A nested polyhedra model of turbulence

A nested polyhedra model of turbulence

Ö. D. Gürcan CNRS, Laboratoire de Physique des Plasmas, Ecole Polytechnique, Palaiseau Sorbonne Universités, UPMC Univ Paris 06, Paris

A discretization of the wave-number space is proposed, using nested polyhedra, in the form of alternating dodecahedra and icosahedra that are self-similarly scaled. This particular choice allows the possibility of forming triangles using only discretized wave-vectors when the scaling between two consecutive dodecahedra is equal to the golden ratio, and the icosahedron between the two dodecahedra is the dual of the inner dodecahedron. Alternatively, the same discretization can be described as a logarithmically spaced (with a scaling equal to the golden ratio), nested dodecahedron-icosahedron compounds. A wave-vector which points from the origin to a vertex of such a mesh, can always find two other discretized wave-vectors that are also on the vertices of the mesh (which is not true for an arbitrary mesh). Thus, the nested polyhedra grid can be thought of as a reduction (or decimation) of the Fourier space using a particular set of self-similar triads aranged approximately in a spherical form. For each vertex (i.e. discretized wave-vector) in this space, there are either 9 or 15 pairs of vertices (i.e. wave-vectors) with which the initial vertex can interact to form a triangle. This allows the reduction of the convolution integral in the Navier-Stokes equation to a sum over 9 or 15 interaction pairs. Transforming the equation in Fourier space, to a network of “interacting” nodes, that can be constructed as a numerical model, which evolves each component of the velocity vector on each node of the network. Such a model gives the usual Kolmogorov spectrum of . Since the scaling is logarithmic, and the number of nodes for each scale is constant, a very large inertial range (i.e. a very high Reynolds number) can be considered with a much lower number of degrees of freedom. Incidentally, by assuming isotropy and a certain relation between the phases, the model can be used to systematically derive shell models.

I Introduction

Turbulence is a complex phenomenon involving chaotic behaviour over a range of scales. Yet, it has important underlying symmetries and regularities. Both its unpredictable nature, and its regular hierarchical structure is a result of the form of the nonlinear interactions. Therefore, the study of turbulence is a study of the nonlinear interaction, and the attempt to understand the hierarchical structure of the underlying symmetries it implies and their limitationsFalkovich (2009).

While the simple picture of a turbulent cascade, introduced by Kolmogorov, involves interactions between different “scales” (i.e. wave-number magnitudes ) of a conserved quantity, the Navier-Stokes equation does not readily uphold this picture. One usually has to write down the equation for a conserved quadratic quantity, such as energy or kinetic helicity, and assume statistical isotropy, homogeneity etc. in order to arrive at a description that is literally consistent with the basic cascade pictureFrisch (1995). However, it is clear that the nonlinear cascade happens in the original equation, even without these assumptions. For instance, even without any assumption of isotropy, the energy is transferred from wave-number to wave-number. If one uses a representation of the wave-vector in spherical polar coordinates in -space [i.e. using , , , such that = ], one could describe how the energy would be transfered from to , which is closely related, to what we call the “cascade”, even if the cascade as such, is not the only thing that is implied by the nonlinear interaction.

Figure 1: Alternating dodecohedron-icosahedron shells covering the Fourier space. Each starts at the origin and ends at one of the vertices of this object.

It is therefore tempting to imagine a discretization of the space using some form of spherical polar coordinates, which would assign the phenomenon of nonlinear cascade to a particular direction . Furthermore, one may introduce a logarithmic discretization in this direction, so that with only a small number of points, one may cover a large range in . In the study of weak wave turbulence, where the frequency and the wave-number can be linked using a dispersion relation, such logarithmic grid may be used without any difficulty (e.g. Galtier et al. (2000) for weak MHD turbulence). It is also commonly used in isotropic cascade modelling using closures, such as eddy damped quasi-normal Markovian approximation Leith (1971); Bowman (1996); Meldi and Sagaut (2012), or differential approximation models Lilly (1989); L’vov and Nazarenko (2006); Thiagalingam and Sagaut (2012). Cascade models that use a logaritmic discretization, or shell models (see for example Ref. Biferale (2003)), are also simplified models that try to exploit this particular aspect of the geometry of the turbulent cascade. Logarithmically discretized models (LDMs) for two dimensional turbulence, which can be derived from a systematic self-similar reduction of the Fourier spaceGürcan et al. (2016), can also be counted among these models.

