2.1 Craya equation

In this part, the evolution equation of the spectral two-point velocity correlation is obtained in the framework of HHT, without any mean-velocity gradient nor rotation. The evolution equation of the spectral fluctuating velocity $\hat{u} _{i}$ reads

which is the spectral counterpart of the Navier–Stokes equation, with $\unicode[STIX]{x1D708}$ the kinematic viscosity, $k_{i}$ the wavenumber, $\hat{p}$ the spectral fluctuating pressure, $\text{i}^{2}=-1$ and $\hat{()}$ denotes the Fourier transform. The pressure term is cancelled by projecting this equation onto the plane perpendicular to $\hat{u} _{i}$ , by virtue of the incompressibility condition $k_{i}\hat{u} _{i}=0$ . From this point, one needs to define the two-point second-order velocity correlation $\hat{\unicode[STIX]{x1D619}}_{ij}$ , also called the spectral Reynolds tensor

where $\boldsymbol{k}$ and $\boldsymbol{p}$ are wave vectors, $()^{\ast }$ is the complex conjugate, $\langle \cdot \rangle$ an ensemble average and $\unicode[STIX]{x1D619}_{ij}$ is the second-order velocity correlation in physical space

where $\boldsymbol{r}$ is the separation vector, and with in particular $\unicode[STIX]{x1D619}_{ii}(0)=\langle u_{i}u_{i}\rangle =2K$ . The evolution equation of $\hat{\unicode[STIX]{x1D619}}_{ij}$ , the so-called Craya equation, is

where $T_{ij}^{NL}$ is the total nonlinear and conservative transfer. This nonlinear transfer can be expressed as a function of $S_{ijn}$ , the three-point third-order velocity correlation

so that (Sagaut & Cambon Reference Sagaut and Cambon2008)

where $P_{imn}$ is the Kraichnan operator defined as

with $\unicode[STIX]{x1D6FC}_{i}=k_{i}/k$ . In the following § 2.2, the spectral Reynolds tensor is expressed as a function of the energy and helical densities ${\mathcal{E}}$ and ${\mathcal{H}}$ , in a procedure similar to what was done recently in the framework of homogeneous anisotropic turbulence (Mons et al. Reference Mons, Cambon and Sagaut2016). Then, in § 2.3, the nonlinear transfer $T_{ij}^{NL}$ is derived with a classical EDQNM procedure.

2.2 ${\mathcal{E}}$ – ${\mathcal{H}}$ decomposition

In HHT, the spectral Reynolds tensor is not symmetric and real anymore, since it contains an imaginary antisymmetric part (Cambon & Jacquin Reference Cambon and Jacquin1989) reflecting the breakdown of mirror symmetry. Consequently, $\hat{\unicode[STIX]{x1D619}}_{ij}$ can be written

where $\unicode[STIX]{x1D716}_{ijk}$ is the Levi-Civita permutation tensor. Similar decompositions were used by Borue & Orszag (Reference Borue and Orszag1997), Chen et al. (Reference Chen, Chen and Eyink2003). From this rather simple decomposition, one could extend to more complex cases containing polarization and directional anisotropies, such as helical-shear driven turbulence, whose equations are given in Cambon et al. (Reference Cambon, Danaila, Godeferd and Scott2013). The energy and helical densities ${\mathcal{E}}$ and ${\mathcal{H}}$ are linked to the spectral Reynolds tensor $\hat{\unicode[STIX]{x1D619}}_{ij}$ through

The helical density ${\mathcal{H}}$ can also be defined in terms of the spectral fluctuating velocity and vorticity

where $\unicode[STIX]{x1D619}_{ij}^{H}$ is the second-order velocity–vorticity correlation in physical space

with in particular $\unicode[STIX]{x1D619}_{ii}^{H}(0)=2K_{H}$ . The evolution equations of the kinetic energy and helical densities are then

Afterwards, to express $T_{{\mathcal{E}}}$ and $T_{{\mathcal{H}}}$ as a function of $T_{ij}^{NL}$ , one has to use the tensor

where $S_{ijn}$ is the three-point third-order velocity correlation defined in (2.5). This procedure, used recently in Briard et al. (Reference Briard, Gomez and Cambon2016), Mons et al. (Reference Mons, Cambon and Sagaut2016) in the framework of homogeneous anisotropic turbulence, permits us to express the total nonlinear transfer as

One obtains

and the latter expression was notably derived in Cambon & Jacquin (Reference Cambon and Jacquin1989).

The kinetic energy and helical spectra $E(k,t)$ and $H(k,t)$ are defined as the spherical average of the kinetic energy and helical densities

and in HHT, the spherical integration amounts to a multiplication by $4\unicode[STIX]{x03C0}k^{2}$ . The evolution equations of both $E$ and $H$ are given in § 2.3 along with the spherical average of $T_{{\mathcal{E}}}$ and $T_{{\mathcal{H}}}$ . The kinetic energy and its dissipation rate are computed as

Similarly, one has for the helicity and its dissipation rate

The kinetic energy and helicity evolution equations in freely decaying turbulence are

and the decay of these quantities for high Reynolds numbers is investigated in § 4. We define as well the kinetic and helical integral scales

Finally, from the definition of the vorticity itself, it can be shown that the helical spectrum $H(k,t)$ must satisfy the following condition (Kraichnan Reference Kraichnan1973)

with $K_{H}=\langle u_{i}\unicode[STIX]{x1D714}_{i}\rangle /2$ . At this level, one must point out that unlike kinetic energy which is always positive, helicity is not positive definite and consequently can be negative, so that $H(k,t)$ can take negative values.

2.3 EDQNM closure and nonlinear transfers

The EDQNM approximation is now used to close the nonlinear transfers $T_{{\mathcal{E}}}$ and $T_{{\mathcal{H}}}$ that call into play the triple correlation $S_{ijn}$ . The complete procedure is detailed and discussed in numerous references (Pouquet et al. Reference Pouquet, Frisch and Léorat1976; André & Lesieur Reference André and Lesieur1977; Lesieur Reference Lesieur2008; Sagaut & Cambon Reference Sagaut and Cambon2008) and has been used recently in homogeneous isotropic turbulence (Briard et al. Reference Briard, Gomez, Sagaut and Memari2015). Consequently, only a brief description is presented here.

The EDQNM model was firstly introduced by Orszag (Reference Orszag1970), Leith (Reference Leith1971), Orszag (Reference Orszag, Balian and Peube1977) for hydrodynamic turbulence. This approximation is used here to solve the equations of dynamics in order to obtain the time evolution of the kinetic and helical spectra. The EDQNM model is a quasi-normal closure: this approximation assumes that the fourth-order moments of the velocity field appearing in the evolution equation of the third-order correlation are related to the second-order moments as if the field was Gaussian. The realizability of the model is then enforced by adding an eddy-damping term

that reflects departure from Gaussianity, where the Kolmogorov constant $C_{E}\simeq 1.30$ with $A=0.355$ . The Markovianization leads to define the relaxation time of the third-order correlations in the EDQNM model as

When it comes to the study of helicity, an additional assumption is made, as in André & Lesieur (Reference André and Lesieur1977): the characteristic time of the third-order correlations $\unicode[STIX]{x1D703}_{kpq}$ is chosen to be the same for the kinetic and helical fields. It is shown in § 3.1 that such an hypothesis is consistent with joint cascades for the kinetic and helical spectra, and can be a posteriori justified with physical arguments. Beyond these physical justifications, it seems consistent to choose $\unicode[STIX]{x1D703}_{kpq}$ for both the kinetic and helical fields since the closure comes from the same evolution equation, which is the one of the third-order correlation defined in (2.5).

