A closure theory for the split energy-helicity cascades in homogeneous isotropic homochiral turbulence

# A closure theory for the split energy-helicity cascades in homogeneous isotropic homochiral turbulence

## Abstract

We study the energy transfer properties of three dimensional homogeneous and isotropic turbulence where the non-linear transfer is altered in a way that helicity is made sign-definite, say positive. In this framework, known as homochiral turbulence, an adapted eddy-damped quasi-normal Markovian (EDQNM) closure is derived to analyze the dynamics at very large Reynolds numbers, of order based on the Taylor scale. In agreement with previous findings, an inverse cascade of energy with a kinetic energy spectrum like is found for scales larger than the forcing one. Conjointly, a forward cascade of helicity towards larger wavenumbers is obtained, where the kinetic energy spectrum scales like . By following the evolution of the closed spectral equations for a very long time and over a huge extensions of scales, we found the developing of a non monotonic shape for the front of the inverse energy flux. The very long time evolution of the kinetic energy and integral scale in both the forced and unforced cases is analyzed also.

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## I Introduction

Since the discovery that helicity is an inviscid invariant of the three-dimensional Navier-Stokes equations Moffatt (1969), the possibility of inverse cascades in homogeneous isotropic turbulence (HIT) without mirror symmetry has been greatly investigated: indeed, two-dimensional turbulence possesses as well two inviscid invariants, kinetic energy and enstrophy, and in this configuration, energy cascades towards large scales (Boffetta and Ecke, 2012).

In 3D, the first theoretical considerations date back to the study of Brissaud and coworkers (Brissaud et al., 1973) where two different scenarios were proposed: (i) a joint direct cascade of kinetic energy and helicity where , and , where and are the kinetic and helical spectra, and and the kinetic energy and helicity dissipation rates; (ii) a split cascade with a direct transfer of helicity combined with an inverse cascade of kinetic energy. In fact, the second scenario was proven to be impossible by André and Lesieur (1977) within the eddy-damped quasi-normal Markovian (EDQNM) approximation (Orszag, 1970; Lesieur, 2008; Sagaut and Cambon, 2008). In subsequent papers like Borue and Orszag (1997); Chen et al. (2003) and more recently Briard and Gomez (2017), the joint direct cascade of helicity was observed in HIT without mirror symmetry, with strictly zero inverse energy transfer. This stems from the fact that helicity, the scalar product of velocity and vorticity, is not positive-definite, unlike kinetic energy and enstrophy.

Strangely enough, inverse cascades with EDQNM were rarely investigated in the past and mainly for two configurations only: the inverse cascade of kinetic energy in 2D turbulence (Pouquet et al., 1975), and the inverse cascade of magnetic helicity in isotropic magnetohydrodynamics turbulence (Pouquet et al., 1976). Consequently, it appears interesting in terms of modelling to check that a sophisticated model such as EDQNM can handle complex and particular configurations when only specific triadic interactions are kept. In the present work, we aim at further studying these features when helicity is made sign-definite by deriving a new EDQNM model. We show for the first time that a split cascade scenario develops and that it can be studied for very large Reynolds numbers and for very long times. In what follows, the adapted EDQNM closure for homochiral turbulence is derived, and then numerical results are presented for cases with and without forcing.

## Ii EDQNM closure for homochiral turbulence

The spectral counterpart of the Navier-Stokes equation for the fluctuating velocity reads in homogeneous isotropic turbulence

 (∂∂t+νk2)^uj(k)=−iPjmn(k)∫k=p+qum(p)un(q)d3p, (1)

where is the kinematic viscosity, , the operator , with the projector and , and are respectively the wavenumber and wavevector, and denotes the Fourier transform. In isotropic turbulence without reflexion symmetry, the spectral second-order velocity-velocity correlation , where denotes the complex conjugate and an ensemble average, can be decomposed as