More generally, the reduction of a continuous, but self-similar system, to a finite number of interacting modes Zeitlin (1991), respecting the original self-similar structure, has been studied in the past for cascade models based on the structure of the Burger’s equation Kerr and Siggia (1978) or Navier-Stokes Eggers and Grossmann (1991), especially in the context of earlier high resolution simulation efforts Vázquez-Semadeni and Scalo (1992). Such reduction procedures played an important role, beyond simple numerical convenience, in turbulence studies by providing theoretical insight into the underlying hierarchical structure of the dynamics of turbulence Frisch et al. (1978); Biferale et al. (1995); L’vov et al. (1998). Since one of the primary goals of turbulence study, is a reduction of the degrees of freedom in turbulent dynamics as faithfully as possible to its essential features, such models were studied for various aspects. It was found for instance that, severely reduced models, such as shell models, which rely on a truncation of the original system to keep only local interactions, recover both the wave-number spectrum and its intermittency Ohkitani and Yamada (1989); Jensen et al. (1991); Pisarenko et al. (1993). Even though how the shell model gets the intermittency correction is still contraversial.

In the same spirit, here we present a direct discretization of the Navier-Stokes equation in -space on a special mesh constructed from self-similarly scaled dodecahedron-icosahedron compounds (i.e. Wenninger model index 47 Wenninger (1974)). As will be shown below, choosing the scaling between two consecutive dodecahedron-icosahedron compounds equal to the golden ratio, allows the possibility of forming triangles using only discretized wave-vectors. The same grid can be obtained by considering nested, alternating icosahedra and dodecahedra where the scaling between the inner dodecahedron and the outer icosahedron is the square-root of the golden ratio times the factor so that the two consecutive polyhedra are the the duals of one another.

The model as introduced here may appear artificial as it is composed of rather complicated sounding polyhedra. However it is a minimal model in the sense that the icosahedron is the minimal basis for the icosphere, which is an approximation to the sphere with roughly equal vertex density everwhere on its surface (unlike a straightforward discretization of angles, which results in higher vertex density near the poles), and that its dual polyhedron, the dodecahedron is necessary for completing the triads formed by the position vectors (in -space) of its vertices.

The method of reduction, which could be generalized, can be thought of a discretization based on wave-vector-triads instead of a more classical discretization based on wave-vectors themselves. Notice that, the emposed self similar structure of the model enforces a uniform triad density as a function of scale, which is known to be incorrect in the case of turbulence, and appears as an important weakness of the model. However the model gives the correct wave-number spectrum and it can describe anisotropy in three dimensions. Furthermore, a higher order method, based on algorithmic construction of the grid may be imagined where wave-vectors picked from two distant (i.e. non-consecutive) polyhedra could be used to deduce a third polyhedron by completing the triads.

It can be speculated that, such a structure may be meaningful beyond the model itself as a crystalline state in wave-number space, which could be tailored by choosing the initial conditions and the driving to fall on the vertices of a compound polyhedron and the vertices of such an object would interact with eachother to drive other objects that are self-similar scalings of the initial object as well as other higher order compound objects.

Ii The Nested Polyhedra Model

Consider the Navier-Stokes equation in Fourier space:


We propose a discretization of the space using a logarithmic alternating icosahedral/dodecahedral basis (see figure 1):

where is the logarithmically spaced wavenumber magnitude with ,



where and are to be picked from the angles corresponding to the icosahedral and the dodecahedral vertices, listed in table 1. It is shown below that this choice comes from the condition of forming triads with the vertices of the three consecutive polyhedra.

0 10 0
1 11 1
2 12 0 2
3 13 3
4 14 4
5 15 5
6 16 6 .
7 17 0 7
8 18 8
9 19 9
Table 1: Polar and azimuthal angles and of a dodecahedron (left) and an icosahedron (right) are listed. Here and and with .
: : : : : :
0 5 10 3 0 11
6 11 18 16
7 12 14 9
8 13 15 17
9 14 3 12
1 1 10 4 1 12
18 15 19 17
12 7 10 5
16 19 16 18
3 14 4 13
2 4 10 5 2 13
17 15 15 18
13 8 11 6
19 16 17 19
2 11 0 14
Table 2: Two dodecahedral vertices, from the neighbouring shells (i.e. and ) that form a perfect triad with the icosahedral vertex at shell .