Thanks to the EDQNM approximation, one can compute explicitly the spherical average of the nonlinear kinetic and helical transfers, as done in André & Lesieur (Reference André and Lesieur1977), which yields

with in detail

with $\unicode[STIX]{x1D6E5}_{k}$ the domain where $k$ , $p$ and $q$ are the lengths of the sides of the triangle formed by the triad $\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q}=\mathbf{0}$ , and where $x$ , $y$ and $z$ are the cosines of the angles formed by $\boldsymbol{p}$ and $\boldsymbol{q}$ , $\boldsymbol{q}$ and $\boldsymbol{k}$ , and $\boldsymbol{k}$ and $\boldsymbol{p}$ respectively. For the sake of clarity ${\mathcal{E}}=E(k,t)/4\unicode[STIX]{x03C0}k^{2}$ , ${\mathcal{E}}^{\prime }=E(p,t)/4\unicode[STIX]{x03C0}p^{2}$ and ${\mathcal{E}}^{\prime \prime }=E(q,t)/4\unicode[STIX]{x03C0}q^{2}$ , and ${\mathcal{H}}=H(k,t)/4\unicode[STIX]{x03C0}k^{2}$ , ${\mathcal{H}}^{\prime }=H(p,t)/4\unicode[STIX]{x03C0}p^{2}$ and ${\mathcal{H}}^{\prime \prime }=H(q,t)/4\unicode[STIX]{x03C0}q^{2}$ . These four nonlinear transfers $S^{(iso)}$ , $S^{(hel)}$ , $S_{H}^{(dir)}$ , and $S_{H}^{(inv)}$ are conservative (with zero integral over the whole wavenumber space) and have been obtained previously (André & Lesieur Reference André and Lesieur1977). Here is some additional information: $S^{(iso)}$ is the classical kinetic energy nonlinear transfer appearing in HIT. Moreover, it is clear from (2.25) that helicity directly impacts the nonlinear transfers of $E(k,t)$ through a purely helical transfer $S^{(hel)}$ . One can further remark that $S_{H}^{(dir)}$ has a form very similar to the isotropic nonlinear transfer of a passive scalar field (Briard et al. Reference Briard, Gomez and Cambon2016). In addition to the original work of André & Lesieur (Reference André and Lesieur1977), it will be shown numerically in § 3.1 that both $S^{(iso)}$ and $S_{H}^{(dir)}$ correspond to direct transfers from large to small scales, whereas $S^{(hel)}$ and $S_{H}^{(inv)}$ are inverse transfers.

Finally, with (2.25) and (2.26), it is now possible to write the Lin evolution equations for $E(k,t)$ and $H(k,t)$ in homogeneous helical turbulence

In the next part, some details are provided about the numerical resolution of these two equations.

2.4 Numerical set-up and initial conditions

The time evolution of the kinetic and helical spectra $E(k,t)$ and $H(k,t)$ is obtained by solving two coupled integro-differential equations using a third-order Runge–Kutta scheme with implicit treatment of viscous terms. The wavenumber space is discretized using a logarithmic mesh $k_{i+1}=rk_{i}$ for $i=1,\ldots ,n$ , where $n$ is the total number of modes and $r=10^{1/f}$ , $f=17$ being the number of points per decade. This mesh spans from $k_{min}=10^{-6}k_{L}$ to $k_{max}=10k_{\unicode[STIX]{x1D702}}$ , where $k_{L}=1/L$ is the integral wavenumber and $k_{\unicode[STIX]{x1D702}}=(\unicode[STIX]{x1D716}/\unicode[STIX]{x1D708}^{3})^{1/4}$ is the Kolmogorov wavenumber (the possibility of a different viscous cutoff wavenumber for the helical spectrum will be discussed as well). If not mentioned otherwise, the initial Reynolds number based on the Taylor microscale is $Re_{\unicode[STIX]{x1D706}}(0)\simeq 5\times 10^{4}$ : this allows us to investigate the decay of helicity and kinetic energy in regimes that cannot be reached yet by DNS, but that nevertheless involve typical Reynolds numbers of atmospheric flows. The characteristic time used for normalization is the eddy turnover time $\unicode[STIX]{x1D70F}_{0}=K(0)/\unicode[STIX]{x1D716}(0)$ .

Figure 1.

Early decay of the kinetic energy $K(t)$ with and without the additional presence of helicity, for Batchelor turbulence ( $\unicode[STIX]{x1D70E}=4$ ). The results are similar for Saffman turbulence ( $\unicode[STIX]{x1D70E}=2$ ).

The choice of initial conditions is important, and for consistency, the same as in André & Lesieur (Reference André and Lesieur1977) are used

with $\unicode[STIX]{x1D70E}$ the large-scale spectral exponent of the kinetic energy spectrum: $E(k<k_{L})\sim k^{\unicode[STIX]{x1D70E}}$ . Consequently, the helical infrared slope is $\unicode[STIX]{x1D70E}_{H}=\unicode[STIX]{x1D70E}+1$ and satisfies (2.22). It can be checked numerically that if one chooses $E(k,0)=H(k,0)$ , this violates the previous condition at large scales. Moreover, with the choice (2.33), both spectra evolve conjointly, and helicity affects the filling of the small scales during the cascade of kinetic energy: this is illustrated in figure 1 as well, where the filling of the small scales with kinetic energy is delayed by helicity at the early times of the decay. This is consistent with the results of André & Lesieur (Reference André and Lesieur1977). The authors insist on the fact that the first effect of helicity is to slightly slow down the filling of small scales with energy, as already noticed by Polifke & Shtilman (Reference Polifke and Shtilman1989), Polifke (Reference Polifke1991), Moffatt & Tsinober (Reference Moffatt and Tsinober1992). Once the spectra $E$ and $H$ are fully developed, the effects of helicity on the kinetic energy decay are more subtle, as addressed in §§ 3.2 and 4.

3.1 Joint cascades – direct and inverse nonlinear transfers

Since the numerical work of Chen et al. (Reference Chen, Chen and Eyink2003), the joint cascades of the kinetic energy and helicity in $k^{-5/3}$ inertial ranges have been well assessed. This section aims at exploring large Reynolds numbers with EDQNM, as done in the pioneering study of André & Lesieur (Reference André and Lesieur1977): more precisely, higher Reynolds numbers are reached, the assumptions and physics behind the $k^{-5/3}$ scaling of the kinetic energy and helical spectra are discussed and the spectral nonlinear transfers of kinetic energy and helicity are investigated in order to exhibit some inverse transfer mechanisms.

Firstly, the $k^{-5/3}$ scaling for both $E$ and $H$ is presented in figure 2. The inertial kinetic and helical ranges both extend from their respective integral wavenumbers $k_{L}=1/L$ and $k_{H}=1/L_{H}$ to the Kolmogorov wavenumber $k_{\unicode[STIX]{x1D702}}$ . It was argued in Ditlevsen & Giuliani (Reference Ditlevsen and Giuliani2001) that there could be a different cutoff for the helical spectrum, defined as

The opposite was proposed by Chen et al. (Reference Chen, Chen and Eyink2003), i.e. there is only one viscous cutoff $k_{\unicode[STIX]{x1D702}}$ . The latter is clearly what is obtained in figure 2: the $k^{-5/3}$ ranges for $E$ and $H$ both reach $k_{\unicode[STIX]{x1D702}}$ . It is worth noting that at moderate Reynolds numbers $Re_{\unicode[STIX]{x1D706}}\sim 50$ , $k_{\unicode[STIX]{x1D702}}$ is only three to four times higher than $k_{\unicode[STIX]{x1D702}}^{H}$ : consequently, large Reynolds numbers are needed to assess that the helical inertial range reaches the Kolmogorov wavenumber. Another interpretation of $k_{\unicode[STIX]{x1D702}}^{H}$ is proposed in appendix E. Helicity being a quantity not positive definite, a change of sign can be observed in $H(k,t)$ : more precisely here, negative values are obtained at small scales. Such a phenomenon also occurred at large Reynolds numbers in André & Lesieur (Reference André and Lesieur1977), and was observed as well at moderate Reynolds numbers in DNS (Polifke & Shtilman Reference Polifke and Shtilman1989; Polifke Reference Polifke1991).

