1. Introduction

Wave attractor is a peculiar but important phenomenon of internal/inertial waves self-focusing under proper conditions [1], leading to the formation of a stable wave of a specific shape (called a wave attractor) which is prone to accumulate the energy incoming into the system. Due to the increase of the amplitude on an attractor, there appears an opportunity for the instability formation, that hides the coherent structure formed [2]. The energy overpumping into the system, with the consequent production of secondary waves, is named one of the main reasons for why attractors are so hard to discover in the real ocean [3] where they are supposed to present respectively frequently [4].

The experimental investigation of the phenomenon has been mainly concentrated on the simple-shaped geometry [3,5–12]. Simultaneously, attractors have been successfully studied via numerical simulation [11, 13–18]. The main method of the visualization has remained to be velocity-based: slices or isosurfaces of a component/modulus [19].

However, these methods become inapplicable for visualization with the growth of instabilities. This makes one to seek for new visualization approaches. The latest works concerning attractor in enlarged geometries [18], with modified density gradient [20], in specifically-shaped domains [21] revealed effects that require more precise visualization.

On the other hand, in gas dynamics, there is a method called numerical Schlieren used for the visualization of the internal structure of a flow. In experiments, the Schlieren method is an optic method; for numerical simulation, considering density gradient () yield the same picture in binary colormap as black-and-white photography in experiment [22, 23]. Since attractors are considered in liquid, it is an incompressible medium; hence, considering density gradient is senseless. Because of it, we propose to consider a pressure gradient instead of density one:

To apply this method we selected the following computational cases from [24, 14, 18, 20, 21]:

• Attractor in trapezoid geometry

• Attractor in basin with non-linear stratification

• Attractor in large aspect ratio domain

• Attractor in geometry with underwater peak

• (2,1) attractor in domain with underwater plateau

All these cases were simulated using Nek5000 [25] software, solving the system of Navier-Stokes equation in Boussinesq approximation, salt transport equation (with Schmidt number being 700) and incompressibility condition.

Here is a fresh water density, is density increase caused by dissolved salt, is salt diffusivity (was taken so that ).

The external forcing was provided via a wave-maker [26] located on the top side (numerically it was a boundary condition for velocity). The visualization is tested for 2D cases, after the dimension-reduction approach being approved in [11, 13, 14, 17, 27].

 

Fig. 1: Spectral element computational grids: a) for base trapezoidal case and halocline profile, b) for geometry with underwater step, c) for geometry with mid-depth plateau, d) for large aspect ratio geometry

 

Nek5000, being a spectral-element method, provides a high-order solution. The element grids for cases considered hereinafter are depicted on Fig. 1. However, the data were written in regular post-processing grid (300x101 for the large aspect ratio domain, 150x101 for the rest). For better understanding how do these meshes correspond to each other, one can take a look at Fig. 2. It is an example with reduced element/points number to be readable (e.g., spectral mesh is shown to be 5x5 instead of 48x48 really used for this geometry). The spectral mesh consists of inclined elements, with data being written in Gauss-Lobatto-Legeandre points (cross-points of the thinnest lines); the post-processing grid is orthogonal, with more points to use the post-processing data “as is”, without after-reading interpolation.

It should be noted that the gradient was calculated on the post-processing grid with 2nd order finite-difference method, not by internal Nek5000 gradient method, despite the latter being finer. This was done to prove the ability of the visualization method proposed to work in post-processing mode on a secondary grid.

 

Fig. 2: Principle meshes types used in this work: a) spectral-element mesh for CFD computation (5x5 example for better view; thinnest lines for Gauss-Lobatto-Legeandre nodes) and b) regular post-processing grid on the same geometry (25x20 example)

2. The method application

To test the new method, we will compare it with convenient visualization with a vertical velocity slice. We will not plot the colorbar (values range scale) for pressure-gradient method, hence the values plotted are mostly contingent and vary from case to case. Velocity snapshots will be accompanied by colorbars with values in cm/s, hence they have real physical sense.

For all the cases, we must note a thing. Hence we drop the data on a post-processing grid, for a faster work it is useful to make the grid regular. Since one of the necessary conditions for the presence of an attractor is a slope [1], some points appear to be outside the computational domain. We are ready to put up with it to maintain the regularity of the history points grid, but while the plotting slice, we have to separately cut the under-slope regions. In the figures below, we deliberately did not do that, to show the pressure-gradient method’s advantage: for it, it is not necessary.

Results of pressure-gradient will be represented in black-and-white palette (black for higher values), following the Schlieren method. As one can see in what follows, it will be useful, because out-of-bounds regions (appeared in secondary post-processing mesh, but in fact not belonging to computational domain, like right white triangle on Fig. 2(b)) will automatically be colored white in this painting scheme. It is important, for instance, in case of complex-shaped domain like those with plateau or underwater peak (see below).