Note that by enumerating the vertices of the icosahedron and the dodecahedron, we reduce the number of indices necessary to describe a given wave-vector from to (i.e. using only and , we can define a unique wave-vector). It is also important to mention that since the original velocity field is real, its Fourier transform has the symmetry that . The assignment of numbers to vertices are picked such that where is the number of vertices of the polyhedron in consideration (i.e. for the icosahedron, while =20 for the dodecahedron). This symmetry can be used to reduce the number of degrees of freedom by half. Otherwise, one must pay attention that the initial conditions as well as all the terms in the equation (such as forcing, dissipation etc.) respect this symmetry.

Three dimensional turbulence requires solving three vector components of the velocity. Here we use cartesian coordinates . In this representation, the Navier-Stokes equation becomes:


where and can be inferred from ,, and using the fact that the corresponding wavenumbers form a triad:

consider three consecutive spherical shells such that and , and take alternating spheres to be discretized as dodecahedrons and icosahedrons (i.e. is an icosahedron, is a dodecahedron, is icosahedron and so on), we can write:


which can be shifted in (i.e. and ) to cover all the necessary triads. Note that other interactions don’t correspond to grid points and will be dropped. It is interesting to note that this does not lead to leaking of conserved quantites.

Now consider , , for the first equation:

Where the coefficients of and can be made to vanish by choosing , while in order to make the coefficient of vanish, we need to choose the scaling of the radius of the icosahedron with respect to as . This way we can satisfy the condition of the triad. The icosahedron that is constructed this way is actually nothing but the dual icosahedron of the inner dodecahedron with the radius . While these two can be thought of as being on separate shells in -space, since their radii are different. They could also be thought of sampling a single shell together in the form of a dodecahedron-icosahedron compound (the shell boundary in this case could be thought to be in between the two consecutive compounds).

: : : : : :
0 4 6 5 3 8
9 7 5 7
11 8 6 4
1 5 6 6 1 8
7 9 4 9
10 8 6 5
2 1 6 7 2 9
11 9 5 10
8 10 6 1
3 2 6 8 3 10
9 11 1 11
7 10 6 2
4 3 6 9 2 7
8 11 4 11
10 7 6 3
Table 3: Two icosahedral vertices, from the neighbouring shells (i.e. and ) that form a perfect triad with the dodecahedral vertex at shell .

Of course one can rotate the triangle around the primary vector to obtain another interacting pair. For instance for the node , the condition of the triad will be satisfied by the pairs . Since each point of the icosahedron is equivalent, we can compute the interacting pairs of dodecahedral vertices for each vertex of the icosahedron using the same algorithm. The node-pair connections obtained this way are given in tables 2 and 3.

Now consider (5), and choose , , . This gives:

which is automatically satisfied by the earlier choice and . Note again that the above condition for is also satisfied by the pairs: . Similarly as the case where the icosahedron was in the middle, we can find the rest of the interacting pairs of icosahedral vertices for each vertex of the dodecahedron by rotating the mesh.

Figure 2: Pairs of nodes interacting with the node number . (i.e. node on the shell ) where the first shell is an icosahedron, which is shown here as an example.
Figure 3: All the k-space triads corresponding to an icosahedron squeezed in between two dodecahedra.
Figure 4: All the k-space triads corresponding to a dodecahedron squeezed in between two icosahedra.