The $k^{-5/3}$ scaling can be deduced from dimensional analysis. The main hypothesis is that the typical characteristic time in the inertial range $\unicode[STIX]{x1D70F}(k)=(k^{2}\unicode[STIX]{x1D716})^{-1/3}$ is the same for both the kinetic and helical spectra. From this assumption results an ‘Obukhov-like’ scaling, $\unicode[STIX]{x1D716}_{H}(t)=kH(k,t)/\unicode[STIX]{x1D70F}(k)$ , which directly yields in the inertial range

The constant $C_{H}\simeq 2$ is obtained at $Re_{\unicode[STIX]{x1D706}}=2\times 10^{4}$ , whereas the Kolmogorov constant remains unchanged with respect to HIT simulations. The present value $C_{H}\simeq 2$ is in agreement with André & Lesieur (Reference André and Lesieur1977). It is worth noting that one could change $C_{E}$ and $C_{H}$ to fit better with DNS by modifying the eddy-damping constants: this is out of the scope of the present work, but could be of interest for the development of simplified models.

One can remark that the scaling (3.2) is very similar to the one of a scalar field being passively advected by a turbulent flow: indeed, a similar assumption regarding the nonlinear time and the scalar dissipation rate is made in this case (see Lesieur (Reference Lesieur2008) for a review). This is one of the reasons why it is often said that helicity cascades linearly with the kinetic energy (André & Lesieur Reference André and Lesieur1977; Ditlevsen & Giuliani Reference Ditlevsen and Giuliani2001).

The scaling (3.2) can also be obtained by considering that nonlinear transfers in the inertial range are mainly local, i.e. $k\sim p\sim q$ in (2.27)–(2.30). By dimensional analysis, and dropping all geometric factors $x$ , $y$ and $z$ , this yields for the kinetic and helical fluxes $\unicode[STIX]{x1D6F1}(k)\sim \unicode[STIX]{x1D703}E(k)^{2}k^{4}$ and $\unicode[STIX]{x1D6F1}_{H}(k)\sim \unicode[STIX]{x1D703}E(k)H(k)k^{4}$ , where the flux is $\unicode[STIX]{x1D6F1}\sim kS_{E}^{NL}$ , and $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}_{kkk}$ is the ‘local’ third-order correlation characteristic time. Using the hypothesis for the eddy damping of the helicity field, which is that $\unicode[STIX]{x1D703}_{kpq}$ is identical for both $E$ and $H$ , and assuming that at high Reynolds numbers in the inertial range one has $\unicode[STIX]{x1D6F1}\simeq \unicode[STIX]{x1D716}$ and $\unicode[STIX]{x1D6F1}_{H}\simeq \unicode[STIX]{x1D716}_{H}$ , yields

The latter assumption $\unicode[STIX]{x1D6F1}\simeq \unicode[STIX]{x1D716}$ is often made at large Reynolds numbers and is well assessed in figure 2. Finally, using (3.3), one obtains two equivalent relations

The scaling (3.2) is further recovered by replacing the kinetic energy spectrum by $E(k)\sim \unicode[STIX]{x1D716}^{2/3}k^{-5/3}$ . Thus, the joint cascade of kinetic energy and helicity is recovered, and simple arguments showed that the assumption behind the $k^{-5/3}$ scaling of $H(k,t)$ is that the velocity and helical fields have the same characteristic inertial time $\unicode[STIX]{x1D70F}(k)=(k^{2}\unicode[STIX]{x1D716})^{-1/3}$ .

Figure 3.

(a) Helical nonlinear transfers $S_{H}^{(dir)}$ and $S_{H}^{(inv)}$ . (b) Purely helical and isotropic nonlinear transfers $S^{(hel)}$ and $S^{(iso)}$ . (c) Total nonlinear kinetic and helical transfers $kS_{E}^{NL}$ and $kS_{H}^{NL}$ . All for Batchelor turbulence ( $\unicode[STIX]{x1D70E}=4$ ) at $Re_{\unicode[STIX]{x1D706}}=5\times 10^{3}$ , along with the integral and Kolmogorov wavenumbers $k_{L}$ and $k_{\unicode[STIX]{x1D702}}$ .

The total nonlinear kinetic and helical transfers $S_{E}^{NL}$ and $S_{H}^{NL}$ , defined in (2.25) and (2.26) respectively, are now investigated in figure 3 for Batchelor turbulence (results are similar for Saffman turbulence). In § 2.3, $S_{E}^{NL}$ was decomposed into the sum of a purely kinetic contribution $S^{(iso)}$ , identical to the nonlinear transfer in HIT, and a purely helical contribution $S^{(hel)}$ . It is found in the EDQNM simulations that the latter part corresponds in fact to a transfer of energy from small to large scales. This inverse transfer is nevertheless less intense than the direct one, so that the total kinetic transfer $S_{E}^{NL}$ is direct. Moreover, whereas the direct nonlinear transfer $S^{(iso)}$ spans all scales of the wavenumber space, the inverse transfer $S^{(hel)}$ is very localized at large scales, which creates a small region where $S^{(iso)}$ is positive at large scales, which is different from HIT. Nevertheless, the total kinetic energy transfer $S_{E}^{NL}$ is completely similar to the one in HIT.

Similarly, $S_{H}^{NL}$ was decomposed into two contributions $S_{H}^{(dir)}$ and $S_{H}^{(inv)}$ , which both span the entire inertial range. It is revealed in figure 3 that these two parts correspond to direct and inverse transfers respectively. Once again, the inverse transfer is weaker than the direct one, so that the total transfer of helicity $S_{H}^{NL}$ goes from large to small scales. One can observe that $S_{H}^{NL}$ is negative around $k_{\unicode[STIX]{x1D702}}$ , and this explains the negative values of the helical spectrum $H(k,t)$ at the dissipative scales observed in figure 2: this can be interpreted as viscous production of helicity, since $-2\unicode[STIX]{x1D708}k^{3}H(k)$ is positive in this region. The fact that inverse transfers of kinetic energy and helicity are hidden in the total direct cascade is in agreement with the recent results of Alexakis (Reference Alexakis2017).

Figure 4.

Decaying spectra in Batchelor turbulence ( $\unicode[STIX]{x1D70E}=4$ ) with $Re_{\unicode[STIX]{x1D706}}(0)=3400$ . (a) Kinetic energy spectrum $E(k,t)$ : at $t=10^{6}\unicode[STIX]{x1D70F}_{0}$ , $E$ for HIT is displayed as well in grey. (b) Helical spectrum $H(k,t)$ .

In this part, it was recovered that in freely decaying HHT, there is a joint cascade of kinetic energy and helicity towards small scales. The main assumption behind the $k^{-5/3}$ inertial scaling is that the kinetic and helical fields have the same inertial characteristic time. Finally, it was shown numerically that despite a direct cascade, some inverse nonlinear transfers occur, which are less intense than direct ones.

3.2 Non-local interactions – infrared range dynamics

This section focuses on the permanence of large eddies (PLE) in the presence of helicity, and on non-local interactions between small and large scales. Consequently, at this point, one needs to recall the definition of the PLE hypothesis. Both Saffman and Batchelor turbulence are investigated in this work, whose infrared exponents are respectively $\unicode[STIX]{x1D70E}=2$ and $\unicode[STIX]{x1D70E}=4$ , where $E(k<k_{L},t)=Bk^{\unicode[STIX]{x1D70E}}$ and $H(k<k_{H},t)=B_{H}k^{\unicode[STIX]{x1D70E}_{H}}$ . Due to the initial conditions (2.33) where $\unicode[STIX]{x1D70E}_{H}=\unicode[STIX]{x1D70E}+1$ , one has for instance in Batchelor turbulence, $E(k<k_{L},0)\sim k^{4}$ and $H(k<k_{H},0)\sim k^{5}$ .