 ^Rij(k,t)=E(k,t)Pij(k)+iϵijkαkH(k,t)k, (2)

with the Levi-Civita permutation tensor, and where and are respectively the kinetic energy and kinetic helicity densities. Such a decomposition involving the kinetic energy density and helical density was previously used in Cambon and Jacquin (1989); Borue and Orszag (1997); Chen et al. (2003). More recently Briard and Gomez (2017), this decomposition was applied to investigate with EDQNM the large Reynolds numbers dynamics of the kinetic energy and helical spectra defined as

 E(k,t)=∫SkE(k,t)d2k,H(k,t)=∫SkH(k,t)d2k, (3)

where is a sphere of radius . The total helicity, which is the scalar product of velocity and vorticity, is then obtained by . In what follows, we analyze within an adapted EDQNM approximation at large Reynolds numbers, the configuration proposed in Biferale et al. (2012, 2013), namely the homochiral decimated Navier-Stokes.

To select only helical modes of identical sign the classical Lin equation for (von Kármán and Lin, 1949) and its non-linear transfer of HIT cannot be used anymore. One needs a new closure, adapted to homochiral turbulence where helicity is made sign-definite. First, the spectral fluctuating velocity is decomposed in positive and negative modes using the helical decomposition (Cambon and Jacquin, 1989; Waleffe, 1992; Sagaut and Cambon, 2008) so that

 ^u(k,t)=u+(k,t)N(k)+u−(k,t)N∗(k), (4)

where are the helical modes, which verify , , , and , so that the spectral fluctuating vorticity reads

 ^ω(k,t)=k[u+(k,t)N(k,t)−u−(k,t)N∗(k,t)]. (5)

Considering only the positive helical modes we get for the kinetic energy spectrum:

 E+(k,t)=∫SkE+(k,t)d2k=∫Sk⟨u∗+(k,t)u+(k,t)⟩d2k, (6)

and the helical spectrum is thus simply given by . The evolution equation of is obtained by contracting (1) with which gives

 (∂∂t+νk2)u+(k,t)=−12i(kiN∗j(k)+kjN∗i(k))∫k=p+qu+(p)u+(q)Ni(p)Nj(q)d3p (7)

The evolution equation of the kinetic energy density is then

 (∂∂t+2νk2)E+(k,t)=T+(k,t). (8)

It is very important to stress that the non-linear terms of the Navier-Stokes equations conserve both energy and helicity triad-by-triad, and therefore the same is true for the dNS. In other words, the non-linear transfer is conservative, takes into account triadic interactions of positive helical modes, and reads after some algebra

 T+(k,t)=∫R[(kiN∗j(k)+kjN∗i(k))S+(k,p,t)Ni(p)Nj(q)]d3k, (9)

where is the spectral triple velocity correlation

 S+(k,p,t)δ(k+p+q)=i⟨u+(k,t)u+(p,t)u+(q,t)⟩. (10)

Then, after some technical manipulations typical of the EDQNM approximation (Orszag, 1970; Lesieur, 2008), one obtains the decimated Lin equation of the (positive) kinetic energy spectrum, namely

 (∂∂t+2νk2)E+(k,t)=SE+(k,t), (11)

where the spherically-averaged non-linear transfer is

 SE+(k,t) =∫SkT+(k,t)d2k=16π2∫Δkk2pqθkpq(1+z)E′′+[pE+(y−z)(y+z−2x) +(1−x−2yz)(k(1+y)E′+−p(1+x)E+)]dpdq, (12)

with the domain where , and are the lengths of the sides of the triangle formed by the triad , and where , and are the cosines of the angles formed by and , and , and and respectively. For the sake of clarity , , and . is the characteristic time of the triple correlations where the eddy-damping term is given by . Here we will start by taking for consistency with previous studies (Pouquet et al., 1976; Lesieur and Ossia, 2000; Briard et al., 2015; Briard and Gomez, 2017), the impact of changing is discussed later on. The previous equations of the EDQNM closure for the sign-definite helicity originating from the decimated Navier-Stokes equation are the main theoretical contributions of this work.

