2. Exact decomposition of the kinetic helicity flux

To derive the aforementioned exact decomposition of the helicity flux and relations between the resulting subfluxes, we begin with the three-dimensional (3-D) incompressible Navier–Stokes equations, here written in component form:

(2.1)\begin{gather} \partial_t u_i + \partial_j\left (u_i u_j \right)={-} \partial_j p \delta_{ij} + 2 \nu \partial_j S_{ij} + f_i , \end{gather}

(2.2)\begin{gather}\partial_j u_j = 0 , \end{gather}

where $\boldsymbol {u} = (u_1, u_2, u_3)$ is the velocity field, $p$ the pressure divided by the constant density, $\nu$ the kinematic viscosity, $S_{ij}$ the rate-of-strain tensor and $\boldsymbol {f} = (f_1, f_2, f_3)$ an external solenoidal force that may be present. To define the helicity flux across scales, we introduce a filtering operation to separate large- and small-scale dynamics (e.g. Germano Reference Germano1992). Specifically, for a generic function $\phi$, the filtered version at scale $\ell$ is $\bar {\phi }^\ell = G^\ell * \phi$, where $G^\ell$ is a filter kernel with filter width $\ell$ and the asterisk denotes the convolution operation. Applying the filter to the Navier–Stokes equations (2.1)–(2.2) results in

(2.3)\begin{equation} \partial_t \bar{u}^\ell_i + \partial_j\left( \bar{u}^\ell_i \bar{u}^\ell_j + \bar{p}^\ell \delta_{ij} - 2 \nu \bar{S}^\ell_{ij} + \tau_{ij}^\ell \right) = \bar{f}_i^\ell,\\ \end{equation}

where $\tau _{ij}^\ell = \tau ^\ell (u_i, u_j) = \overline {u_i u_j}^\ell - \bar {u}^\ell _i \bar {u}^\ell _j$ is the SGS stress tensor. Here, we follow the notation of Germano (Reference Germano1992) in defining the generalised second moment for any two fields as $\tau ^\ell (a,b) = \overline {ab}^\ell - \bar {a}^\ell \bar {b}^\ell$. We also require the filtered vorticity equation

(2.4)\begin{equation} \partial_t \bar{\omega}^\ell_i + \partial_j\left( \bar{\omega}^\ell_i \bar{u}^\ell_j - \bar{u}^\ell_i \bar{\omega}^\ell_j - \nu \partial_j \bar{\omega}^\ell_i \right) - \bar{g}_i^\ell ={-} \partial_j\left(\epsilon_{imn} \partial_m \tau^\ell_{nj} \right) , \end{equation}

where $\boldsymbol {g} = \boldsymbol {\nabla } \times \boldsymbol {f}$. The large-scale helicity density, $H^\ell = \bar {u}^\ell _i \bar {\omega }^\ell _i$, then evolves according to

(2.5)$$\begin{gather} \partial_t H^\ell + \partial_j \left[ H^\ell \bar{u}^\ell_j + (\bar{p}^\ell - \tfrac{1}{2} \bar{u}^\ell_i \bar{u}^\ell_i) \bar{\omega}^\ell_j - \nu \partial_j H^\ell \right] + 2\nu ( \partial_j \bar{u}^\ell_i )( \partial_j \bar{\omega}^\ell_i ) - \bar{\omega}_i^\ell\bar{f}_i^\ell - \bar{u}_i^\ell\bar{g}_i^\ell \nonumber\\ ={-} \partial_j \left[ 2 \bar{\omega}^\ell_i \tau^\ell_{ij} + \epsilon_{ijk} \bar{u}^\ell_i \partial_m \tau^\ell_{km} \right] + 2 \tau^\ell_{ij} \partial_j \bar{\omega}^\ell_i. \end{gather}$$

The last term in this equation is the helicity flux

(2.6)\begin{equation} \varPi^{H, \ell} ={-} 2\tau^\ell_{ij} \partial_j \bar{\omega}^\ell_i , \end{equation}

and is the central focus herein. It has an alternative form (Yan et al. Reference Yan, Li, Yu, Wang and Chen2020),