2.1. Base case

 

Fig. 3: Attractor in model trapezium domain: a) snapshot (cm/s), b) pressure gradient method

 

Before using the new method for specific cases, let us make sure that it works properly on classical cases. For this reason we apply it to the well-studied attractor flow [14,28,29] in trapezium geometry with linear stratification profile. The laminar attractor is clearly visible even if the velocity-component visualization is used; we consider it to validate that the new-proposed method works for a known case not worse than the methods used before.

It is so; moreover, one can see an advantage of the pressure-gradient method: the bottom right ray looks brighter (Fig. 3(b)), indicating the more energy accumulation on the first attractor ray. It is known from the precious studies [28, 30], but is not visible on the velocity snapshot on Fig. 3(a).

2.2. Attractor in media with a halocline

 

a)	b)

Fig. 4: Attractor in domain with halocline salinity profile: a) snapshot (cm/s), b) pressure gradient method

 

The next case is non-linear salinity. Vertical stratification is principal for wave attractor formation hence it requires buoyancy [1, 5]. In model set-ups it widely considered to be linear [1, 2, 14, 28]. For real systems it is no more so [31–33], and the attractor’s properties change. For instance, as it was found for a salinity profile with sharp-gradient density range (halocline), typical for subequatorial basins [32], attractors in such system accumulate more energy than those in the same geometry with linear stratification [20]. The instability develops non-homogeneously in basin space (on Fig. 4(a) one can see that the upper part is more disturbed than the lower one). Pressure gradient visualizes clearly the attractor contour as well as the secondary waves of instability (Fig. 4(b)), which is incapable for snapshot.

2.3. Large aspect ratio

 

a)

b)

Fig. 5: Attractor in domain with large aspect ratio: (a) snapshot (cm/s), (b) pressure gradient method

 

The real systems where attractors can exist are typically enlarged horizontally [35, 36], thus, it is reasonably to consider attractors in domains with large aspect ratio. Attrac-tors of this type are characterized by a greater energy accumulation, comparatively to those in domains with comparable sizes [18] (even in a greater degree than a halocline attractors from previous subsection [20]); and, subsequently, the more energy accumulation leads to earlier instability production (with slighter forcing) that ’disturbs’ the main attractor view. On Fig. 5(a) one can hardly guess the presence of an attractor. The pressure-gradient method allows revealing its contour (Fig. 5(b)) considerably clearer.

2.4. Underwater mountain

 

Fig. 6: Attractor in domain with underwater mountain: (a) snapshot (cm/s), (b) pressure gradient method

 

Despite the model attractor basic setup presumes that an attractor occupies the whole domain (thus the side walls the attractor reflects from can be supposed to extend bot-tom to surface), normally it is not so easy, and a real attractor reflects from underwater structure which has a shape of a mountain or plateau slope which top not necessary reaches the surface [34]. This fact made us to investigate an attractor in such peculiar geometry in [21]. It was discovered that a domain with an underwater peak may lead to a qualitatively new flow pattern: in such geometry there can appear an additional ”viscous” coherent structure. The main feature is that it cannot be predicted using the inviscid approximation (ray-tracing), and disappears if the viscous in the simulation tends to zero; thus, for small viscosities it is hardly seen (see Fig. 6(a): the main attractor on the left, with the ”viscous” structure poorly visible on the background). Pressure gradient allows to fully visualize both main and collateral attractor structures better than slice (cf. Fig. 6(a) and 6(b)).

2.5. Underwater mid-depth plateau

 

Fig. 7: Attractor in domain with underwater plateau: (a) snapshot (cm/s), (b) pressure gradient method

 

No less important geometry is that with the mid-depth underwater plain. This case is interesting by a new instability formation type [24]. For this geometry type, we simulate an asymmetric (2,1)-type attractor [1] with moderate instability intensiveness (Fig. 7(a)). Despite the structure is resolved quite good by v_y spashot, pressure-gradient Schlieren method reveals also the background structure (Fig. 7(b)), which is visible worse on the snapshot. Herewith, this case shows the limitations of the method: too heavy turbulence can still overlay the attractor, despite for moderately-turbulent flows it is better than velocity component snapshot.

3. Conclusions and Discussions

Previously, wave attractors were visualized by velocity. However, this method shows clearly enough only laminar attractors; such things as secondary instability waves, inhomogeneity of their production over volume or another coherent substructure presence complicates the visualization. The recent investigations of wave attractors faced with the phenomenon mentioned, motivated the reconsideration of approaches for wave attractor visualization.

Following the gas dynamics that widely uses the numerical Schlieren method, we adopted the latter for an incompressible flow to apply to wave attractor problems. In stead of density gradient, we consider the pressure gradient. By an example of a set of known cases, it was shown that the method proposed can be successfully used, revealing both attractor and instability structure. It was shown that for moderate instability pressure gradient Schlieren analogue visualizes the structure more clearly, which makes this method to be preferable for wave attractors.

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