Since exchanging , another interaction is obtained, we will consider this explicitly by symmetrizing the equations as:



This way we can go over each node-pair connection once, without paying attention to the sign, and all possible interactions will be covered. Defining as the flattened node number (e.g. , if the first shell is an icosahedron), instead of the shell number as before.


where the sum is computed over the interacting pairs of a node (e.g. is the list of the pairs of nodes shown in figure 2). These connections can be obtained using the tables 2 and 3 and the flattening rule (and then ), where is the number of vertices of the first shell, and can be thought of as a regular network model (see figure 2). Note that, in (7) the interaction matrix is also flattened the same way as the vector .

ii.1 Further simplifications

As discussed earlier, in order to reduce the degrees of freedom of the nested polyhedra model of turbulence, one may consider only half of each polyhedra (for instance the hemisphere), and obtain the other half using the relation . In order to achieve this, in practice one has to keep in mind wheter a node in an interaction is conjugated or not. (i.e. in tables 2 and 3 if the node number falls into the upper hemisphere it would be conjugated, if it falls into the lower hemisphere it would not be conjugated in eqn. 7). This can be done by keeping a “conjugated flag” for each interaction between nodes. In addition to decreasing the number of degrees of freedom to half, this approach has the advantage of automatically imposing the reality condition of the velocity field as a function of space, which is important and is not exact in the more straightforward formulation.

Another more interesting simplification is to consider the helical decomposition, which allows the reduction of the velocity 3-vector to two scalars representing right handed and left handed helicities with respect to the wavenumber. This is possible because of the fact that the velocity field is divergence free in the Navier-Stokes equation, and therefore its projection onto the wavenumber has to vanish. This formulation allows us to reduce the number of degrees of freedom to 2/3 times the initial number and it guarantees that the velocity field remains divergence free, in contrast to the straightforward method, which does not guarantee this numerically.

Together these two simplifications would permit a reduction of the number of degrees of freedom to 1/3 times the initial number, and guarantee a real and divergence free velocity field as function of space.

Note that a simple spherical representation of the velocity field in space (i.e. ) also reduces the velocity field to two dimensions since the “radial” component vanihses (i.e. ). However the simplification of the interaction coefficients as well as the direct physical interpretation in terms of helicity are lost in this representation. Nonetheless, this representation allows us to see that the Fourier transformed velocity field is everywhere tangential to the spherical shells in k-space.

Further reduction can be achieved by associating one kind of helicity (i.e. Right), with one type of polyhedron (i.e. dodecahedron) and the other kind of helicity (i.e. Left) with the other type of polyhedron (i.e. icosahedron). This reduces the number of degrees of freedom, further by half. It also assigns a physical sense to the different types of polyhedra in the model. The reduced model obtained this way can be written as:



and if :even, and if :odd, with:

This gives

for even , and

for odd . It is interesting to note that these two equations have the same form (apart from the additional resolution) as the model discussed in Ref. De Pietro et al. (2015). In particular the above form corresponds to the model SM1 as discussed in that paper.

ii.2 Conservation Laws

Energy conservation can be shown by considering the energy of a single triad (e.g. , and ):

by using the form of and the facts that and . The total energy can then be written as a sum of the energy over triads. Since each closed triad conserves energy, any discretization of the Fourier space using a reduced set of “triads” automatically respects energy conservation. In fact this should be true for all the conservation laws of the system even without an explicit knowledge of the conservation laws. This is important, as we will see consecutively, since it provides a “derivation” of shell models and their generalizations without a detailed knowledge of the conservation laws. A similar effort was discussed earliler for 2D turbulence Gürcan et al. (2016).

ii.3 Connection to Shell Models

When the sum over different values are written explicitly, the model takes the form:


regardless of wheter the th shell is an icosahedron or a dodecahedron. Note that and sum is computed over pairs of interacting nodes of the consecutive shells as given in tables 2 and 3. The basic form of the equation (9) is consistent with the Gledzer-Ohkitani-Yamada (GOY) modelOhkitani and Yamada (1989). In fact one can arrive at a form very similar to the GOY model by arranging a certain (rather particular) choice of phases and signs of helicities for each node. Taking

and imposing the resulting coefficients to be independent of (see below for a discussion of this), we get:




The fact that (10) loses its dependence on (as the system is summed over and ) means that the , and as defined above should be identical for all of a polyhedron.