The PLE hypothesis is said to be true in decaying turbulence for $E(k,t)$ (for $H(k,t)$ ) when both $B$ and $\unicode[STIX]{x1D70E}$ ( $B_{H}$ and $\unicode[STIX]{x1D70E}_{H}$ ) remain constant throughout the decay. If a kinetic energy spectrum with $\unicode[STIX]{x1D70E}\leqslant 4$ is imposed initially, then $\unicode[STIX]{x1D70E}$ remains constant (Lesieur & Ossia Reference Lesieur and Ossia2000). Regarding $B$ , this prefactor is time dependent for values $\unicode[STIX]{x1D70E}$ $\in [\unicode[STIX]{x1D70E}_{c},4]$ where the critical value is located in the range $3\leqslant \unicode[STIX]{x1D70E}_{c}\leqslant 4$ (Eyink & Thomson Reference Eyink and Thomson2000; Lesieur & Ossia Reference Lesieur and Ossia2000). Consequently, for Saffman turbulence ( $\unicode[STIX]{x1D70E}=2$ ), the PLE hypothesis is verified, whereas it is not for Batchelor turbulence ( $\unicode[STIX]{x1D70E}=4$ ): this has been assessed numerically in HIT for $E(k,t)$ (Meldi & Sagaut Reference Meldi and Sagaut2013) and for a passive scalar $E_{T}(k,t)$ (Briard et al. Reference Briard, Gomez, Sagaut and Memari2015).

In what follows, it is proposed to investigate both numerically and theoretically what are the effects of helicity on the strong backscatter of energy that classically occurs in Batchelor HIT, and to address the permanence of large eddies at the level of the helical spectrum. The starting point is figure 4, where the time evolutions of the kinetic energy and helical spectra $E(k,t)$ and $H(k,t)$ are displayed for Batchelor turbulence. Two features need to be underlined: firstly, it appears that $H$ experiences no backscatter, so that the PLE hypothesis is verified in Batchelor HHT for the helical spectrum, unlike $E$ . Secondly, the backscatter for $E$ is weaker in the presence of helicity than in HIT, as revealed by the grey curve corresponding to a HIT simulation for $E$ , at the same time. These two features can be explained analytically, using non-local expansions. These important results, and especially the fact that the PLE hypothesis is verified for $H$ even in Batchelor turbulence, are applied in the next section to determine theoretical decay exponents for $K(t)$ and $K_{H}(t)$ .

One can show that the non-local kinetic energy transfer acting in the infrared range, i.e. coming from small scales, is

Such expressions were derived in Lesieur & Schertzer (Reference Lesieur and Schertzer1978), Lesieur (Reference Lesieur2008). The first term on the right-hand side is responsible for the backscatter of energy that breaks the PLE hypothesis. The second term on the right-hand side is classically written under the eddy-viscous form $-2\unicode[STIX]{x1D708}_{t}k^{2}E$ , and represents a pseudo kinetic energy dissipation, in other words the damping of large scales by small-scale turbulence. To understand why back transfers of energy on $E(k,t)$ are decreased with helicity, one has to expand $S^{(hel)}$ when $k\ll p\sim q$ , which gives

The first term on the right-hand side modifies the backscatter of energy whereas the second one can also be interpreted as a pseudo-helical dissipation term in $-2\unicode[STIX]{x1D708}_{t}^{H}k^{2}H$ . Combining this expression with (3.5) reveals the impact of helicity on the total inverse non-local kinetic transfer in HHT

The second term indicates a change in the effects of viscous damping at large scales, which are difficult to quantify. However, for the first term on the right-hand side, one has, using the realizability condition (2.22), i.e. $0\leqslant |H|/kE\leqslant 1$ ,

Consequently, the $k^{4}$ backscatter of energy of $E(k,t)$ is decreased by helicity, with respect to HIT.

Then, similar non-local expansions for $S_{H}^{NL}$ yield

This inverse non-local helical transfer indicates that there is no strong $k^{4}$ backscatter that breaks the PLE hypothesis. The first term can be written $-2\unicode[STIX]{x1D708}_{t}k^{2}H$ (and was found also in Baerenzung et al. Reference Baerenzung, Politano, Ponty and Pouquet2008), and the second one is quite original since it involves a pseudo-enstrophy dissipation and could consequently be written $-2\unicode[STIX]{x1D708}_{t}^{\unicode[STIX]{x1D714}}k^{4}E$ .

The method used now to describe analytically the infrared dynamics of the kinetic energy and helical spectra in HHT – at first order – is inspired from Lesieur (Reference Lesieur2008), where a similar reasoning is performed for HIT. When the turbulence is fully developed, both spectra scale with $E(k,t)\sim B(t)k^{\unicode[STIX]{x1D70E}}$ and $H(k,t)\sim B_{H}(t)k^{\unicode[STIX]{x1D70E}_{H}}$ at large scales. In this infrared range, viscous dissipation is negligible, and inverse non-local transfers $T_{E}^{-}$ and $T_{H}^{-}$ dominate with respect to local ones $S_{E}^{NL}$ and $S_{H}^{NL}$ . Thus, the simplified Lin equations in the infrared range are $\unicode[STIX]{x2202}_{t}E=T_{E}^{-}$ and $\unicode[STIX]{x2202}_{t}H=T_{H}^{-}$ , which yields

With the present initial conditions (2.33), one has $\unicode[STIX]{x1D70E}_{H}=\unicode[STIX]{x1D70E}+1$ , but one could think of different initial conditions that would result in a more complex infrared dynamics for $E$ and $B$ : this is the object of § 4.4. For now, one has $\unicode[STIX]{x1D70E}_{H}=\unicode[STIX]{x1D70E}+1$ , and in this case, whatever the infrared slope $\unicode[STIX]{x1D70E}$ is, the dissipation terms involving the turbulent eddy viscosities $\unicode[STIX]{x1D708}_{t}$ , $\unicode[STIX]{x1D708}_{t}^{\unicode[STIX]{x1D714}}$ and $\unicode[STIX]{x1D708}_{t}^{H}$ are negligible because $k\ll 1$ , so that

$\text{d}B_{H}/\text{d}t$ is not rigorously zero, since the non-local expansion is kept at the lowest order in $k/p\ll 1$ (see appendix A for details) in the infrared range, as done in Lesieur & Schertzer (Reference Lesieur and Schertzer1978), but is equal to some subdominant terms for the dynamics of $B_{H}$ , as assessed by figure 4. This means that whatever the large-scale initial conditions $\unicode[STIX]{x1D70E}$ are, the PLE hypothesis holds for the helical spectrum $H(k,t)$ when $\unicode[STIX]{x1D70E}_{H}=\unicode[STIX]{x1D70E}+1$ .

Regarding $E(k,t)$ : with the same reasoning, for $\unicode[STIX]{x1D70E}\leqslant 3$ , the terms on the right-hand side of (3.10) are negligible, meaning that the PLE hypothesis holds for $E(k,t)$ . Whereas for $\unicode[STIX]{x1D70E}=4$ , $B(t)$ truly depends on time since the first term on the right-hand side is stronger than the two others $-2\unicode[STIX]{x1D708}_{t}k^{2}B$ and $-2\unicode[STIX]{x1D708}_{t}^{H}k^{3}B_{H}$ , so that

The latter result contains the fact that helicity decreases the backscatter of $E(k,t)$ in Batchelor HHT. The different turbulent spectral viscosities introduced previously are summarized in table 1. Details about the non-local expansions are provided in appendix A.

Table 1.

Summary of the different eddy viscosities $\unicode[STIX]{x1D708}_{t}$ , $\unicode[STIX]{x1D708}_{t}^{H}$ and $\unicode[STIX]{x1D708}_{t}^{\unicode[STIX]{x1D714}}$ , that intervene in the non-local expansions of the nonlinear transfers $S_{E}^{NL}$ and $S_{H}^{NL}$ in HHT.