(2.7)\begin{equation} \tilde{\varPi}^{H, \ell} ={-} \tau^\ell_{ij} \partial_j \bar{\omega}^\ell_i - \left[ \tau^\ell(\omega_i, u_j) - \tau^\ell(u_i, \omega_j) \right] \partial_j \bar{u}^\ell_i , \end{equation}

and it can be shown that the right-hand side of (2.6) and (2.7) differ by an expression that can be written as a divergence and therefore vanishes after averaging spatially, at least for statistically homogeneous turbulence (Yan et al. Reference Yan, Li, Yu, Wang and Chen2020). This implies $\langle \varPi ^{H, \ell } \rangle = \langle \tilde \varPi ^{H, \ell } \rangle$. Eyink (Reference Eyink2006) links the first term in (2.7) – which is proportional to $\varPi ^{H, \ell }$ – to vortex twisting and Yan et al. (Reference Yan, Li, Yu, Wang and Chen2020) attribute the second term to vortex stretching. In what follows we discuss an exact decomposition of $\varPi ^{H, \ell }$, and show that both effects can be identified therein. We also use $\varPi ^{H, \ell }$ for our numerical evaluations (cf. Chen et al. Reference Chen, Chen and Eyink2003a; Eyink Reference Eyink2006).

2.1. Gaussian filter relations for the helicity flux

So far all expressions are exact and filter-independent. To derive exact decompositions of the helicity flux in both representations, we now focus on Gaussian filters. For that case, Johnson (Reference Johnson2020, Reference Johnson2021) showed that the SGS stresses can be obtained as the solution of a forced diffusion equation with $\ell ^2$ being the time-like variable, resulting in

(2.8)\begin{equation} \tau_{ij}^\ell = \tau^\ell(u_i, u_j) = \ell^2 \bar{A}_{ik}^\ell \bar{A}_{jk}^\ell + \int_{0}^{\ell^2} \,\mathrm{d}\theta\ \tau^\phi\left( \bar{A}_{ik}^{\sqrt{\theta}}, \bar{A}_{kj}^{\sqrt{\theta}} \right) , \end{equation}

where $\phi (\theta ) = \sqrt {\ell ^2 - \theta }$ and $A_{ij} = \partial _j u_i$ are the velocity-field gradients. Since the SGS stress tensor $\tau ^\ell _{ij}$ is symmetric, for the first form of the helicity flux we obtain in analogy to the energy flux

(2.9)\begin{equation} \varPi^{H, \ell} ={-} 2 \tau^\ell_{ij} \bar{S}^\ell_{\omega,ij} , \end{equation}

where $S_\omega$ is the symmetric component of the vorticity gradient tensor, with components $S_{\omega, ij} = (\partial _j \omega _i + \partial _i \omega _j)/2$. Employing (2.8) this yields

(2.10)\begin{equation} \varPi^{H,\ell} ={-} 2 \ell^2 \bar{S}^\ell_{\omega,ij} \bar{A}_{ik}^\ell \bar{A}_{jk}^\ell - 2 \int_{0}^{\ell^2} \,\mathrm{d}\theta\ \bar{S}^\ell_{\omega,ij} \tau^\phi\left( \bar{A}_{ik}^{\sqrt{\theta}}, \bar{A}_{kj}^{\sqrt{\theta}} \right) . \end{equation}

The first term involves a product of gradient tensors filtered at the same scale, $\ell$; hence we refer to it as being single-scale, and denote it $\varPi _s^{H,\ell }$. In mean, it coincides with the nonlinear LES model for the SGS stresses (Eyink Reference Eyink2006). In contrast, the second term encodes the correlation between resolved-scale vorticity-field gradients and (summed) velocity-field gradients at each scale smaller than $\ell$, so that we refer to it as multi-scale.

Splitting the velocity gradient tensors into symmetric and antisymmetric parts, that is, into the rate-of-strain tensor $\boldsymbol{\mathsf{S}} = (\boldsymbol{\mathsf{A}} + \boldsymbol{\mathsf{A}}^t)/2$ and vorticity tensor ${\boldsymbol{\mathsf{\Omega}}} = (\boldsymbol{\mathsf{A}}-\boldsymbol{\mathsf{A}}^t)/2$, where $\boldsymbol{\mathsf{A}}^t$ is the transpose of $\boldsymbol{\mathsf{A}}$, the helicity flux can be decomposed into six subfluxes:

(2.11)\begin{equation} \varPi^{H,\ell} = \varPi_{s, SS}^{{H},\ell} + \varPi_{s,\varOmega\varOmega}^{{H},\ell} + \varPi_{s,S\varOmega}^{{H},\ell} + \varPi_{m,SS}^{{H},\ell} + \varPi_{m,\varOmega\varOmega}^{{H},\ell} + \varPi_{m,S\varOmega}^{{H},\ell} , \end{equation}

where the single-scale terms are

(2.12)\begin{gather} \varPi_{s,SS}^{H,\ell} ={-} 2 \ell^2 \bar{S}^\ell_{\omega,ij} \bar{S}_{ik}^\ell \bar{S}_{jk}^\ell ={-} 2 \ell^2 {\rm tr}\left\{ {(\bar{\boldsymbol{\mathsf{S}}}^\ell_\omega)^t \bar{\boldsymbol{\mathsf{S}}}^\ell (\bar{\boldsymbol{\mathsf{S}}}^\ell)^t} \right\} , \end{gather}

(2.13)\begin{gather}\varPi_{s,\varOmega\varOmega}^{H,\ell} ={-} 2 \ell^2 \bar{S}^\ell_{\omega,ij} \bar{\varOmega}_{ik}^\ell \bar{\varOmega}_{jk}^\ell ={-}2 \ell^2 {\rm tr}\left\{ {(\bar{\boldsymbol{\mathsf{S}}}^\ell_\omega)^t \bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell (\bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell)^t} \right\} , \end{gather}

(2.14)\begin{gather}\varPi_{s, S\varOmega}^{H,\ell} ={-} 2 \ell^2 \bar{S}^\ell_{\omega,ij} \left ( \bar{S}_{ik}^\ell \bar{\varOmega}_{jk}^\ell - \bar{\varOmega}_{ik}^\ell \bar{S}_{jk}^\ell \right ) ={-}4 \ell^2 {\rm tr}\left\{ {(\bar{\boldsymbol{\mathsf{S}}}^\ell_\omega)^t \bar{\boldsymbol{\mathsf{S}}}^\ell (\bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell)^t} \right\} , \end{gather}

and ${\rm tr}\left \{ {{\cdot }} \right \}$ denotes the trace. We purposefully retained the transposition operation here also for symmetric tensors to show what the approach needs to be when the tensors involved are not symmetric. Similarly, the multi-scale terms are

(2.15)\begin{align} \varPi_{m,SS}^{H,\ell} &={-} 2 \int_{0}^{\ell^2}\, \mathrm{d}\theta\ \bar{S}^\ell_{\omega,ij} \tau^\phi\left( \bar{S}_{ik}^{\sqrt{\theta}}, \bar{S}_{kj}^{\sqrt{\theta}} \right) , \end{align}

(2.16)\begin{align} \varPi_{m,\varOmega\varOmega}^{H,\ell} &= 2 \int_{0}^{\ell^2} \,\mathrm{d}\theta\ \bar{S}^\ell_{\omega,ij} \tau^\phi\left( \bar{\varOmega}_{ik}^{\sqrt{\theta}}, \bar{\varOmega}_{kj}^{\sqrt{\theta}} \right), \end{align}

(2.17)\begin{align} \varPi_{m,S\varOmega}^{H,\ell} &={-} 2 \int_{0}^{\ell^2} \,\mathrm{d}\theta\ \bar{S}^\ell_{\omega,ij} \left[ \tau^\phi\left( \bar{S}_{ik}^{\sqrt{\theta}}, \bar{\varOmega}_{jk}^{\sqrt{\theta}} \right) + \tau^\phi\left( \bar{\varOmega}_{ik}^{\sqrt{\theta}}, \bar{S}_{jk}^{\sqrt{\theta}} \right) \right] \nonumber\\ &={-} 4 \int_{0}^{\ell^2} \,\mathrm{d}\theta\ \bar{S}^\ell_{\omega,ij} \tau^\phi\left( \bar{S}_{ik}^{\sqrt{\theta}}, \bar{\varOmega}_{jk}^{\sqrt{\theta}} \right) . \end{align}

We recall that $\langle \varPi ^{H,\ell }_{s,\varOmega \varOmega }\rangle$, the spatial average of the contribution to the helicity flux due to coupling of resolved-scale vorticity strain with resolved-scale vorticity, vanishes,