Assuming that the phases repeat after each shells (i.e. ), gives independent nodes and having independent vector components for velocity each, we have independent phases. The idea that the coefficients of (10) be independent of can be written as:

for any . In total this would give equations (here is the number of coefficients per node and is the number of nodes. Note that number of components does not enter, since the coefficients , and are already summed over , and ). However the equations for , and involves the phases , and , which are the same as , and respectively (due to the assumption of 4-fold periodicity), resulting in some of the same equations as before. If we assume that the the shell is an icosahedron, the number of repeated equations are which reduces the number of independent equations to . If the th shell is a dodecahedron on the other hand, the number of repeated equations are , which results in independent equations. This means that we can actually pick the independent phases in such a way that the or independent equations that guarantees a GOY-like model, are satisfied, and we would still have or undetermined phases, which could be taken for example as the phases of the independent components of st polyhedron (i.e. ).

Notice that this analytical excercise should qualify as an actual, rigorous derivation of the GOY model, starting from the nested polyhedra model, assuming isotropy and imposing some particular phase relations. Since the nested-polyhedra model comes from a systematic reduction of the triads in a self similar way, the derivation provides a rigorous path from the initial field equations to the shell model. Of course both the coefficients and the shell spacing are not free parameters as they are imposed by the constraints of self similarity of the nested polyhedra model.

ii.4 Velocity field as function of space

The three dimensional velocity field in real space implied by a -space discretization using a set of vertices (assuming the list of vertices contain their reflections):


where is the unit vector in the th direction (typically , and directions). Now if we consider a single unit icosahedron (i.e. of radius ), we can write the velocity field as:


which is defined by the 12 complex coefficients . These coefficients correspond to weights of right and left handed helicities in 6 different directions that are defined by the icosahedron. The same can be done for a unit dodecahedron:

In order to see the real space structure of such a flow, we have considered an icosahedron dodecahedron compound with randomly picked values for the coefficients , and constructed the flow using (12) and (II.4), and plotted the result in Fig 5.

Figure 5: The three dimensional velocity field corresponding to a single icosahedron-dodacoheron compound in Fourier space, with random phases for the Fourier coefficients.

ii.5 Transition to two dimensions

There are various limiting cases, such as rotating turbulence, MHD with a strong mean magnetic field, turbulence in a thin film etc. where the turbulent dynamics become two dimensional. However there are some peculiar aspects of two dimensional turbulence and other peculiar aspects of 2D shell models (or logarithmically discretized models). Two dimensional turbulence is generally believed to result in a dual cascade, where the enstrophy cascades in the forward and energy cascades in the backward directions. However, logarithmic discretization of 2D space leads to a numerical inconvenience that the equipartition of energy between shells dominate over the inverse energy cascade. Thus, 2D shell models can not describe the inverse energy cascade (unless there are additional terms in the equation that keeps the system away from this tendency to equipartition). Interestingly this is less of an issue in three dimensional models. When the model is isotropic, however, it is impossible to study the transition from 3D to 2D, since all the directions are, by construction, the same. However, a nested polyhedra model can in principle be scaled in one direction (say the direction) and spherical shells can be flattened to pancake-like forms and therefore the transition to 2D can be studied.

The limiting form of 2D turbulence can be obtained by simply setting in the nested-polyhedra structure of the grid. However an interesting phenomenon takes place in this limit. The projection of the nodes to of the th icosahedron (which gives a regular pentagon) is the same as those of the nodes to of the th and to of the th dodecahedra. Similarly the nodes to of the th icosahedron is the the same as the nodes to of the th and to of the th dodecahedra. This means that the th “shell” (really a circular ring or annulus) in a 2D model corresponds to at least three different “shells” , and in a 3D model. As they fall exactly on the same points on the 2D space and they are indistiguishable and lead to degeneracy.

Note that is equivalent to the form given in Table 1 for . The smaller pentagon which results from the projection of the inner points of a dodacohedron on the plane has a circumscribed circle of radius:

The larger pentagon which comes from the projection of the outer points of the dodecahedron and that which comes from those of its dual icosahedron has a circumscribed circle of radius:

This gives a scaling factor of between two consecutive circles. In other words, the wavenumbers of the shells can now be defined with:

where and .

The equations on the shells can be obtained by projection also. The helicity directions become:

which allows us to write seperate equations for and for each scale simply by projecting the corresponding three dimensional equations to two dimensions. This means that now, we need to solve equations for each scale, or noting that half of these points are simply reflections of the rest of the points, equations for each shell.

