2.1 Helical mode decomposition

Each velocity Fourier coefficient is split into helical modes (Craya Reference Craya1958; Herring Reference Herring1974; Cambon & Jacquin Reference Cambon and Jacquin1989; Waleffe Reference Waleffe1992; Lessinnes et al. Reference Lessinnes, Plunian, Stepanov and Carati2011),

(2.4)$$\begin{eqnarray}\boldsymbol{u}(\boldsymbol{k})=u^{+}(\boldsymbol{k})\boldsymbol{h}^{+}(\boldsymbol{k})+u^{-}(\boldsymbol{k})\boldsymbol{h}^{-}(\boldsymbol{k})\equiv \boldsymbol{u}^{+}(\boldsymbol{k})+\boldsymbol{u}^{-}(\boldsymbol{k}),\end{eqnarray}$$

where $u^{\pm }(\boldsymbol{k})$ are complex scalars and $\boldsymbol{h}^{\pm }(\boldsymbol{k})$ are the two eigenvectors of the curl operator, satisfying $\text{i}\boldsymbol{k}\times \boldsymbol{h}^{\pm }(\boldsymbol{k})=\pm k\boldsymbol{h}^{\pm }(\boldsymbol{k})$. The latter are defined as

(2.5)$$\begin{eqnarray}\boldsymbol{h}^{\pm }(\boldsymbol{k})=\frac{1}{\sqrt{2}}\frac{(\boldsymbol{z}_{\boldsymbol{k}}\times \boldsymbol{k})\times \boldsymbol{k}}{k|\boldsymbol{z}_{\boldsymbol{k}}\times \boldsymbol{k}|}\pm \frac{\text{i}}{\sqrt{2}}\frac{\boldsymbol{z}_{\boldsymbol{k}}\times \boldsymbol{k}}{|\boldsymbol{z}_{\boldsymbol{k}}\times \boldsymbol{k}|},\end{eqnarray}$$

where at each time step the vector $\boldsymbol{z}_{\boldsymbol{k}}$ is generated randomly for each wavenumber $\boldsymbol{k}$, keeping $\boldsymbol{z}_{\boldsymbol{k}}\times \boldsymbol{k}\neq 0$ (Waleffe Reference Waleffe1992). Note that $\boldsymbol{h}^{+}(\boldsymbol{k})$ and $\boldsymbol{h}^{-}(\boldsymbol{k})$ are complex conjugate of each other, $\boldsymbol{h}^{\pm }(\boldsymbol{k})=\boldsymbol{h}^{\mp }(\boldsymbol{k})^{\ast }$. They are vectors of unit norm, $|\boldsymbol{h}^{\pm }(\boldsymbol{k})|^{2}=\boldsymbol{h}^{+}(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{h}^{-}(\boldsymbol{k})=1$, and the scalar product of each of them with itself is equal to zero, $\boldsymbol{h}^{+}(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{h}^{+}(\boldsymbol{k})=\boldsymbol{h}^{-}(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{h}^{-}(\boldsymbol{k})=0$. Then the helical modes $u^{\pm }(\boldsymbol{k})$ can be derived from (2.4) according to

(2.6)$$\begin{eqnarray}u^{\pm }(\boldsymbol{k})=\boldsymbol{u}(\boldsymbol{k})\boldsymbol{\cdot }\boldsymbol{h}^{\mp }(\boldsymbol{k}).\end{eqnarray}$$

The vorticity can also be expressed in terms of helical modes,

(2.7)$$\begin{eqnarray}\boldsymbol{w}(\boldsymbol{k})=k(u^{+}(\boldsymbol{k})\boldsymbol{h}^{+}(\boldsymbol{k})-u^{-}(\boldsymbol{k})\boldsymbol{h}^{-}(\boldsymbol{k})).\end{eqnarray}$$

Then, the energy and helicity take the form

(2.8a,b)$$\begin{eqnarray}\displaystyle E(\boldsymbol{k})=E^{+}(\boldsymbol{k})+E^{-}(\boldsymbol{k}),\quad H(\boldsymbol{k})=k(E^{+}(\boldsymbol{k})-E^{-}(\boldsymbol{k})), & & \displaystyle\end{eqnarray}$$

with $E^{\pm }(\boldsymbol{k})=\frac{1}{2}|u^{\pm }(\boldsymbol{k})|^{2}$.

