1. Introduction
A wave attractor is a result of internal/inertial waves self focusing
[1][2][3][4][5]
In this investigation, the corresponding review of the methods used for wave-attractors flow processing will be conducted. For this purpose, the numerical simulation result will be used following
[2][6][7][10][2][7][9][11]
2. Research methods and principles
The simulation is conducted in open-source package
[12][10][13][14][15]
These are Navier-Stokes equation in Boussinesq approximation, dissolved salt transport and continuity equation. Here
For the focusing, internal waves require a slope
[1]
Domain principle outline
[LATEX_FORMULA]a<<H[/LATEX_FORMULA]
[16][17][18]
The other specific parameters (like viscosity, startification profile, wave-maker amplitude and frequency etc.) are varied from case to case; the specific values can be found in the article describing each particular case (the link will be provided in the corresponding sections).
3. Main results
3.1.
Kinetic energy is one of the most frequently appearing characteristics when discussing wave attractors. Its popularity is based on its huge values typical for this flow type.
The total kinetic energy is defined as
where
[19]
In wave attractor flows, the
[LATEX_FORMULA]\overline{E}>>1[/LATEX_FORMULA]
[5][20][21][3]
Since both energy and relative energy differ to a constant, their behaviour is the same, and the principles considered in what follows are applicable to both of them to the same extent.
The energy behaves specifically in time: at the formation process it grows, that reach the saturation, after which begins to "noise" (it is conditioned by the additional harmonics from the secondary waves). The energy has a tendency to oscillate even in a linear regime
[3][21][22]
To describe the system, the local average value is used (the oscillations are excluded); however, because of the rapid increase at the beginning and non-linear oscillations, the average obtaining becomes a non-trivial problem. Local average is required to maintain the following principle properties: to start from zero (as the real kinetic energy) and to not start to oscillate while the instability develops.
Energy behaviour in a non-linear regime (red) with different methods of the local average estimation
Energy at the attractor fomation time range
Energy at the developed instability
[3][23][24]
The method does not require special fitting and is governed by the number of filtration modes. The increase of
The next idea is the spectral filtration; the higher frequencies of the spectrum are suppressed, and then the inverse transform is conducted:
The different filter action on a real energy spectrum is represented on Fig. 5; the "box" filter is
[LATEX_FORMULA]f_{thr}<0.5f_0[/LATEX_FORMULA]
Energy spectrum with different filtartion
where
Energy local average calculation with simple moving average
Moving average action at the attractor formation time: the window width influence
Window influence on the moving average calculation: formation times
Moving average calculation: developed instability
The temporal spectra, while calculated for an attractor flow, are investigated in a point; as a representative point, the attractor's ray middle (Point 4 on Fig. 10) is usually considered
[3][21][22][25]
Liner attractor regime, vy snapshot and spectrum calculation locations
Linear attractor regime, spectra in different locations
[3]
Non-linear attractor regime in a large-aspect ration domain, vy snapshot and spectrum calculation locations
Spectra in different locations for large-aspect ratio attractor
[26]
Attractor in a layer, vy snapshot and spectrum calculation locations
Attractor in a layer spectra in different locations
[27]
(2,1) linear-regime attractor in a domain with underwater plateau, vy snapshot and spectrum calculation locations
Spectra in different locations for attractor in basin with underwater plateau, linear regime
(2,1) non-linear attractor in a domain with underwater plateau, vy snapshot and spectrum calculation locations
Spectra in different locations for attractor in basin with underwater plateau, non-linear regime
To improve the spectrum, windows are often applied. There is a set of windows most frequently used (for normalized window range from 1 to -1):
In fact, they can be approximated accurately enough by one of the Kaiser windows with the corresponding
where
Different spectral windows
The correspondence between classical window types and parametric Kaiser ones
| Window | Similar to, β |
| boxcar | 0 |
| Hamming | 5 |
| Hanning | 6 |
| Blackman | 8.6 |
The main consequence of the window using is the reduction of the "background" (the intervals between the peaks) and the revealing of the minor peaks, as shown on Fig. 21. This property is useful when one calculates the ratio between different spectral components; also, the narrow windows eliminate the smooth transition between peak and background, allowing to determine easily the borders of the peak. Note that the background level for Kaiser with high
The other side of the coin is peaks widening, that may affect the minor peaks resolution when triadic resonance instability subharmonics are studied
[4][3][21][22][27]
Wave attractor spectrum calculated using different windows
[22]
Wave attractor in two-layered stratification (vy Hilbert transform, cm/s)
fx-fy diagram in two-layered stratification, no window
fx-fy diagram in two-layered stratification, Kaiser window with β=5
fx-fy diagram in two-layered stratification, Kaiser window with β=14
Wave attractor flows are known to have a specific form of the coherent structure, which is frequently considered as a marker of a wave attractor presence, and one strives to visualize it
[26][28][30][31][4][26][28][4][26][27][31][3][32][33][34][35][29]
The different visualizations are represented on Fig. 26; the plotted is the momentary velocity amplitude. The colormap labels are the same as in the
[36]
The discrete (quantitative) colormaps can also be used for a wave-attractor visualization (Fig. 27). These ones, however, represent the background instability very poorly, which nevertheless may be useful for the structure visualization. "Accent" seems to be better for the structure emphasizing, and none of them reveal the background waves.
A wave-attractor in a weakly-nonlinear regime visualization (|v|, cm/s): continuous colormaps
A wave-attractor in a weakly-nonlinear regime visualization (|v|, cm/s): discrete colormaps
4. Conclusion
Wave attractors as complex flows require a thorough selection of the methods for their processing. In this article, a number of methods were discussed. For the energy mean calculation, the best method was found that is stable for the sharp initial increase as well as for developed instability oscillations. For the spectra, different windows application were discussed and revealed that different windows should be used for spatial and temporal spectra; the influence of the spectrum spatial position is investigated for the different setups, found that it is minimal. Additionally, the field map representation is discussed, the different colormaps are applied, their aspects are shown for the selection in further works.