The 4-fold degeneracy which appear in going from 3D to 2D is remarkable, since in the standard 2D formulation one would consider only vorticity (not helicities of both signs) and only one kind of polygon as a representation of the 2D -spaceGürcan et al. (2016). The effort discussed in this work for going to a 2D formulation may be worthwhile however due to the issue of unphysical shell equipartition overwhelming the inverse cascade in 2D logarithmic discretization. A model which has the same topological structure as the 3D one, may actually be able to reproduce the inverse cascade spectrum without requiring the expensive hierarchical tree approachAurell et al. (1994, 1997). However, this particular issue is not the focus of the current article, and therefore the concentrated effort neccessary for establishing the implications of such a model is left to a future publication.

Iii Numerical Results

The model, solves for all three components of the velocity field, with an interaction matrix representing the Navier-Stokes equation. In order to implement it, an object oriented approach can be used, where each node has a list of its connecting pairs, as can be inferred from Tables 2 and 3 (as seen in ure 2), and so that a sum over these pairs can be computed rapidly. The resulting model is a stiff set of ordinary differential equations (ODEs) on an exponantially coarse grid, somewhat similar to the 2D model discussed in Ref. Gürcan et al. (2016).

Figure 6: The resulting instantaneous 3D -spectrum at on the left and averaged over the range on the right for the run with and . Here we used a spherical log-log representation, where is plotted with respect to . The resulting spectrum is consistent with the Kolmogorov spectrum as shown in figure 7.
Figure 7: Log-log plot of the spectral energy density as a function of . The blue (if in color) curve that spans from to is a run with and averaged in the range , whereas the red curve (if in color) that spans all the way up to is a run at the limit of currently available resolution with and averaged in the range .
Figure 8: Log-log plot of spectral energy density as a function of (see figure 7 for definition) for . The blue (if in color) curve is with , while the red curve corresponds to , both averaged in the range .

We have implemented the nested polyhedra model in Python with no parallelization, which is distributed as an open source solver at http://github.com/gurcani/nestp3d. It solves (where is the number of -space shells) complex system of equations, for the three components of the velocity field for each half-polyhedra as discussed in section II.1 above. We performed several runs ranging from to .

The three dimensional spectra for the highest resolution case (i.e. ) are shown in figure 6. As expected, with no source of anisotropy, the resulting spectrum remains perfectly isotropic. Nonetheless, it shows the capability of the model to resolve three dimensional anisotropy accross many decades. The results of the two limiting cases with and with are shown in figure 7, where the case (which takes several weeks to compute on a pc workstation) covers almost 6 decades in -space and has almost no need of a dissipative range to reach steady state. This seems to be a feature of the model, which may facilitate development of large eddy simulation (LES) versions of itself. As can be seen in figure 7, the truncation from 60 to 30 shells while varying accordingly, has almost no effect on part of the spectrum that is resolved by the run. In order to study the effect of the existence of a dissipation range, we have also varied the viscosity coefficient . The results for , and are considered for , and shown in figure 8. As expected, increasing causes a dissipative range to appear, but it doesn’t change neither the saturation level, nor the slope of the spectrum.

Finally, we have performed some preliminary studies of intermittency using this model, whose results are shown in figure 9. Curiously, the model shows no sign of intermittency as it follows the scaling in the inertial range (i.e. where denotes average over time). On one hand, this is surprising, since for instance the GOY model, which can be derived from the nested polyhedra model by additional assumptions about the way the phases are organised, displays dynamical multiscaling and intermittencyPisarenko et al. (1993); Benzi et al. (1996), while on the other hand, it is natural, since the model is perfectly self-similar and made of a single three dimensional “fractal”. In order to further verify that our method of obtaining intermittency is valid, we double-checked our results by repeating the exercise of Ref. Pisarenko et al. (1993) with standard GOY model as well as a model with alternating shells (i.e. for even and for odd n), and found that while their results are rather robust for shell models, our result of “no intermittency” is similarly robust for our model. We repeated this exercise many times, and remain puzzled by these results, since it seems to us paradoxal that the reduction of a reduction can recover a property of the original system that was lost in the first level of reduction. One possible explanation is the fact that shell models do not rely on a single type of triad, but is a result of a net transfer of conserved quantity from shell to shell, where many similar triads may play a role. This sense of “net flux” computed over a set of similar triads may be the reason why these models can somehow have intermittent dynamics, while our model, which relies on a single triad family accross many scales, does not.