In conclusion, it has been shown numerically and analytically in this part that firstly the helical spectrum $H(k,t)$ does not experience any strong backscatter that breaks the PLE hypothesis, for initial conditions such that $\unicode[STIX]{x1D70E}_{H}=\unicode[STIX]{x1D70E}+1$ : in particular, the PLE holds in Batchelor turbulence for $H(k,t)$ . And secondly that helicity reduces the backscatter of kinetic energy for $E(k,t)$ . These theoretical results regarding the infrared dynamics of $H(k,t)$ and the impact of helicity on the infrared dynamics of $E(k,t)$ are both directly applied in the following section which aims at predicting the decay of the kinetic energy and helicity at large Reynolds numbers: indeed, the derivation of theoretical decay exponents require an accurate knowledge of the infrared dynamics.

4.1 About the interest of the helicity decay

The prediction of the decay of helicity is made possible by the determination, at large Reynolds numbers, of theoretical decay exponents: the method, originally applied in Comte-Bellot & Corrsin (Reference Comte-Bellot and Corrsin1966) (hereafter CBC) for kinetic energy, relies essentially on dimensional analysis. Similar theoretical results regarding the decay of the kinetic energy have also been obtained with self-preservation analysis (George Reference George1992).

Recently, EDQNM has been intensively used to determine these theoretical decay exponents in different configurations, at high Reynolds numbers and after a large number of turnover times: the decay of kinetic integrated quantities in HIT (Meldi & Sagaut Reference Meldi and Sagaut2013), the extension to the decay of a passive scalar field in HIT (Briard et al. Reference Briard, Gomez, Sagaut and Memari2015) and finally the decay of the velocity–scalar correlation in an isotropic turbulence with a mean scalar gradient (Briard et al. Reference Briard, Gomez and Cambon2016). The study of the helical case is therefore a natural extension and our predictions are compared to those of Levshin & Chkhetiani (Reference Levshin and Chkhetiani2013) in appendix B.

As a starting point, let us recall that figure 1 revealed that helicity was slowing down the filling of small scales with energy at the early times of the simulation. This is a transitory result for the early stage of the decay, in agreement with Polifke & Shtilman (Reference Polifke and Shtilman1989), and it will be shown hereafter that the impact of helicity on kinetic energy in fully developed turbulence is more subtle.

Figure 5.

Decay laws for the kinetic (–) and helical (– –) fields, in Saffman (black) and Batchelor (grey) turbulence. Symbols refer to the theoretical predictions given in (4.1) for $\unicode[STIX]{x1D6FC}$ , (4.3) for $\unicode[STIX]{x1D6FC}_{H}$ and (4.2) for $L$ and $L_{H}$ . (a) Growth exponents of the kinetic and helical integral scales $L$ and $L_{H}$ . At a given $\unicode[STIX]{x1D70E}$ , the kinetic and helical theoretical exponents indicated by ○ cannot be distinguished. (b) Decay exponents of the kinetic energy and helicity $K$ and $K_{H}$ , where ○ and ▫ refer to the kinetic and helical theoretical predictions respectively.

Table 2.

Theoretical predictions for the kinetic and helical decay exponents $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FC}_{H}$ as a function of the infrared slope $\unicode[STIX]{x1D70E}$ .

4.2 Effect of helicity on the kinetic energy $K(t)$ decay

The effects of helicity on the kinetic energy decay are firstly addressed. Simulations show that the decay exponent $\unicode[STIX]{x1D6FC}$ of the kinetic energy, where $K(t)\sim t^{\unicode[STIX]{x1D6FC}}$ , is not modified by helicity, except in the case of Batchelor turbulence ( $\unicode[STIX]{x1D70E}=4$ ), because of the reduction of the non-local inverse transfers analysed in § 3.2. Hence, one would expect the decay of $K(t)$ to be more rapid in Batchelor HHT than in Batchelor HIT: indeed, the non-local inverse transfers bring back less energy to the large scales. This is recovered in figure 5.

To analytically take into account the breakdown of the PLE hypothesis, a backscatter parameter $p$ is usually introduced (Lesieur & Ossia Reference Lesieur and Ossia2000; Meldi & Sagaut Reference Meldi and Sagaut2013; Briard et al. Reference Briard, Gomez, Sagaut and Memari2015). In HIT, one has $p(\unicode[STIX]{x1D70E}=4)=0.55$ and $p(\unicode[STIX]{x1D70E}\leqslant 3)=0$ : in particular, in Batchelor HIT $K(t)\sim t^{-1.38}$ . Here, in Batchelor HHT, $K(t)\sim t^{-1.417}$ is obtained, and a least-square fit provides a new backscatter parameter $p_{H}=0.14$ for HHT. Consequently, with respect to HIT, only the backscatter parameter changes from $p$ to $p_{H}$ in HHT for the decay of kinetic energy

These decay exponents for the kinetic energy, whose values are gathered in table 2, are assessed in figure 5 in Saffman and Batchelor turbulence: only the case $\unicode[STIX]{x1D70E}=4$ differs from HIT, where here in HHT the decay of $K(t)$ is slightly more rapid. The fact that helicity does not influence much the energy cascade once the turbulence is fully developed is in good agreement with conclusions drawn by Polifke (Reference Polifke1991), Borue & Orszag (Reference Borue and Orszag1997).

4.3 The decay of helicity $K_{H}(t)$

In the previous part, helicity was shown to impact the kinetic energy decay only in Batchelor turbulence. The emphasis is now put on the decay of $K_{H}$ itself. The method to predict the decay of helicity is similar to the one of an advected passive scalar, and even more simple. Indeed, as revealed in figure 4 and explained in § 3.2, the helical spectrum $H(k,t)$ experiences no strong back transfers, so that the PLE hypothesis is verified, from $\unicode[STIX]{x1D70E}=1$ to $\unicode[STIX]{x1D70E}=4$ . Therefore, there is no need to introduce a backscatter parameter. It is recalled that given the initial conditions (2.33), the spectral infrared slope of $H(k,t)$ is $\unicode[STIX]{x1D70E}_{H}=\unicode[STIX]{x1D70E}+1$ , where $\unicode[STIX]{x1D70E}$ is the infrared slope of $E(k,t)$ .

Then, it is reasonable to assume that the kinetic and helical integral scales $L(t)$ and $L_{H}(t)$ decay similarly, so that their algebraic exponents $n_{L}$ and $n_{L_{H}}$ are equal

This assumption is completely assessed in figure 5 in both Saffman and Batchelor turbulence (the same kind of hypothesis for the scalar integral scale is made when it comes to predicting the decay of the scalar variance). Then, using either the continuity of $H(k,t)$ in $k_{H}=1/L_{H}$ to determine the decay law for $\unicode[STIX]{x1D716}_{H}$ , or dimensional analysis with $K_{H}\sim K/L_{H}$ , or equation (3.4), one obtains

Theoretical values of this expression for $\unicode[STIX]{x1D6FC}_{H}$ , gathered in table 2, are in excellent agreement with simulations presented in figure 5 for Saffman and Batchelor turbulence: one can remark that the more $\unicode[STIX]{x1D70E}$ increases, the more $K_{H}$ decays rapidly, similarly to the dynamics of $K(t)$ . In addition, the decay exponent of $\unicode[STIX]{x1D716}_{H}$ is $\unicode[STIX]{x1D6FC}_{H}-1$ , according to (2.20). Interestingly, the theoretical decay exponent of $K_{H}$ is equivalent to the one of the scalar variance in HIT with an infrared slope $\unicode[STIX]{x1D70E}_{T}=\unicode[STIX]{x1D70E}+1$ without any backscatter. Finally, our predictions for $K_{H}$ are compared to those of Levshin & Chkhetiani (Reference Levshin and Chkhetiani2013) in appendix B.

In conclusion, the present theoretical predictions for the decay of helicity and kinetic energy have been assessed numerically. This is the first time such a result is proposed, with a careful attention to initial conditions and infrared dynamics.

