Geoscience Reference
In-Depth Information
Detailed 3D flow measurements by
Akkermans et al.
[2008a, 2008b, 2010],
Cieslik et al.
[2009a, 2009b, 2010],
Lacaze et al.
[2010] and
Sous et al.
[2004] on electromag-
netically generated vortex structures in a shallow fluid
layer have clearly revealed that such flows are generally
not quasi-2D: Vertical motion is observed to occur
throughout the evolution of the flow. These experimental
observations were confirmed by 3D numerical flow simu-
lations. In the case of a flow field generated by an array of
magnets, large-scale meandering motions were observed
instead of larger coherent vortex structures, that would
emerge when the flow would be (quasi-)2D. This behavior
is attributed to 3D effects: The PIV flow measurements
and the numerical simulations showed that significant ver-
tical motions were present throughout the flow evolution.
Upward motion was mainly observed in areas around
the elliptical points in the horizontal flow field, while
downward motion was concentrated in elongated regions
in strain-dominated parts of the horizontal flow field.
The relation between vertical, secondary motion and the
strain-vorticity characteristics of the primary horizontal
flow field has been analyzed in detail by
Kamp
[2012].
Background rotation is commonly believed to promote
the two dimensionality of flows by virtue of the Taylor-
Proudman theorem valid for geostrophic motion. In a
recent experimental and numerical study,
Duran-Matute
et al.
[2012] have investigated the effects of background
rotation on shallow-layer flows for the specific case of an
axisymmetric swirl flow. It was found that two dimension-
ality was enforced only in extreme cases: for very high
rotation rates and in very shallow fluid layers. In partic-
ular, two dimensionality was promoted for the Rossby
number Ro = 0.2 (defined as Ro =
ω
z
/
2
, with
ω
z
the
axial vorticity of the primary vortex structure), indepen-
dently of the fluid depth. Shallowness of the fluid layer
only promotes two dimensionality when the layer depth
is small compared to the typical (Ekman) layer thickness.
Duran-Matute et al.
[2012] have carried out a systematic
study, and they report on a number of different regimes in
the relevant parameter space.
Duran-Matute et al.
[2010a] demonstrated that a more
subtle scaling argument is appropriate. These authors con-
sidered the case of a single axisymmetric shallow vortex
flow, and we will here summarize their analysis. The initial
state is an axisymmetric swirl flow in terms of cylindrical
coordinates
(r
,
θ
,
z)
given by
t
=0:
v
=
(v
r
,
v
θ
,
v
z
)
=
(
0,
v
θ
,0
)
(7.28)
with
∂/∂θ
= 0. During the subsequent evolution a sec-
ondary flow in the
(r
,
z)
plane may develop, but it is
assumed that this flow remains axisymmetric. The conti-
nuity equation then becomes
1
r
∂r
(rv
r
)
+
∂v
z
∂
∂z
= 0,
(7.29)
implying that no relation between the primary swirl veloc-
ity
v
θ
and the vertical velocity
v
z
can be derived. This is
in contrast to argument (7.27) based on the continuity
equation in Cartesian formulation. Apparently, the scal-
ing ratio
v
z
/v
θ
must depend on the flow dynamics, which
is governed by the Navier-Stokes equation. The
θ
compo-
nent of this equation is essentially an evolution equation
for
v
θ
(with terms containing
v
r
and
v
z
), and the
r
and
z
components can be combined into an evolution equation
for the azimuthal vorticity
ω
θ
=
∂v
r
/∂z
∂v
z
/∂r
, which
describes the secondary circulation in the
(r
,
z)
plane. The
latter equation contains the following viscous term:
−
∂v
θ
ν
r
∂z
=
2
ν
∂v
θ
∂z
r
v
θ
(7.30)
Apparently, a vertical gradient in the primary flow field
v
θ
implies a nonzero
ω
θ
and hence a secondary flow in
the
(r
,
z)
plane. Again, the problem is nondimensionalized
by adopting the length scales
L
and
H
and the velocity
scales
U
and
W
. In the nondimensional equations for
v
θ
and
ω
θ
the aspect ratio
δ
and the Reynolds number Re =
UL/ν
now appear as dimensionless quantities. Under the
assumption of a shallow layer, i.e.,
δ
1, the flow quanti-
ties were expanded in powers of
δ
. It was shown that the
radial and vertical velocities scale as
v
r
v
θ
v
z
v
θ
=
O(
Re
δ
2
)
,
=
O(
Re
δ
3
)
.
(7.31)
7.5.2. Scaling Arguments
This scaling is in remarkable contrast to the commonly
adopted scaling (7.27) based on the continuity equation.
Duran-Matuteetal.
[2010a] have performed 3D numerical
simulations of a decaying Lamb-Oseen vortex and com-
pared these with the analytical solution for the case of
zero secondary circulation (i.e.,
ω
θ
= 0). Good agreement
between the analytical and numerical results was found for
δ
Re
1
/
2
It is commonly argued that the shallowness of the flow
domain implies that shallow flows are quasi-2D. This
argument is based on the continuity equation: By using
L
and
H
as horizontal and vertical length scales and
U
and
W
as typical horizontal and vertical velocity scales,
respectively, one derives
1. For these values of
δ
Re
1
/
2
and
δ
Re
1
/
3
the shallow swirling flow can hence be considered
as quasi-2D.
In a separate study
Duran-matute et al.
[2010b] have
analyzed the evolution of decaying dipolar vortices in
3and
δ
Re
1
/
3
≤
≤
U
L
∼
W
H
→
U
H
W
∼
L
=
Uδ
,
(7.27)
with
δ
=
H/L
the domain aspect ratio. As
δ
1 in shal-
low flows, this would imply that
W
U
. In a recent study,