Figure 9: The index of the power law for the structure function of order [i.e. ] as a function of , displaying a clear scaling. The case that is shown here corresponds to , with , which was averaged only from to over shells to . However another case with was integrated up to so that an average could be computed from to over shells to , and it gives virtually the same result.

Iv Conclusion

A dodecahedron-icosahedron compound discretization of the Navier-Stokes equation, which is proposed in this paper, gives the expected wave-number spectrum of Kolmogorov and can possibly be used to study the spectra in three dimensional fluid turbulence. The advantage of a formulation based on logarithmic scaling with a constant number of nodes per scale is that a very large inertial range (i.e. very high Reynolds numbers) can be considered with a much lower number of degrees of freedom. Further simplifications of the model were proposed using the symmetry in -space and the helical decomposition. It was shown that, when isotropy and a particular relation between phases are imposed, the nested polyhedra model reduces to a GOY model. This provides an actual “systematic derivation” of the latter, since the nested polyhedra model itself was obtained by a systematic reduction/decimation of the continuous wave-number space to a finite set of self-similar triads. The straightforward issue of reconstruction of the velocity field from the nested polyhedra description is also discussed in order to demonstrate the richness of the types of flows that the highly reduced model can sustain. Transition to two dimensions is briefly discussed as a reference for future work. It is noteworthy that one can derive a model for describing two dimensional turbulence, which has the same topological network structure of the three dimensional one. Preliminary studies show that the model as described in this paper, show no sign of intermittence since it follows the scaling in the inertial range. This is curious, since the GOY model, which can be derived from the nested polyhedra model by additional assumptions about the way the phases are organised, displays dynamical multiscaling and intermittencyPisarenko et al. (1993); Benzi et al. (1996). On the other hand, it is natural, since the model is perfectly self-similar and made of a single three dimensional “fractal”.

Note that the icosahedron and the dodecahedron (i.e. its dual), together forms a compound polyhedron called “dodecahedron-icosahedron compound” (i.e. Wenninger model index 47 Wenninger (1974)). The nested polyhedra model that we introduced here can be seen as a discretization of the -space using these objects. A faceting of this compound polyhedron is a Catalan solid called “rhombic triacontahedron”, which is also the dual of an Archimedean solid called “icosidodecahedron”. One could use this connection to “refine” the k-space, that is divided in self-similar nested polyhedra, by introducing an icosidodecahedron between two dodecahedron-icosahedron compounds that constitute the nested polyhedra model. The resulting model would use nested “icosidodecahedron-rhombic triacontahedron compounds” as the building blocks for the nested polyhedra model. It would be interesing to determine, using wavenumber triad matching conditions, if there exist more of these complex polyhedra, which can be used to develop more and more complex nested polyhedra models. One could then speculate that if one would initialize the turbulence on an icosahedron in -space, the full system of Navier stokes equations (with no truncation) would result in the turbulence energy going from one type of complex polyhedron to another, without ever leaving the space of compound polyhedra. This is remarkable as it would transform the infinite system to a set of nested polyhedra. Incidentally, the use of multiple types of polyhedra, would also introduce multifractality naturally.

When the approach detailed in this study is applied to a system that supports waves (i.e. linearly), the resulting network model can be suitable for the study of various different phenomena including synchronizationKuramoto (1984), small world or scale freedomWang and Chen (2003) since it can be thought of as a complex network of coupled “oscillators” Strogatz (2001). The structure of the model is somewhat similar to those describing food web networks Pimm et al. (1991) or supply chains, which implies a complex adaptive system. Such analogies are used regularly in plasma turbulence, especially in the presence of large scale flow structures called zonal flowsDiamond et al. (2005); Gürcan and Diamond (2015). It would be interesting to see if a similar analogy can be extended to fluid turbulence and to what extent a rigorous mathematical relation can be established between such reduced models of turbulence and analogous ones from biology and other fields.

The author would like to thank P. Morel, P. H. Diamond, R. Grappin and attendants and the organizers of the Festival de Théorie, Aix en Provence in 2015.


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