Geoscience Reference
In-Depth Information
7.5.1. Experiments on Shallow-Layer Flows
the oppositely signed vortices separate when approach-
ing the coast, subsequently moving away from each other
in opposite directions along the coast. This behavior is
similar to that of two oppositely signed point vortices
approaching a wall [see
Lamb
, 1932, Section 155.3], the
motion of which is governed by the “image principle.”
Inclusion of viscous effects drastically changes this behav-
ior, as shown in the numerical simulations by
Orlandi
[1990] of a 2D dipole structure colliding against a no-slip
wall. The boundary layer formed at this wall contains
vorticity with a sign opposite to that of the neighboring
vortex core. When the dipole moves closer to the wall,
the oppositely signed vorticity is removed by advection
induced by the dipole and moves around the dipole cores,
which leads to a widening of the primary structure, a
splitting of the dipole, and eventually a rebound from
the wall.
The shallow-layer experiments by
vanHeijstetal.
[2012]
have revealed that 3D effects may play a significant role in
shallow flows with bottom topography, for both the dipo-
lar vortex approaching an upward step and the dipole
climbing a linearly sloping bottom. In the case of the
step topography, it was observed that the no-slip sidewall
of the step gives rise to generation of vertical vorticity
ω
z
, the signature of which is also clearly observed in the
upper parts of the fluid column. As a result, a shield of
oppositely signed vorticity is formed around the dipole,
causing its arrest at the deeper side of the step. Before this
takes place, a portion of the dipole in the upper part of
the fluid column might cross the step, although it decays
quickly because of the shallowness of the fluid layer on
that side. The dipole moving uphill over the sloping bot-
tom experiences a similar effect. The boundary layer at the
no-slip bottom implies the generation of vorticity, that is
strongest in the shallower part of the fluid layer. The vor-
ticity vector pointing locally parallel to the bottom implies
a vertical component
ω
z
that is strongest near the front
of the dipole. This bottom-induced vorticity
ω
z
has a sign
opposite to that of the primary vortex, and its signature is
clearly observed even at the free surface. As in the case of
step topography, the bottom-induced vorticity component
becomes concentrated in a band around the vortex dipole,
which causes its widening and its slowdown, as visible in
the measured flow evolution shown in Figure 7.8.
This exploratory study has revealed that viscous effects,
i.e., no-slip conditions at the nonhorizontal parts of the
bottom, play an important role in shallow flows with
bottom topography. The combination of these viscous
effects and the 3D nature of the bottom topography gives
rise to generation of vorticity (including a vertical com-
ponent
ω
z
), which significantly influences the flow evo-
lution. Clearly, such flows cannot be described simply
by the two-dimensional Navier-Stokes equation with an
additional “bottom friction” term.
Flows in shallow fluid layers have been studied quite
intensively in the laboratory by a number of researchers
[e.g.,
Tabeling et al.
, 1991;
Xia et al.
, 2009], because they
were assumed to mimick quasi-2D turbulence to a good
approximation. Essentially, the experimental arrangement
consists of a thin layer of electrolytic fluid (salt solution)
contained in a square tank. The fluid depth
H
(typically
5-10 mm) is chosen much smaller than the horizontal
length scale
L
(typically 20 cm or more), the shallow-
ness implicitly assumed to result in planar flows with
negligible vertical motions. The fluid is set in motion by
electromagnetic forcing: The combination of an electrical
current running through the fluid between two electrodes
mounted along opposing walls and a magnetic field intro-
duced by one or more magnets underneath the tank bot-
tom results in a Lorentz force that acts on the fluid. For
example, by placing a single disk-shaped magnet under-
neath the tank bottom, the locally vertically oriented mag-
netic field, in combination with the horizontal electrical
current, gives rise to a locally horizontal Lorentz force. By
applying an electrical current pulse of short duration, one
may thus generate a dipolar vortex structure that propels
itself in the direction of its axis. By a suitable arrangement
of the magnets, more complicated initial flow conditions
can be created, e.g., colliding dipoles, arrays of dipoles,
and an irregular arrangement of vortex structures, leading
to a turbulent field.
This generation technique has been used to create dipo-
lar vortices that were forced to move over topographic
features such as a sloping bottom or a step in the bottom
[see
van Heijst et al.
, 2012]. Such a situation is observed
in the case of an inhomogeneous obliquely incident break-
ing wave train on a beach, where two vortices of opposite
circulation are generated [see, e.g.,
Bühler and Jacobson
,
2001]. As a first step to analyze this problem, a single vor-
tex over a linearly sloping bottom may be considered as
being a segment of a larger 3D vortex ring. In this invis-
cid approach the motion of the vortex is identical to the
self-propulsion speed of the ring, and in the case of a
vortex ring with a uniform vorticity distribution and a cir-
cular cross section, it will thus move without change of
shape in a direction parallel to the coast [see
Bühler and
Jacobson
, 2001]. As a slightly better model, one could take
into account the stretching/squeezing of vortex tubes in
the shallow vortex, according to Kelvin's circulation the-
ory, leading to changes in its vorticity distribution. This
“modulation” of the vorticity structure implies a shape
change and hence a more complicated motion of the
vortex.
For the case of a vortex pair moving over a sloping bot-
tom toward the coast, one could argue that each vortex
behaves according to the “vortex-ring model” and that