4.4 Robustness of the decay exponents – altered infrared dynamics

In § 3.2, for kinetic energy and helical spectra scaling in $E=Bk^{\unicode[STIX]{x1D70E}}$ and $H=B_{H}k^{\unicode[STIX]{x1D70E}_{H}}$ in fully developed turbulence, the evolution equations of $B(t)$ and $B_{H}(t)$ were derived in (3.10)–(3.11) in the infrared range, assuming the dominance of inverse non-local transfers $T_{E}^{-}$ and $T_{H}^{-}$ on local ones. For the initial conditions (2.33) where $\unicode[STIX]{x1D70E}_{H}=\unicode[STIX]{x1D70E}+1$ , it was notably found that the permanence of large eddies holds for Batchelor turbulence ( $\unicode[STIX]{x1D70E}=4$ ).

Nevertheless, one can wonder if other initial conditions would produce a different infrared dynamics, and this could be relevant to test the robustness of the decay exponents $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FC}_{H}$ derived in the previous sections. One can remark that in (3.10), the term responsible for the backscatter of $E$ is the one where the $k^{()}$ factor vanishes. Thus, it is legitimate to wonder if in (3.11) one could have $4+\unicode[STIX]{x1D70E}-\unicode[STIX]{x1D70E}_{H}=0$ in order to obtain $d_{t}B_{H}\neq 0$ , and consequently to create backscatter for the helical spectrum. It is worth noting that $2+\unicode[STIX]{x1D70E}_{H}-\unicode[STIX]{x1D70E}=0$ is impossible in (3.10) because of the realizability condition (2.22).

In order to simplify this study, the maximal helicity condition $H=kE$ is kept, but the initial kinetic energy spectrum is now changed into a sharply peaked Gaussian one

with $C_{1}$ so that one has a unit initial kinetic energy, and $C_{2}=0.1$ . After a few turnover times, the kinetic energy spectrum scales in $E\sim Bk^{4}$ in the infrared range (Lesieur & Ossia Reference Lesieur and Ossia2000). Unlike (2.33), the helical infrared slope $\unicode[STIX]{x1D70E}_{H}$ is now different from $\unicode[STIX]{x1D70E}+1$ . As a consequence of (3.11), it follows that $\unicode[STIX]{x1D70E}_{H}=8$ , and that the helical spectrum experiences backscatter with

The theoretical infrared scaling prediction $H\sim k^{8}$ is assessed in figure 6, along with the $k^{4}$ infrared scaling of $E(k,t)$ : in this case, the permanence of large eddies is not verified anymore for $H(k,t)$ for the initial conditions (4.4). Nevertheless, it is shown in figure 6 as well that the previous theoretical decay exponents for kinetic energy and helicity are still valid, which implies that the backscatter of $H(k,t)$ is negligible in the decay. This case further illustrates the robustness of the theoretical predictions for the decay exponents.

Figure 6.

(a) Helical spectrum $H(k,t)$ (black) for the initial condition (4.4) (– –) at various times $t/\unicode[STIX]{x1D70F}_{0}=10$ , $10^{3}$ and $10^{5}$ ; the kinetic energy spectrum $E(k,t)$ is displayed as well (grey) at $t/\unicode[STIX]{x1D70F}_{0}=10^{5}$ . (b) Decay exponents $\unicode[STIX]{x1D6FC}$ (–) and $\unicode[STIX]{x1D6FC}_{H}$ (– –), where ○ and ▫ refer to the kinetic and helical theoretical predictions (4.1) and (4.3) respectively.

Simulations not presented here show that for Saffman turbulence ( $E\sim k^{2}$ ), a $k^{6}$ infrared scaling for $H$ could be created, but this is out of the maximal helicity framework: $H(k,t=0)$ should be Gaussian with $E(k,t=0)\sim k^{2}\exp (-k^{2})$ , and the theoretical decay exponents are still verified.

5.1 Inertial scaling for $S(r)$ and $D^{(uu\unicode[STIX]{x1D714})}(r)$

This part aims at recalling the main steps of the derivation of the laws (5.1) and (5.2), and linking $S(r)$ and $D^{(uu\unicode[STIX]{x1D714})}(r)$ . The two-point second-order Reynolds tensor reads in HHT

where $f=\langle u_{L}u_{L}^{\prime }\rangle /u^{2}$ is the second-order longitudinal correlation, with $u^{2}=2K/3$ and $h(r)=R_{23}(r)$ . The mixed velocity–vorticity correlation is given by

Also,

The latter expression shows that $h(0)=0$ . Then, the two-point triple velocity correlation contains an additional antisymmetric part with respect to HIT, so that (Chkhetiani Reference Chkhetiani1996; Gomez et al. Reference Gomez, Politano and Pouquet2000)

where $k(r)=\langle u_{L}u_{L}u_{L}^{\prime }\rangle$ and $S(r)=\langle u_{L}u_{2}u_{3}^{\prime }\rangle$ : in particular, $\langle u_{2}u_{3}{u_{L}}^{\prime }\rangle =0$ and $\langle u_{L}u_{3}u_{2}^{\prime }\rangle =-S(r)$ , which gives $S(0)=0$ . Thus, from the evolution equation of $\unicode[STIX]{x1D619}_{ij}(r)$

it is possible to compute the evolution equation of the antisymmetric part $(\unicode[STIX]{x1D619}_{ij}-\unicode[STIX]{x1D619}_{ji})/2=h\unicode[STIX]{x1D716}_{ijl}r_{l}/r$ linked to helicity (Chkhetiani Reference Chkhetiani1996; Kurien Reference Kurien2003). The spatial derivative $\unicode[STIX]{x2202}_{r_{j}}$ erases the part which contains the third-order longitudinal correlation $k(r)$ , so that, after some algebra and using $\unicode[STIX]{x2202}_{t}K_{H}=-\unicode[STIX]{x1D716}_{H}$ , one obtains

Further, neglecting the viscous effects in the inertial range and integrating over $r$ , one obtains (5.1). Then, it is possible to derive the equation for $D^{(uu\unicode[STIX]{x1D714})}$ starting from

where $\unicode[STIX]{x2202}/\unicode[STIX]{x2202}_{x_{j}^{\prime }}=\unicode[STIX]{x2202}/\unicode[STIX]{x2202}_{r_{j}}=-\unicode[STIX]{x2202}/\unicode[STIX]{x2202}_{x_{j}}$ has been used (see Antonia et al. (Reference Antonia, Ould-Rouis, Anselmet and Zhu1997) for further details on the method). Combining these two equations yields

Then, $\langle \unicode[STIX]{x1D6FF}u_{j}\unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D714}_{i}\rangle =r_{j}\langle \unicode[STIX]{x1D6FF}u_{L}\unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D714}_{i}\rangle /r$ , and $\unicode[STIX]{x2202}_{r_{j}}(D^{(uu\unicode[STIX]{x1D714})}r_{j}/r)=\unicode[STIX]{x2202}_{r}(r^{2}D^{(uu\unicode[STIX]{x1D714})})/r^{2}$ . Further neglecting the time derivative and the viscous term in the inertial range, integration over $r$ yields (5.2).

Interestingly, equations (5.1) and (5.2) can be linked: remarking that $\langle \unicode[STIX]{x1D6FF}u_{i}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D714}_{i}\rangle =2\langle u_{i}\unicode[STIX]{x1D714}_{i}\rangle -2\langle u_{i}\unicode[STIX]{x1D714}_{i}^{\prime }\rangle$ , $\unicode[STIX]{x2202}_{t}R_{11}^{H}=-2\unicode[STIX]{x1D716}_{H}/3$ , with $R_{11}^{H}=-2h/r$ , this provides

and one gets by identification

This important relation is used in the next part, where (5.1) and (5.2) are assessed numerically at large Reynolds numbers.