(2.18)\begin{equation} \langle \varPi^{H,\ell}_{s,\varOmega\varOmega}\rangle ={-} \frac{\ell^2}{4} \left \langle \left( \partial_j \bar{\omega}^\ell_i + \partial_i \bar{\omega}^\ell_j\right)\bar{\omega}^\ell_i \bar{\omega}^\ell_j \right \rangle ={-} \frac{\ell^2}{4} \left \langle \partial_j (\bar{\omega}^\ell_i \bar{\omega}^\ell_i \bar{\omega}^\ell_j) \right \rangle = 0 , \end{equation}

due to periodic boundary conditions and the divergence-free nature of the vorticity field, as previously discussed by Eyink (Reference Eyink2006) in the context of a multi-scale gradient expansion of the SGS stress tensor.

The physics encoded in these transfer terms may be understood in terms of three effects: (i) ‘vortex flattening’ – compression and stretching of a vortex tube into a vortex sheet by large-scale straining motion, with the principal axes of the vorticity deformation tensor $S_\omega$ aligning with that of the strain-rate tensor at smaller scale, see (2.12) and (2.15); (ii) ‘vortex twisting’ – a twisting of small-scale vortex tubes by large-scale differential vorticity into thinner tubes consisting of helical vortex lines, and subsequent small-scale alignment between the resulting vorticity vectors and the extensile stress generated thereby (Eyink Reference Eyink2006), see (2.14) and (2.17); and (iii) ‘vortex entangling’ – twisting of entangled vortex lines, see (2.13) and (2.16). Interpreting helicity as the correlation between velocity and vorticity, a change in this correlation (or alignment) across scales occurs by vorticity deformation through straining motions or differential vorticity. This results in decorrelation at large scales and an increase in small-scale correlation.

The interpretation of vortex entangling, or twisting of entangled vortex lines, relies on the topological interpretation of helicity in terms of link, twist and writhe of vortex tubes. The terms we interpret in this way do not involve strain-rate tensors; they only involve vorticity and differential vorticity. As such they describe vortex–vortex interactions driven by a twisting and stretching of a bundle of vortex tubes by differential vorticity. If we consider a single bundle of vortex tubes, writhe is a global (or large-scale) quantity in the sense that one needs to consider the entire bundle to describe it. In contrast, twist is a local (or small-scale) concept that describes the winding of the entangled vortex tubes that constitute the bundle around each other (Scheeler et al. Reference Scheeler, van Rees, Kedia, Kleckner and Irvine2017). What we loosely describe as ‘twisting of entangled vortex lines’ can be thought of as a writhe-to-twist transfer of helicity, by deformation of the vortex bundle, see e.g. Scheeler et al. (Reference Scheeler, van Rees, Kedia, Kleckner and Irvine2017). This would necessarily be a multi-scale process, and indeed we see in (2.18) that the single-scale term $\langle \varPi ^{H,\ell }_{s,\varOmega \varOmega }\rangle$ vanishes in mean.

2.2. An exact Betchov-type relation for the helicity flux

In homogeneous turbulence, the Betchov (Reference Betchov1956) relation is an exact expression connecting the contributions associated with vortex stretching and strain self-amplification to the mean energy flux across scales. Here we show that there is an analogous exact expression relating two (single-scale) mean helicity subfluxes: $3\langle \varPi ^{H, \ell }_{s,SS} \rangle = \langle \varPi ^{H, \ell }_{s,S\varOmega } \rangle$. These subfluxes are associated with vortex flattening, $\langle \varPi ^{H, \ell }_{s,SS} \rangle$, and vortex twisting, $\langle \varPi ^{H, \ell }_{s,S\varOmega } \rangle$. Written in terms of the definitions given in (2.12) and (2.14), this expression reads

(2.19)\begin{equation} 3\, \left \langle {\rm tr}\left\{ {\bar{\boldsymbol{\mathsf{S}}}^\ell_\omega \bar{\boldsymbol{\mathsf{S}}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell} \right\} \right \rangle = 2\, \left \langle {\rm tr}\left\{ {\bar{\boldsymbol{\mathsf{S}}}^\ell_\omega \bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell} \right\} \right\rangle . \end{equation}

Before proceeding to a proof of this expression, we point out that it is a relation between the respective traces of two tensors, that is, geometric objects. Consequently, (2.19) is independent of the filter kernel and indeed also holds for the un-filtered case. Here, we embed the derivation within the framework developed in § 2.1 so as to remain focused on the physical interpretation of the terms in (2.19) as subfluxes across scales.