For consistency and clarity, it is recalled that in isotropic helical turbulence (without decimation), the kinetic energy spectrum evolves according to , where is the usual isotropic spherically-averaged non-linear transfer, namely

 SE(k,t)=16π2∫Δkθkpqk2p2q(xy+z3)E′′(E′−E)dpdq, (13)

where , , and . The eddy-damping term is given by the complete energy spectrum , unlike in the decimated version where it is given by . The expression (12) is more complicated than (13) in the sense that it is less compact and symmetric: this is due to the further contraction with the helical modes to select only specific triadic interactions.

In what follows, we wish to recover two features of homochiral turbulence: (i) the direct helicity cascade of Biferale et al. (2013) where the kinetic energy spectrum scales as and (ii) the inverse cascade of energy in . For the EDQNM simulations, the wavenumber space is discretized using a logarithmic mesh for , where is the total number of modes and , being the number of points per decade. It has been checked that increasing , for instance up to , does not modify the slopes of the spectra, nor the asymptotic values of the integrated quantities (like kinetic energy) more than . This mesh spans from to , where is the integral wavenumber and the Kolmogorov wavenumber. The initial kinetic energy spectrum is given by , where the infrared slope is , corresponding to Saffman turbulence. Simulations not presented here revealed that the results of this work are independent of the infrared slope, in particular the findings are the same for Batchelor turbulence (). The initial is normalized so that is unit at .

## Iii Results at large Reynolds numbers

First, homogeneous homochiral turbulence without any forcing is considered. The scaling is recovered in figure a at large Reynolds numbers, which corresponds to the forward helicity cascade (Brissaud et al., 1973): this inertial range grows with time and spans more than 4 decades at the largest Reynolds number. The noteworthy feature is that even without forcing, if large Reynolds numbers are reached, strong inverse transfers mechanisms occur since the peak of , given at the integral wavenumber , increases with time, unlike in fully isotropic decaying turbulence. A further evidence for intense inverse non-local transfers is that, even though the infrared scaling of the kinetic energy spectrum is initially , it rapidly becomes , which is a signature of strong back transfers (Eyink and Thomson, 2000; Lesieur and Ossia, 2000; Meldi and Sagaut, 2012; Briard et al., 2015). It follows from that the helical spectrum scales in .

In addition, one can remark from figure a that the peak of the kinetic energy spectrum seems to evolve as with time. This has nothing to do with inertial range scaling considerations, since the inertial range scaling clearly remains . This can be briefly justified as follows. Since we have mainly a direct helicity cascade, one can roughly assumes that , consistently with arguments given in Brissaud et al. (1973). Thus, it follows that the kinetic energy , whose evolution is given by , remains constant, which is well verified numerically (see later figure b). Then, using the definition of the kinetic energy, one can reasonably assume that it is mainly given by the integral scale:

 ⟨u2+⟩=∫∞0E+(k,t)dk≃∫kL0Epeak+(t)dk=kLEpeak+(t). (14)

The kinetic energy being constant, one obtains that the time evolution of the kinetic energy spectrum peak is given by . Note that for dimensional reasons the integral scale evolves like in the unforced case. It will be shown later for the forced case that the time evolution of and are quite different. In figure b we show the evolution of the two fluxes, and , where and . As said earlier, both energy and helical transfers with homochiral triadic interactions are conservative: indeed, one has . Furthermore, there is a direct cascade of helicity since is mostly positive in the inertial range and spans more and more decades with time, which is consistent with the scaling of of figure a. In addition, figure b illustrates that there is an inverse cascade occurring on a small range around for .