5.2 Numerical assessment of $S(r)$ and $D^{(uu\unicode[STIX]{x1D714})}(r)$

Now that the main theoretical results have been recalled, the relations linking spectral quantities to helical structure functions are derived. First, one needs to obtain the evolution equations of $\langle u_{i}\unicode[STIX]{x1D714}_{i}^{\prime }\rangle /2$ : this is done starting from (5.8) since

One obtains

This equation needs to be identified with the helical Lin equation (2.32), so that

After some algebra, one obtains

Notice that these two formulae are valid for both decaying and forced HHT, and the latter expression for $D^{(uu\unicode[STIX]{x1D714})}$ is very similar to the ones for $\langle \unicode[STIX]{x1D6FF}u_{L}\unicode[STIX]{x1D6FF}q^{2}\rangle$ and $\langle \unicode[STIX]{x1D6FF}u_{L}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}^{2}\rangle$ which can be found in appendix C, which is expected since they all refer to conservation laws. The relevance of the two previous formulae (5.17) and (5.18) is illustrated in figure 7, where the compensated helical third-order correlations $-D^{(uu\unicode[STIX]{x1D714})}/(r\unicode[STIX]{x1D716}_{H})$ and $S/(r^{2}\unicode[STIX]{x1D716}_{H})$ are displayed at high Reynolds numbers. The theoretical values $4/3$ and $1/30$ are almost recovered at $Re_{\unicode[STIX]{x1D706}}=3\times 10^{4}$ : the slight difference is comparable to the difference observed for the $-4/5$ law in decaying turbulence (Bos et al. Reference Bos, Chevillard, Scott and Rubinstein2012). Interestingly, $D^{(uu\unicode[STIX]{x1D714})}$ is slightly closer to $4/3$ than $\langle \unicode[STIX]{x1D6FF}u_{L}\unicode[STIX]{x1D6FF}q^{2}\rangle$ in HHT. Simulations not presented here revealed that at the same high Reynolds numbers, $-\langle \unicode[STIX]{x1D6FF}u_{L}\unicode[STIX]{x1D6FF}q^{2}\rangle /(\unicode[STIX]{x1D716}r)$ is as close to $4/3$ in HIT and in HHT. Let us mention that an equivalent scaling for $S$ (the $2/15$ law) was already assessed in DNS (Kurien et al. Reference Kurien, Taylor and Matsumoto2004b ). But so far, to our knowledge, the scaling for $D^{(uu\unicode[STIX]{x1D714})}$ was not verified numerically, at least for freely decaying turbulence: the present simulations show a very good agreement between EDQNM results and the theoretical expectations. Finally, it is revealed in figure 7 that $S\sim r^{4}$ at small scales, which is straightforward using (5.13), unlike $\langle \unicode[STIX]{x1D6FF}u_{L}^{3}\rangle$ , $\langle \unicode[STIX]{x1D6FF}u_{L}\unicode[STIX]{x1D6FF}q^{2}\rangle$ , $\langle \unicode[STIX]{x1D6FF}u_{L}\unicode[STIX]{x1D6FF}\unicode[STIX]{x1D703}^{2}\rangle$ and $D^{(uu\unicode[STIX]{x1D714})}$ which scale with $r^{3}$ (see figures 12 and 13).

Figure 7.

Third-order helical correlations $D^{(uu\unicode[STIX]{x1D714})}$ and $S$ , for $\unicode[STIX]{x1D70E}=2$ at $Re_{\unicode[STIX]{x1D706}}=3\times 10^{4}$ , along with the integral and Kolmogorov scales $L$ and $\unicode[STIX]{x1D702}$ . (a) $-D^{(uu\unicode[STIX]{x1D714})}/(r\unicode[STIX]{x1D716}_{H})$ and $-\langle \unicode[STIX]{x1D6FF}u_{L}\unicode[STIX]{x1D6FF}q^{2}\rangle /(r\unicode[STIX]{x1D716})$ ; (b) $S/(r^{2}\unicode[STIX]{x1D716}_{H})$ . (c) Different scalings of $D^{(uu\unicode[STIX]{x1D714})}$ and $S$ : the scale $\unicode[STIX]{x1D706}_{H}$ is defined later on in (5.24).

5.3 Evolution equation of $\unicode[STIX]{x1D716}_{H}$

In this part, the evolution equation of the helicity dissipation rate is addressed. The objective is to simplify this equation, in a manner similar to what is usually done for the kinetic energy dissipation rate $\unicode[STIX]{x1D716}$ in HIT (Pope Reference Pope2000). More precisely, $\unicode[STIX]{x1D716}_{H}$ itself and its production term are greatly simplified, and expressed as simple functions of the derivatives of the fluctuating velocity and vorticity fields. The final expressions (5.22) and (5.31) constitute one of the main new theoretical contributions of the present work, and applications are proposed as well. The calculations being rather lengthy, the intermediate steps are gathered in appendix D for the sake of clarity. The evolution equation of $\unicode[STIX]{x1D716}_{H}$ reads

where the production term $D[u,\unicode[STIX]{x1D714}]$ contains contributions from spatial derivatives of skew-isotropic tensors such as $\langle u_{i}u_{j}\unicode[STIX]{x1D714}_{l}^{\prime }\rangle$ and $\langle \unicode[STIX]{x1D714}_{i}u_{j}u_{l}^{\prime }\rangle$ . In the two following subsections, both $\unicode[STIX]{x1D716}_{H}$ and $D[u,\unicode[STIX]{x1D714}]$ are simplified. This procedure consists into two steps: first, expressing $\unicode[STIX]{x1D716}_{H}$ and $D[u,\unicode[STIX]{x1D714}]$ as functions of the derivatives of $h(r)$ and $S(r)$ respectively; and then expressing these derivatives as functions of particular components of the fluctuating velocity and vorticity fields. Obviously, this is much more lengthy for $D[u,\unicode[STIX]{x1D714}]$ since it is a third-order moment, composed of three different terms. All results of this part remain valid whether the turbulence is forced or decaying.

6.1 Evolution equation of the quadrature spectrum

The scalar flux can be decomposed similarly to the spectral Reynolds tensor $\hat{\unicode[STIX]{x1D619}}_{ij}$ in (2.8), as

where both $E_{j}^{F}$ and $E_{j}^{Q}$ are real. The first part of the decomposition is usual (see Briard et al. (Reference Briard, Gomez and Cambon2016) and references therein) and is linked to the cospectrum ${\mathcal{F}}$ when a mean scalar gradient is present

Whereas the second part is linked to the quadrature spectrum $Q$ , which is non-zero in the presence of helicity

The phenomenon creating this additional contribution to the scalar flux, parallel to the mean gradient in the presence of helicity, is called ‘skew diffusion’ in Moffatt & Tsinober (Reference Moffatt and Tsinober1992). It is worth noting that from the decomposition (6.1) and (6.3), imaginary components of the scalar flux, perpendicular to the mean scalar gradient, are non-zero. Nevertheless, these components vanish after spherical averaging.

The mixed vorticity–scalar correlation is then $\langle \unicode[STIX]{x1D714}_{3}\unicode[STIX]{x1D703}\rangle =\int _{0}^{\infty }Q(k,t)\,\text{d}k$ , and its evolution equation reads

In the framework of homogeneous helical turbulence with a mean scalar gradient, the spherically averaged Lin equations for the cospectrum and the quadrature spectrum are

where helicity is the production term of the quadrature spectrum. The nonlinear transfer of the cospectrum $S_{3}^{F}$ is always present when there is a mean scalar gradient, whereas $S_{3}^{F(hel)}$ is the additional contribution arising from the presence of helicity

where $\unicode[STIX]{x1D703}_{kpq}^{F}$ is the characteristic relaxation time of the triple velocity–velocity–scalar correlations, given in Briard et al. (Reference Briard, Gomez and Cambon2016). The remarkable feature is that $S_{3}^{F(hel)}$ is a conservative transfer, with zero integral over the whole wavenumber space, similarly to $S^{(hel)}$ in the evolution equation of $E(k,t)$ . The spherically averaged quadrature nonlinear transfer $S_{Q}^{NL}=S_{3}^{Q}$ finally reads

and is not a conservative transfer, similarly to $S_{3}^{F}$ . Its integral over the whole wavenumber space yields, according to (6.4), the term $\langle \unicode[STIX]{x1D703}\unicode[STIX]{x1D714}_{j}\unicode[STIX]{x2202}_{j}u_{3}\rangle$ .

