects and bottom topography variations are
taken into account. An interesting example is the spin
up of fluid over a sloping bottom in a rectangular tank
observed by
van Heijst et al.
[1994], who discuss the influ-
ence of the topography on the formation of a regular
pattern of vortices along the container. When random
variable topography is present, the inverse energy cascade
is halted as the flow tends toward a steady state, aligned
with topography contours with shallow water to the right.
This implies anticyclonic structures over topographic
bumps and cyclonic vorticity over hollows. This process
was examined within the quasi-geostrophic context by
Bretherton and Haidvogel
[1976], who showed that the
quasi-steady state of the flow aligned along topographic
contours is characterized by a linear relationship between
potential vorticity and the stream function. Using the
more general shallow-water approximation,
Zavala San-
són et al.
[2010] studied numerically the long-term evolu-
tion of turbulent flows over random topography. In their
results a linear relationship between potential vorticity
and transport function was verified, equivalent to that
found by Bretherton and Haidvogel.
We have already described some experiments using iso-
lated topographic features (submarine obstacles, linear
slopes, ridges, etc.); however, irregular topographies cover-
ing the whole container have not been sufficiently studied
1 m, respec-
tively) in which the bottom was divided in two regions,
deep and shallow. This is perhaps the most simple topo-
graphic variation that can be analyzed by experimental
methods, because both regions have a flat bottom over
which the self-organization process is expected to occur.
However, the presence of the step that divides both sides
of the tank implied rather complicated and unexpected
results. Let us first describe the general characteristics of
the observed phenomena: Initially the flow evolves almost
as if the step was not present, generating larger structures
according to the inverse energy cascade. After a few rota-
tion periods, the step leads to a flow along the topography
that always maintains the shallow region on its right.
Depending on the strength of the flow and the step height,
at some time the vortical structures are no longer able to
cross the topography. As a result, the flow evolves almost
independently in the shallow and deep regions.
Let us illustrate the long-term state of the flow by
means of numerical simulations based on the vorticity
equation (7.17) and omitting Ekman friction. It is impor-
tant to emphasize that the results are radically different
when the aspect ratio
δ
C
of the container is changed
from rectangular (
δ
C
=2,
Tenreiro et al.
[2010]) to square
(
δ
C
=1,
Tenreiro et al.
[2013]). In the rectangular case, the
shallow and deep regions are themselves square regions
×
1mand1m
×