The main steps in a proof of (2.19) are now summarised. Following an argument analogous to that used in proving the Betchov (Reference Betchov1956) relation for the energy flux, and using tensor symmetry properties and (2.18), one obtains (Eyink Reference Eyink2006)

(2.20) \begin{equation} \left \langle {\rm tr}\left\{ {\bar{\boldsymbol{\mathsf{S}}}^\ell_\omega \bar{\boldsymbol{\mathsf{S}}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell} \right\} \right \rangle ={-} \left \langle {\rm tr}\left\{ {\bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell_\omega (\bar{\boldsymbol{\mathsf{S}}}^\ell \bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell + \bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell) } \right\} \right \rangle ={-} 2\left \langle {\rm tr}\left\{ {\bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell_\omega \bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell} \right\} \right \rangle , \end{equation}

where $\varOmega _\omega$ is the antisymmetric part of the vorticity gradient tensor. This yields

(2.21)\begin{equation} \frac{1}{2}\left \langle {\rm tr}\left\{ { \boldsymbol{\nabla} \bar{\boldsymbol{\omega}}^\ell \left (\boldsymbol{\nabla} \bar{\boldsymbol{u}}^\ell \right )^t \left[ \boldsymbol{\nabla} \bar{\boldsymbol{u}}^\ell + \left(\boldsymbol{\nabla} \bar{\boldsymbol{u}}^\ell \right )^t \right] } \right\}\right \rangle = \left \langle {\rm tr}\left\{ {\frac{3}{2} \,\bar{\boldsymbol{\mathsf{S}}}^\ell_\omega \bar{\boldsymbol{\mathsf{S}}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell - \,\bar{\boldsymbol{\mathsf{S}}}^\ell_\omega \bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell} \right\} \right\rangle . \end{equation}

Thus, showing that the left-hand side of this expression vanishes will prove the Betchov relation for the helicity flux, (2.19). To do so, we express the left-hand side of (2.21) using the chain rule and in index notation

(2.22) $$\begin{align} \left \langle \partial_j \bar{\omega}_i^\ell \partial_j \bar{u}_k^\ell \bar{S}_{ki}^\ell \right \rangle &= \left \langle \partial_j \left [\bar{\omega}_i^\ell \partial_j \bar{u}_k^\ell \bar{S}_{ki}^\ell \right] \right \rangle - \left \langle \bar{\omega}_i^\ell \partial_j \partial_j \bar{u}_k^\ell \bar{S}_{ki}^\ell \right \rangle\nonumber\\ &\quad - \left \langle \bar{\omega}_i^\ell \bar{S}_{kj}^\ell \partial_j\bar{S}_{ki}^\ell \right \rangle - \left \langle \bar{\omega}_i^\ell \bar{\varOmega}_{kj}^\ell \partial_j\bar{S}_{ki}^\ell \right \rangle . \end{align}$$

The first term on the right-hand side of this expression vanishes making use of periodic boundary conditions. Using incompressibility and integration by parts it can be shown that the last term also vanishes. The two remaining terms cancel out, which is shown by similar arguments and using the properties of the Levi-Civita tensor. This completes the proof.

The mean single-scale terms also arise as the first-order contribution in a multi-scale expansion of the SGS stress tensor (Eyink Reference Eyink2006), where (2.20) is used to deduce that the full vorticity gradient, not only either its symmetric or antisymmetric component, is involved in the helicity flux across scales. In consequence, (2.19) and (2.20) assert that the mean transfers involving the symmetric or the antisymmetric parts of the vorticity gradient can be related to one another, and thus the single-scale contribution to the mean helicity flux can be written as

(2.23) \begin{equation} \left \langle \varPi^{H,\ell}_s \right \rangle ={-}8\ell^2\, \left \langle {\rm tr}\left\{ {\bar{\boldsymbol{\mathsf{S}}}^\ell_\omega \bar{\boldsymbol{\mathsf{S}}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell} \right\} \right \rangle ={-}\frac{16}{3} \ell^2\, \left \langle {\rm tr}\left\{ {\bar{\boldsymbol{\mathsf{S}}}^\ell_\omega \bar{{\boldsymbol{\mathsf{\Omega}}}}^\ell \bar{\boldsymbol{\mathsf{S}}}^\ell} \right\} \right\rangle . \end{equation}