6.2 Decay of $\langle \unicode[STIX]{x1D714}_{3}\unicode[STIX]{x1D703}\rangle$ and inertial scaling of $Q(k,t)$

For these numerical simulations, the initial condition (2.33) is used, with $Q(k,t=0)={\mathcal{F}}(k,t=0)=0$ . First, the decay of $\langle \unicode[STIX]{x1D714}_{3}\unicode[STIX]{x1D703}\rangle$ is addressed. From the Lin equation (6.6) of $Q(k,t)$ , it follows that the decay exponent of $\langle \unicode[STIX]{x1D714}_{3}\unicode[STIX]{x1D703}\rangle$ is $\unicode[STIX]{x1D6FC}_{Q}=\unicode[STIX]{x1D6FC}_{H}+1$ , where $\unicode[STIX]{x1D6FC}_{H}$ is given in (4.3), because the production term linked to helicity is the one responsible for the creation of the quadrature spectrum. This yields

It is recalled that the decay exponent of $\langle u_{3}\unicode[STIX]{x1D703}\rangle$ is $\unicode[STIX]{x1D6FC}_{{\mathcal{F}}}=\unicode[STIX]{x1D6FC}+1$ , where the backscatter parameter $p_{H}=0.14$ of § 4 should be used for consistency. Both $\unicode[STIX]{x1D6FC}_{{\mathcal{F}}}$ and $\unicode[STIX]{x1D6FC}_{Q}$ are assessed for Saffman and Batchelor turbulence in figure 10 at large Reynolds numbers. For a given infrared slope $\unicode[STIX]{x1D70E}$ , the decay of $\langle \unicode[STIX]{x1D714}_{3}\unicode[STIX]{x1D703}\rangle$ is faster than $\langle u_{3}\unicode[STIX]{x1D703}\rangle$ , similarly to the decay of $K_{H}$ being faster than $K$ .

Figure 10.

Decay exponents $\unicode[STIX]{x1D6FC}_{Q}$ and $\unicode[STIX]{x1D6FC}_{{\mathcal{F}}}$ of $\langle \unicode[STIX]{x1D714}_{3}\unicode[STIX]{x1D703}\rangle$ (– –) and $\langle u_{3}\unicode[STIX]{x1D703}\rangle$ (–) respectively, for $\unicode[STIX]{x1D70E}=2$ (black) and $\unicode[STIX]{x1D70E}=4$ (grey); theoretical predictions, ▫ for $\unicode[STIX]{x1D6FC}_{Q}$ (6.9), and ○ for $\unicode[STIX]{x1D6FC}_{{\mathcal{F}}}$ .

Regarding the nonlinear transfers: the impact of the quadrature spectrum on the cospectrum is rather weak and can be observed through the conservative nonlinear transfer $S_{3}^{F(hel)}$ . This transfer is linked to an inverse cascade of $\langle u_{3}\unicode[STIX]{x1D703}\rangle$ , localized at large scales, between the integral and the helical Taylor scales. This can be qualitatively compared to the impact of helicity on the kinetic energy spectrum dynamics through $S^{(hel)}$ in figure 3. One could conclude that helicity only slightly reduces the nonlinear transfers of the cospectrum at large scales. Then, the quadrature nonlinear transfer $S_{Q}^{NL}$ itself is similar to the cospectrum one $S_{3}^{F}$ , but less intense.

Figure 11.

(a) Cospectrum and quadrature nonlinear transfers. (b) Cospectrum ${\mathcal{F}}(k,t)$ and quadrature spectrum $Q(k,t)$ ; $-Q(k,t)$ is displayed in grey. Both for $\unicode[STIX]{x1D70E}=2$ at $Re_{\unicode[STIX]{x1D706}}=5\times 10^{3}$ , after a hundred turnover times, along with the integral, helical Taylor and Kolmogorov wavenumbers $k_{L}$ , $1/\unicode[STIX]{x1D706}_{H}$ and $k_{\unicode[STIX]{x1D702}}$ .

Finally, in figure 11, the quadrature spectrum itself is presented. In the infrared range, it scales like $Q\sim k^{3}$ for Saffman turbulence, because the helical spectrum itself evolves in $H\sim k^{3}$ : indeed, helicity, through the mean scalar gradient, is the production term of the quadrature spectrum. In the inertial range, $Q(k,t)$ is first positive for scales larger than the helical Taylor scale $\unicode[STIX]{x1D706}_{H}$ , and scales in $k^{-7/3}$ similarly to ${\mathcal{F}}$ . Whereas for scales smaller than $\unicode[STIX]{x1D706}_{H}$ , the spectral slope is close to $k^{-5/3}$ and the quadrature spectrum is negative: it is recalled that around $k_{\unicode[STIX]{x1D702}}$ , $H(k,t)$ is also negative.

One can propose theoretical inertial scalings for the quadrature spectrum. In the positive region, assuming in (6.6) that $\unicode[STIX]{x2202}_{t}Q\sim \unicode[STIX]{x1D6EC}H$ , with the classical characteristic inertial time $(k^{2}\unicode[STIX]{x1D716})^{-1/3}$ , this gives

For the negative region of $Q(k,t)$ , the quadrature spectrum should only depend on $\unicode[STIX]{x1D716}_{H}$ since this is the negative small scales of $H(k,t)$ which are responsible for this inertial–helical range of $Q(k,t)$ . Thus, one gets

This change of slope, from $k^{-7/3}$ for $k\unicode[STIX]{x1D706}_{H}<1$ , to $k^{-5/3}$ for $k\unicode[STIX]{x1D706}_{H}>1$ , observed in figure 11 and justified with dimensional and physical arguments, can also be interpreted in terms of a change in characteristic time scales, from $\unicode[STIX]{x1D70F}=(k^{2}\unicode[STIX]{x1D716})^{-1/3}$ to $\unicode[STIX]{x1D70F}_{H}=(k\unicode[STIX]{x1D716}_{H})^{-1/3}$ . The latter time scale was notably proposed by Kurien, Taylor & Matsumoto (Reference Kurien, Taylor and Matsumoto2004a ) for an alternative scaling of $H(k,t)$ at small scales. This characteristic time scale $\unicode[STIX]{x1D70F}_{H}$ is relevant for the inertial–helical scaling of $Q$ when it is negative. Indeed, for $k\unicode[STIX]{x1D706}_{H}>1$ , the equation (6.6) of $Q$ can be written $\unicode[STIX]{x2202}_{t}Q\sim S_{Q}^{NL}$ . For the nonlinear term, using the inertial scalings (3.2) for $H$ and ${\mathcal{F}}$ , along with $\unicode[STIX]{x1D703}_{kpq}^{F}\sim \unicode[STIX]{x1D70F}_{H}$ , gives $S_{Q}^{NL}\sim \unicode[STIX]{x1D6EC}k^{-4/3}\unicode[STIX]{x1D716}_{H}^{2/3}$ . Finally, one has $Q\sim \unicode[STIX]{x1D70F}_{H}S_{Q}^{NL}$ so that (6.11) is recovered.

As a conclusion, the combined effects of helicity and mean scalar gradient were investigated. Nonlinear transfer terms were derived for the evolution equations of the quadrature spectrum $Q(k,t)$ and of the cospectrum ${\mathcal{F}}(k,t)$ within the EDQNM framework. Fundamental properties of $Q(k,t)$ were tackled: a prediction for the decay of $\langle \unicode[STIX]{x1D714}_{3}\unicode[STIX]{x1D703}\rangle$ was proposed and assessed numerically, $Q(k,t)$ was found to slightly decrease the large-scale nonlinear transfers of ${\mathcal{F}}$ through helicity, and finally, the main theoretical result is that $Q(k,t)$ has two different scalings in the inertial range, which could be associated with two different characteristic time scales.