It is known that there are systems where the
rayfocusing phenomenon leads to the formation of specific structures called
wave attractors. In such systems velocity magnitude (and kinetic energy)
concentrates on a closed curve. This occurs because of specific dispersion
relation (a wave conserves the angle with vertical axis rather than angle with
the normal). Under such reflection law, ray focusing occurs just for
geometrical reasons [1].
Since their discovery attractors
were investigated in terms of main structure. But in recent work [2] vortex
filaments in a system with inertial wave attractor were revealed due to vortex
identification method. This experience makes us think that there can be other
hid hydrodynamical structures.
For that
purpose we will use another method than vortex identification: a mode
decomposition. Being fed with a field, the method decomposes into
characteristic modes, which can themselves be substructures of the flow.
Proper
orthogonal decomposition [3] represents the solution as a series of spatial
modes multiplied by corresponding temporal coefficients. The modes are selected
so that they are eigenvectors of the solution’s covariational matrix (solution
matrix is a temporal slices (”snapshots”) on the discrete spatial positions).
Eigenvectors corresponding different eigenvalues are orthogonal (that’s why the
decomposition is called orthogonal). These modes can be used for dimension
reduction, but we will be concentrated on the investigation of the modes
obtained themselves.
The problem of investigation came
from the ocean dynamics. Comparatively recently the wave attractor phenomenon
was discovered. It is a phenomenon of rayfocusing in liquid in the form of
narrow curve where the fluid motion is concentrated, with the fluid remaining
almost steady outside it. The conditions an attractor appears under are
periodic external force, sloping side and salinity gradient [1].
Soon
after the discovering the phenomenon was examined experimentally. As Leo Maas
showed in [4], the trapezium with one sloping is an object good enough for the describing
of ongoing processes. In the simplest case the wave generated reflects from
each border one times, forming rhomboidshaped internal wave, called (1,1)
attractor, correspondingly the number of reflections. It’s notable that
coordinate of the rhomboid (reflection points) can be calculated analytically
from external frequency and geometry. In [5] the method is introduced for (1,1)
attractors; the recent work [6] generalizes it on (n,1) one. The numerical
investigation turned out to be close to the experimental results [1][7], the
later works [8][9] diverges less than in 10%.
In the pioneer works the external
force was applied to the entire volume. Next, it was transferred to one of the
sides [10][11], which abled to provide a wider range of perturbations applied. The
force was simulated via system called wavemaker that in the case consists of shafts
and eccentrics providing discrete border perturbation.
Figure 1:
Scheme of domain geometry with attractor
That was
the main setup and principles of a problem we base on. We will consider the
model problem as twodimensional one, hence it saves computational resources while
providing enough accuracy [12][13][14].
Let’s
describe the setup to be simulated. The model region is a twodimensional
trapezium with one slope. Wavemaker is situated on the upper side, howbeit its
position is not critical.
The
equations system to solve consists of: the NavierStocks equation in Boussinesq
approximation:

(1)

and salt transport:

(2)


(3)

Here
is a pressure
minus its hydrostatical part at
,
— density of
fresh water,
— dissolved
salt density,
— salt
diffusivity coefficient. Initial stratification was linear:

(4)

The system is
supplemented by the incompressible continuity equation:
The
axis
is
supposed to be directed along the smaller (lower) trapezia base and
— along
the vertical side wall. The scheme of the domain with the dimensions and
schematic attractor curve is shown on Fig. 1.
The initial
condition for velocity is zero:
To emulate tidal forces a
wavemaker was used. It perturbs one of the domain’s side harmonically. Its
position is of no matter, we placed it at the top of the domain.
Equation for
border disturbance:

(5)

where
is an
upper border profile, where
and
are
the external forcing parameters; spatially the perturbance has a form of a half
a sine. As soon as the perturbation is much smaller than the domain height we
can rewrite the condition (5) as a velocity condition:

(6)

This allows us to solve the
problem in a fixed area, which saves computational resources with a minimal
lack of accuracy.
On the other
borders there are Dirichlet conditions for velocity.
For the salt we
have impermeability condition
on all the
boundaries. That’s why we smoothed initial salinity on upper and lower walls.
Sizes of basin are: height
cm,
length
cm,
bottom length
cm.
Hence external force is periodic, we will also consider time values in units of
its period
.
The
problem was solved numerically via spectral element method (Nek5000 [15]);
postprocessing was made using
python3
codes.
The feature of this problem is
that such phenomenon as wave attractor can appear. It comes out as a closed
curve the fluid motion is concentrated on. Fig. 2 shows the characteristical
spatial distribution of the velocity in nonturbulent regime at a = 0.02 cm.
The attractor is clearly visible.
As to
temporal evolution, the velocity has oscillation with amplitude slightly
modulated, with the exit to the saturation (Fig. 3).