To increase the scaling region of the inverse cascade, we add to the decimated Lin equation (11) a forcing term , to have the possibility to study the split cascade scenario with a well developed inverse energy transfer for asymptotically long times. The forcing term is given by

 F(k)=C1exp (−1(C2)2[ln(kkf)]2), (15)

with so that one has , and as proposed in Briard and Gomez (2017). We double checked that the following numerical results are independent of the forcing term by studying also the case with (not shown). The time evolution of the kinetic energy spectrum is shown in figure a, where the forcing term is in grey. It is clear that the system develops a split cascade on both sides of the forcing term. Indeed, as time increases, a range grows at large scales, similar to the one obtained in Biferale et al. (2012), which is a strong evidence for the inverse cascade of kinetic energy. Whereas for wavenumbers larger than , the range is preserved, as in figure a without forcing. It is important to stress that this is the first time where the split energy-helicity simultaneous cascade is observed: this is notably due to the fact that by using EDQNM we can push the resolution on both sides to very large values. It is reasonable to conclude from figure a that the non-linear transfers which are at the origin of the inverse cascade are dominantly local in homochiral turbulence: indeed, because of the logarithmic discretization of the wavenumber space, elongated triads corresponding to non-local transfers cannot be taken into account with EDQNM (Lesieur, 2008). This statement does not mean that there are no non-local interactions: indeed, it has been argued that some inverse non-local transfers are responsible for the change of the infrared slope of from to in figure a (Métais and Lesieur, 1986).

Finally, some one-point statistics are presented in figure b, namely the kinetic energy and the integral scale for both the forced and unforced cases. For the unforced case, the constancy of and , discussed earlier, are recovered. For the forced configuration, the kinetic energy is expected to grow as (Pouquet et al., 1975). The linear dependence with time is recovered in figure b. Then, it follows from dimensional analysis that the integral scale should evolve like , which is also assessed in figure b.

## Iv Conclusions

In conclusion, we addressed a particular kind of helical turbulence, where only specific triadic interactions are kept, so that the helicity is made sign-definite. In this particular homochiral framework, the 3D turbulence possesses two sign-definite inviscid invariants, namely kinetic energy and helicity, like 2D turbulence with kinetic energy and enstrophy. The main objective of this study was to show that a spectral closure method, such as EDQNM, could recover the main findings of Biferale et al. (2012, 2013). To do so, an adapted EDQNM approximation was derived, taking into account only the particular interactions that make helicity sign-definite (positive here). The direct helicity cascade, where , is first obtained in unforced homochiral turbulence, where the kinetic energy flux . In such a configuration, the kinetic energy remains constant and the peak of evolves in with time. When a forcing term is added, in addition to the helicity forward cascade, an inverse energy cascade develops toward smaller wavenumbers, where the helicity flux . In this regime of forced homochiral turbulence, the kinetic energy evolves linearly with time , and the integral scale like .

Our work shows for the first time the possibility to have a stable split cascade in three dimensional turbulence under strong restriction of the helical interactions. It remains to be understood how much this phenomenology might be indeed observed in more realistic flow conï¬gurations, as in the presence of strong rotation or confinement, where also a transition from direct to inverse energy cascade is observed (Mininni and Pouquet, 2009; Celani et al., 2010; Xia et al., 2011; Godeferd and Moisy, 2015; Biferale et al., 2016). It is also interesting to study the inverse cascade regime in the presence of a large scale drag, in order to allow for the formation of a large scale stationary condensate (Chertkov et al., 2007). At difference from the two dimensional case, the condensate will necessarily have strong helical properties and be close to an ABC quasi-stationary solution of the three dimensional Navier-Stokes equations (Dombre et al., 1986; Moffatt, 2014; Sahoo and Biferale, 2015). Work in this direction will be reported elsewhere. Finally, it is important to stress that similar EDQNM studies can be extended to helical MHD (Linkmann et al., 2017).

The research leading to these results has received funding from the European Union’s Seventh Framework Programme (FP7/2007-2013) under grant agreement No. 339032.

### Footnotes

1. preprint: APS/123-QED

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