3. Numerical details and data

Data has been generated by direct numerical simulation of the incompressible 3-D Navier–Stokes equations (2.1) and (2.2) on a triply periodic domain of size $L_{box} = 2{\rm \pi}$ in each direction, where the forcing $\boldsymbol {f}$ is a random Gaussian process with zero mean, fully helical $\boldsymbol {f} = \boldsymbol {f}^+$, and active in the wavenumber band $k \in [0.5,2.4]$. The spatial discretisation is implemented through the standard, fully dealiased pseudospectral method with $1024$ collocation points in each direction. Further details and mean values of key observables are summarised in table 1.

Table 1. Simulation parameters and key observables, where $N$ is the number of collocation points in each coordinate, $E$ the (mean) total kinetic energy, $\nu$ the kinematic viscosity, $\varepsilon$ the mean energy dissipation rate, $\varepsilon _H$ the mean helicity dissipation rate, $L = (3 {\rm \pi}/4 E) \int _0^{k_{max}} \,\mathrm {d}{k} \ E(k)/k$ the integral scale, $\tau = L/\sqrt {2E/3}$ the large-eddy turnover time, ${Re}_\lambda$ the Taylor-scale Reynolds number, $\eta = (\nu ^3/\varepsilon )^{1/4}$ the Kolmogorov microscale, $k_{max}$ the largest wavenumber after de-aliasing, $\Delta t$ the sampling interval which is calculated from the length of the averaging interval divided by the number of equispaced snapshots, and ‘No.’ the number of snapshots. The data corresponds to run 22 of Sahoo et al. (Reference Sahoo, Alexakis and Biferale2017). It is available for download using the SMART-Turb portal http://smart-turb.roma2.infn.it.

Figure 1(a) presents the time series of the total kinetic energy per unit volume, $E(t)$. Time-averaged kinetic energy spectra of positively and negatively helical fluctuations, $E^\pm (k) = \langle \tfrac {1}{2}\sum _{k \leq |\boldsymbol {k}| < k + 1} |\hat {\boldsymbol {u}}^\pm (\boldsymbol {k})|^2 \rangle$ and the total energy spectrum $E(k) = E^+(k) + E^-(k)$, are shown in Kolmogorov-compensated form in figure 1(b). As can be seen by comparison of $E^+(k)$ and $E^-(k)$, the large-scale velocity-field fluctuations are dominantly positively helical, which is a consequence of the forcing. Decreasing in scale, we observe that negatively helical fluctuations increase in amplitude, and approximate equipartition between $E^+(k)$ and $E^-(k)$ is reached for ${k \eta \geq 0.2}$. That is, a helically forced turbulent flow, where mirror symmetry is broken at and close to the forcing scale, restores mirror symmetry at smaller scales through nonlinear interactions (Chen et al. Reference Chen, Chen and Eyink2003a; Deusebio & Lindborg Reference Deusebio and Lindborg2014; Kessar et al. Reference Kessar, Plunian, Stepanov and Balarac2015).

Figure 1. (a) Time evolution of the total energy normalised by its mean value, $E$. Time is given in units of large-eddy turnover time $\tau$. The red dots correspond to the sampled velocity-field configurations. (b) Time-averaged energy spectra in Kolmogorov-compensated form. The grey-shaded area indicates the forcing range. The dashed line indicates a Kolmogorov constant $C_K \approx 1.6$.

