1. Introduction
Internal waves play an important role in climate formation
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The attractor has been studied both experimentally
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In this article, we provide a PCA-decomposition research applyingly to several wave-attractor problems. The questions of numerical application, methods, acceleration and emode number are discussed.
2. Research methods and principles
2.1. Numerical setup
During this research, wave attractor flows were investigated numerically, in 2D formulation, using high-order spectral-element software
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To simulate an attractor flow, the following equations were used:
Navier-Stokes equation in Boussinesq approximation:
salt transport equation:
The salt transport is obligatory since the startification is required for the internal wave propagation.
The last equation is the continuity one for the incompressible fluid:
In the incompressible flow in Bousinesq approximation energy conservation equation is not required. Here the following notations were used:
During the simulation, we used the constant uniform viscosity value
Geometry sketch
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The simulation was run from the steady liquid (i.e., initial velocity condition was zero), initial salt density distribution was linear:
so that buoyancy frequency was constant. The flow was exited by a moving-wall wave maker
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The amplitude
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For
The high-order spectral element meshes used for the simulation are represented in Figures 2–3. The elements shapes can be arbitrary convex quadrilaterals; the orthogonality in this method is not required.
Low-aspect ratio domain mesh 48x48 spectral element
Large-aspect ratio domain mesh 208x48 spectral elements
PCA
Principal component analysis represents the spatio-temporal variable
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where spatial modes
To calculate modes, the eigenvectors of the covariation matrix
Such a procedure saves computational costs (time and memory).
Nevertheless, the direct SVD calculation can also be time-costly, since it calculates all the singular values and vectors (their number corresponds to the number of spatial points of the grid). For the amplification, several numerical methods are used; the most popular are ARPACK and randomized SVD. The first one creates the solution iteratively from the random initial approximation, constructing the Krylov subspace, calculating the required number of the first singular vectors; the latter one projects the original data into a random subspace with the following corrections. All the methods are available in
Despite both of these methods being well-tested and reported to yield results close to the full SVD, the wave attractor flows can have a developed secondary-wave subflow that should be correctly reconstructed, and the convergence of the PCA on these problems was not investigated.
3. Main results
Due to the internal wave focusing, the liquid motion is localized in a specific figure, and the oscillation amplitude increases. Figure 4 represents a typical non-linear regime in the small-aspect ratio domain: the parallelogram-shaped
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Wave attractor in small-aspect ratiodo main, vy snapshot
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Large-aspect ratio domain problem, initial attractor development befor the instability formation
Large-aspect ratio problem, vy snapshot developed non-linear regime
Temporal behaviour of vy in the point selected
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To check the quality of PCA decomposition, full
where the solution reconstruction
where
Residuals in small-aspect ratio problem decomposition
Execution times for different SVD calculation methods
Comparison of reconstruction residuals for different aspect ratio
Large-aspect problem residuals
4. Conclusion
The PCA decomposition was considered for the wave attractor problems. Both full SVD and reduced methods yield the same accuracy of by-mode reconstruction, which allows to use the latter one without significant losses. It was found that their use can accelerate the computation up to 100 times, with the fastest being the
The large-aspect ratio problems demonstrate less residual decrease with the used mode number and thus require more modes for the same accuracy in comparison with the small-aspect ratio problems. This makes the indirect SVD methods application even more preferable.
Withal, PCA method converges slowly and requires about several hundred modes for a precise enough reconstruction, which limits the investigation of secondary waves, forming the instability, with this method.