Geoscience Reference
In-Depth Information
Figure 17.12d shows SV1 at the optimization time
t
=
0
computed by
x
(
0
)
=
G
(
0
)
x
(
0
)
,where
x
(
0
)
is the
SV at
t
= 0 shown in Figure 17.12c. Obviously, the tilted
troughs and ridges have turned up as expected [
Will et al.
,
2006]. The structure does not change much for longer
times. Figure 17.12e shows the SV after
t
= 125
0
=
2500 s computed by
x
(N
0
)
=
G
N
(
0
)
x
(
0
)
with
N
=
125. As expected, this pattern agrees very well with POP1
shown in Figure 17.12b. For
t
spontaneous gravity wave emission is a major issue in
atmospheric research. The differentially heated rotating
annulus is a lab experiment suitable to systematically
study spontaneous gravity wave emission in analogy to
the atmosphere [
Williams et al.
, 2008]. To detect inertial-
gravity waves in the experimental data, it is favorable
to not use the full flow field but instead make use of
the decomposition and analyze just the curl-free part of
the flow.
The primary difficulties with computing the decompo-
sition of the measured horizontal velocity at a given level
of the cylindrical tank is that the PIV data do not line
up on a nice grid, and the data may contain noise. To
handle these two issues, we use a mesh-free reconstruc-
tion method based on radial basis functions (RBFs). The
method employs
matrix-valued
kernels [
Narcowich and
Ward
, 1994] and mimics the Helmholtz-Hodge decompo-
sition of a 2D velocity field. It is similar to the method
described by
Fuselier and Wright
[2009] for the surface of
the sphere but is instead adapted for a 2D annular domain,
for which dealing with boundaries becomes important.
The method also provides a means of filtering the noise
in the measured velocity fields and can be used to recon-
struct the full 3D field in the rotating annulus. The key
ingredients to the mesh-free reconstruction and decompo-
sition technique are
divergence-free
and
curl-free
matrix-
valued kernels. In this study, we construct these kernels
from the scalar-valued Matérn radial kernels, which are
popular for spatial statistics [
Stein
, 1999] and are given by
→∞
the SVs converge to
the corresponding normal modes [
Kalnay
, 2002].
Seelig et al.
[2012] discussed SVs of the simple
Lorenz annulus model [
Lorenz
, 1984] in the con-
text of transitions to irregular flow and compared
some numerically deduced SVs with data-based ones.
The numerical model enabled the construction of a regime
diagram in terms of singular vector growth rates, where
the abscissa was the Taylor number and the ordinate the
thermal Rossby number. Strikingly, the diagram based on
singular vector growth strongly resembles the traditional
bifurcation diagram for annulus flows [
Hide and Mason
,
1970;
Lorenz
, 1984]. The largest growth rates could be
found in the irregular flow regime of the Lorenz model.
The findings from the simple numerical model suggest
that the gradual increase of irregularity in the rather broad
transition region to quasi-geostrophic turbulence might
partly be addressed to singular vector growth. For labora-
tory experiments as well as for natural flows there is always
a certain background noise level. Irregularities in the tran-
sition region might be seen as
extreme events
that arise
from random excitation of singular vectors with unusual
large growth rates [
DelSole
, 2007]. This process, together
with nonlinear wave-wave interaction, could explain the
gradual broadening of the spectrum when the rotating
annulus flow transits to geostrophic turbulence [
Pfeffer
et al.
, 1997]. Whether these ideas, derived from the low-
order Lorenz model, can be transferred to real annulus
flows is not clear yet. More data sets have to be analyzed
by the techniques described above to address this ques-
tion. However, it can be expected that the growth rates
increase for irregular flows since more EOFs have to be
considered to cover, say, 90% of the total variance for
irregular flows.
1
2
ν
+1
(ν
+1
)
(αr)
ν
K
ν
(αr)
,
φ
ν
(r)
=
0,
ν>
5
2
,
α>
0, (17.15)
where
K
ν
is the modified Bessel function of the second
kind of order
ν
. Increasing
ν
in (17.15) increases the
smoothness of the kernel, while increasing
α
increases
its peakedness. Letting
x
=
(x
,
y)
and
x
j
=
(x
j
,
y
j
)
,
the respective divergence-free and curl-free matrix-valued
kernels are then defined as [
Narcowich and Ward
, 1994]
di
ν
(
x
,
x
j
)
=
(
≥
r
2
I
+
T
)φ
ν
(
−∇
∇∇
x
−
x
j
2
)
,
(17.16)
curl
T
φ
ν
(
(
x
,
x
j
)
=
−∇∇
x
−
x
j
2
)
,
(17.17)
ν
17.3.4. Helmholtz-Hodge Decomposition
of Annulus Flows
T
is the Hes-
sian matrix. By construction, the columns of
div
where
I
is the 2
×
2 identity matrix and
∇∇
ν
are
divergence free
, while the columns of
cur
ν
are
curl free
.
Before discussing the exact details on the reconstruc-
tion and decomposition method, we note that since the
present application involves boundaries, it is necessary to
supplement the given data with boundary conditions to
make the decomposition of 2D velocity field unique [
Foias
et al.
, 2008]. We assume that both the divergence-free and
curl-free parts of the field are parallel to the boundaries.
According to the Helmholtz-Hodge decomposition the-
orem, any suitably smooth vector field can be decomposed
into the sum of a divergence-free field and a curl-free
field [
Foias et al.
, 2008]. These two fields can be used to
discriminate different wave types occurring in the annu-
lus. Baroclinic waves and Rossby waves are divergence
free, whereas inertia-gravity waves comprise a significant
part of horizontal divergence. Presently, the process of