4. Numerical results for mean subfluxes and fluctuations

Figure 2 shows the total helicity flux and all subfluxes, normalised by the total helicity dissipation rate $\varepsilon _H$. As can be seen in the figure, the term $\langle \varPi ^{H,\ell }_{s,\varOmega \varOmega }\rangle$ is identically zero, which must be the case according to (2.18). Moreover, the helicity Betchov relation (2.19) derived here is satisfied as it must be – the terms $\langle \varPi ^{H,\ell }_{s,S\varOmega }\rangle$ and $3\,\langle \varPi ^{H,\ell }_{s,SS}\rangle$ are visually indistinguishable, with a relative error between them of order $10^{-6}$ (not shown). A few further observations can be made from the data. The non-vanishing multi-scale terms, $\langle \varPi ^H_{m,S\varOmega } \rangle$, $\langle \varPi ^H_{m,SS} \rangle$ and $\langle \varPi ^H_{m,\varOmega \varOmega } \rangle$ are comparable in magnitude across all scales. They are approximately scale-independent in the interval $10^{-2} \leq k\eta \le 10^{-1}$, with each accounting for approximately 15–20 % of the total helicity flux in this range of scales. Even though clear plateaux are not present for the two non-vanishing single-scale terms, $\langle \varPi ^H_{s,S\varOmega } \rangle$ and $\langle \varPi ^H_{s,SS} \rangle$, one could tentatively extrapolate that at higher $Re$, approximately $30\,\%$ of the mean flux originates from scale-local vortex twisting and $10\,\%$ from vortex flattening. That is, the multi-scale contributions amount to 50–60 % and the scale-local contributions to 40–50 % of the total helicity flux across scales, at least for this particular simulation.

Figure 2. Decomposed helicity fluxes normalised with the mean helicity dissipation rate $\varepsilon _H$. Filled markers correspond to single-scale contributions while empty symbols are related to multi-scale contributions. The error bars indicate one standard error. The subflux $\langle \varPi _{s,S\varOmega }^{H,\ell }\rangle$ has been superposed with $3\langle \varPi _{s,SS}^{H,\ell }\rangle$ in order to highlight the Betchov-type relation (2.19).

Having discussed the mean subfluxes, we now consider the fluctuations of each subflux term, in order to quantify the level of fluctuations in each term and the presence and magnitude of helicity backscatter. Figure 3 presents standardised probability density functions (p.d.f.s) of all helicity subfluxes at ${k \eta = 5.5 \times 10^{-2}}$, which is approximately in the inertial range. These p.d.f.s have wide tails, and are strongly non-Gaussian. Single- and multi-scale terms all have strong fluctuations between 60 and 75 standard deviations. Interestingly, the subflux term $\varPi _{s,\varOmega \varOmega }^{H,\ell }$, which necessarily vanishes in mean (see (2.18)), has the strongest fluctuations (i.e. is the most intermittent).

Figure 3. Standardised p.d.f.s of helicity subfluxes $\varPi _X^{H,\ell }$, where $X$ refers to the subflux identifier, for (a) single-scale and (b) multi-scale contributions; $\sigma _X$ denotes the standard deviation of each respective term. Values of $\sigma _X$ are provided in table 2.

Table 2. Values of variance, skewness and flatness for the subflux p.d.f.s at $k\eta = 0.055$ shown in figure 3.

Values of the variance, skewness and flatness for each of the p.d.f.s shown in figure 3 are provided in table 2. All p.d.f.s are more symmetric than for the kinetic energy fluxes (Johnson Reference Johnson2021) (not shown) with generally low skewness values. The p.d.f.s of $\varPi _{s}^{H,\ell }$, $\varPi _{s,S\varOmega }^{H,\ell }$, $\varPi _{m}^{H,\ell }$ and $\varPi _{m,S\varOmega }^{H,\ell }$ are slightly positively skewed while the p.d.f.s of $\varPi _{s,SS}^{H,\ell }$, $\varPi _{m, SS}^{H,\ell }$ and $\varPi _{m,\varOmega \varOmega }^{H,\ell }$ are slighly negatively skewed. The symmetry is more pronounced in the single-scale rather than the multi-scale terms, as can be seen by comparison of the left and right panels of figure 3, and by comparison of the skewness values reported in table 2. As all averaged fluxes (except $\langle \varPi _{s, \varOmega \varOmega }^{H,\ell } \rangle$ which is zero) transfer positive helicity from large to small scales, symmetry in the p.d.f.s indicates strong backscatter of positive helicity, or forward scatter of negative helicity. The p.d.f.s become even broader with decreasing filter scale (not shown). A comparison between the p.d.f.s of $\varPi ^{H, \ell }$ and the alternative description based on SGS stresses related to vortex stretching, $\tilde \varPi ^{H, \ell }$, has been carried out by Yan et al. (Reference Yan, Li, Yu, Wang and Chen2020), indicating more intense backscatter in the latter compared with the former. Adding or removing a total gradient can strongly reduce the negative tail of the SGS energy transfer (Vela-Martín Reference Vela-Martín2022), and the same may apply to the helicity flux.