Figure 2: Vertical velocity component
distribution,
a = 0.02 cm

Figure 3:
evolution
in the point mapped, a = 0.02 cm

For spatial mode extraction we
applied Proper orthogonal decomposition [3]. The method decompose field into
orthogonal modes timed by temporal coefficients:

(7)

so that the covariance between
modes tends to minimum. For numerical analysis we use intrinsic
python
scikit
library code [16]. It turns out the streamlines of the modes will be
more useful for representation rather than components, thus POD is made for
both velocity component simultaneously.
POD decomposition seems to work suitable
hence the spectrum of the flow is discrete, as Fig. 4 shows, which mean that
the flow consists of a number of harmonics, and the decomposition 7 is
physically justified. This occurs because of triadic resonance cascade in such
system, whose presence in the system was previously proved [17][18][7] [19].
With such small external force amplitude at the harmonic on the externa force
frequency
dominates.
Figure 4:
Energy spectrum of
For the
numerical investigation we used data on an interpolated uniform spatial grid
(150x101 points) and 900 temporal snapshots.
Being
applied to the solution, POD yields curious results. Alongside with the
attractor mode (Fig. 5), which is supposed to be, there is several ordered
vortex structures. The most powerful (in terms of energy share of the total
one) one is those with four vortices oriented diagonally (Fig. 6). It’s energy
is notably high (39.5%), which makes it to be a regular and important structure
rather than a relic.
Remaining
modes are of some interest, and their existence could hardly be proposed. They
presents two to four vortices (Fig. 79) located one over another (’vortex
stack’). They have vanishingly small energy, but their ordered structure does
not allow to consider them as an insignificant noise.


Figure 5: 1st eigenmode. Attractor structure

Figure 6: 2nd eigenmode. Quadrivorticular diagonal
structure.

Figure 7: 5th eigenmode. Divorticular stack.

Figure 8: 6th eigenmode. Trivorticular stack.

Figure 9: 7th eigenmode. Quadrivorticular stack.

With the spatial modes POD
decomposition provides temporal coefficients associated (see 7). Fig. 10 shows
those for modes 1st, 2nd and 5th modes (attractor, quadrivorticular structure
and divorticular stack). If the first two modes are gazed, they turn out to be
’asheared’ by a quarter a period, i.e., maxima of the 2nd coefficient take
place where the first one meets zero. This phase delay can be proved by phase
picture (Fig. 11). This allow to propose this mode to be visible in a pure flow
(without decomposition), hence around the first mode coefficient’s root the
second will be the most powerful mode.
For a
closer investigation spectra of the modes were made. Fig. 12 represents spectra
for 1st, 5th, 6th and 7th modes (i.e., the main one and vortices stacks). The
spectrum of the 2nd mode is very close to that of the 1st one and is
deliberately not shown; their main difference is in phase (see above).
The spectra for the stacklike
structures are discrete as well as that of the general flow, with the peaks
being scattered along the frequency axis. This means that different modes
corresponds different peaks of the spectra (as the modes have several peaks,
they may partly overlap for different modes), which makes to think that the
structures obtained (Fig. 79) are not accidental and result from the resonance
cascade instability, hence this is a mechanism of spectrum saturation
[20][17][18]. Our proposal is that these structures represent spatial
resonances cascade, while the the spectral one is represented by collateral
peaks on the spectrum (Fig. 4).

Figure 10: POD temporal coefficients

Figure 11: 1^{st}, 2^{nd}
temporal coefficients phases

Figure 12: POD
temporal coefficients spectra
As we noticed while discussing POD component
temporal coefficients phases, the coefficient for the 1st mode have a delay
correspondingly that of the 2nd one of some quarter of their common period, and
hence the maxima of the second one occur when the 1st mode coefficient is near
zero, we probably can see the structure like those represented at Fig. 6 in
pure solution (without POD).
This proposal can be
proved investigated flow streamlines picture evolution. Let’s consider
developed flow with attractor formed. The picture supposed to be seen is that
shown on Fig. 13. Here t = 450 s or 45T_{0}
which is enough for the ain
structure establishment. Still, we observe fourvortex structure (Fig. 14)
which is visible every
.
We emphasize
that this is flow itself without any decomposition applied. This is how the
hydrodynamical structures can be revealed after the prediction due to POD.

Figure 13: Streamlines of developed flow,
Main
attractor structure

Figure 14: Streamlines of developed flow,
Quadrivorticular
diagonal structure

Unfortunately, we cannot obtain
the same result for the vortex stacklike structures. The reason is evident —
their modes have frequencies different from those of 1sn and 2nd modes.
Besides, their energies are very low to be directly visible, unlike that of the
2nd mode which is comparable with the energy of the attractor mode. These two
points make their observation in the undecomposed flow nearly impossible.
Revealing hydrodynamical structure in a flow,
especially in a specific one, requires a very detailed investigation, hence the
flow can be noised with a turbulence, or structures themselves may turn out to
be not intense enough, which makes the search to be complicated. Decomposition
tools, like POD, can help in revealing them. In the case of attractor problem
we have decomposed the flow into vortexlike modes. Some of them turned out to
be connected with instability minor frequencies. After a spectral investigation
of POD temporal coefficients we managed to detect one of the structure (beyond
the wellknown rhomboid structure) found visible in the flow without
decomposition. These aspects lead us to an important conclusion: POD modes are
not just formal basis but structures of some physical sense. This allows to use
the decomposition not only for dimension reduction, but for search of real
physical structure attending the flow. Such possibility may be useful for
ordered but lowintense structures that remain hid behind the main structure of
a great energy. This makes POD to be a powerful instrument, especially in
turbulent flows with instability vortex cascade, which on the general plane may
seem only turbulent relics but can turn out to form ordered vorticular
structures.
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