2.2 Dynamical forcing

Looking for a flow forcing of the form

(2.9)$$\begin{eqnarray}\boldsymbol{f}(\boldsymbol{k})=c^{+}(\boldsymbol{k})u^{+}(\boldsymbol{k})\boldsymbol{h}^{+}(\boldsymbol{k})+c^{-}(\boldsymbol{k})u^{-}(\boldsymbol{k})\boldsymbol{h}^{-}(\boldsymbol{k}),\end{eqnarray}$$

from (2.3), (2.4) and (2.7), we obtain

(2.10)$$\begin{eqnarray}c^{\pm }(\boldsymbol{k})=\frac{\unicode[STIX]{x1D700}_{E}(\boldsymbol{k})\pm \unicode[STIX]{x1D700}_{H}(\boldsymbol{k})/k}{4E^{\pm }(\boldsymbol{k})}.\end{eqnarray}$$

As our purpose is to study strongly helical turbulence, one type of helical mode, here $\boldsymbol{u}^{-}(\boldsymbol{k})$, is expected to have a small energy $E^{-}(\boldsymbol{k})$, possibly leading to high values of $c^{-}(\boldsymbol{k})$. To avoid using a very small time step, which would be numerically intractable, we integrate analytically the equations $\unicode[STIX]{x2202}u^{\pm }(\boldsymbol{k},t)/\unicode[STIX]{x2202}t=c^{\pm }(\boldsymbol{k},t)u^{\pm }(\boldsymbol{k},t)$, leading to

(2.11)$$\begin{eqnarray}u_{0}^{\pm }(\boldsymbol{k},t+\unicode[STIX]{x0394}t)=\text{e}^{\text{i}\,arg\,u^{\pm }(\boldsymbol{k},t)}\left(\left[\unicode[STIX]{x1D700}_{E}(\boldsymbol{k},t)\pm \frac{\unicode[STIX]{x1D700}_{H}(\boldsymbol{k},t)}{k}\right]\unicode[STIX]{x0394}t+|u^{\pm }(\boldsymbol{k},t)|^{2}\right)^{1/2}.\end{eqnarray}$$

Then $\boldsymbol{u}_{0}(\boldsymbol{k},t+\unicode[STIX]{x0394}t)=u_{0}^{+}(\boldsymbol{k},t+\unicode[STIX]{x0394}t)\boldsymbol{h}^{+}(\boldsymbol{k})+u_{0}^{-}(\boldsymbol{k},t+\unicode[STIX]{x0394}t)\boldsymbol{h}^{-}(\boldsymbol{k})$ is taken as an initial condition in the calculation of the NSE at time step $t+\unicode[STIX]{x0394}t$. To achieve the desired state of high helicity, we use the following asymptotic quenching:

(2.12)$$\begin{eqnarray}\unicode[STIX]{x1D700}_{H}(\boldsymbol{k},t)=\tilde{\unicode[STIX]{x1D700}}_{H}\left(1-\frac{|E^{+}(\boldsymbol{k},t)-E^{-}(\boldsymbol{k},t)|}{E^{+}(\boldsymbol{k},t)+E^{-}(\boldsymbol{k},t)}\right),\end{eqnarray}$$

where $\tilde{\unicode[STIX]{x1D700}}_{H}$ is the strength of helicity injection rate.

3.1 Integral quantities

The simulations are performed with the pseudo-spectral code TARANG (Verma et al. Reference Verma, Chatterjee, Yadav, Paul, Chandra and Samtaney2013; Stepanov et al. Reference Stepanov, Teimurazov, Titov, Verma, Barman, Kumar and Plunian2018; Teimurazov et al. Reference Teimurazov, Stepanov, Verma, Barman, Kumar and Sadhukhan2018), in a triply periodic domain of size $(2\unicode[STIX]{x03C0})^{3}$. Energy is injected at the permanent rate $\unicode[STIX]{x1D700}_{E}(\boldsymbol{k})=0.2$, over $|\boldsymbol{k}|=k_{E}\in ]\!9,10\!]$. The viscosity and large-scale friction values are taken as $\unicode[STIX]{x1D708}=1.5\times 10^{-3}$ and $\unicode[STIX]{x1D707}=5\times 10^{-2}$, for which a resolution of $512^{3}$ is sufficient. The simulations differ by the values chosen for $\tilde{\unicode[STIX]{x1D700}}_{H}$, which is applied over $|\boldsymbol{k}|=k_{H}\in [1,10^{2}]$. Some variations of the parameters $\unicode[STIX]{x1D708}$ and $k_{H}$ will be discussed in § 4.3.

Figure 1.

Total energy $E_{tot}$, total helicity $H_{tot}$, energy dissipation rates $D_{E}^{\unicode[STIX]{x1D708}},D_{E}^{\unicode[STIX]{x1D707}}$ and $D_{E}^{\unicode[STIX]{x1D708}}+D_{E}^{\unicode[STIX]{x1D707}}$, and helicity dissipation rates $D_{H}^{\unicode[STIX]{x1D708}},D_{H}^{\unicode[STIX]{x1D707}}$ and $D_{H}^{\unicode[STIX]{x1D708}}+D_{H}^{\unicode[STIX]{x1D707}}$, plotted versus time for $\unicode[STIX]{x1D700}_{E}=0.2$ and $\tilde{\unicode[STIX]{x1D700}}_{H}=25$. The time range $t\in [180,280]$ over which the statistics are calculated is indicated, for each curve, by a thicker line.

In figure 1, several integral quantities are plotted versus time for $\tilde{\unicode[STIX]{x1D700}}_{H}=25$. We define $X_{tot}=\int X(\boldsymbol{k})\,\text{d}\boldsymbol{k}$, with $X=E$ for total energy and $X=H$ for total helicity. The molecular dissipation rate of $X$ is given by $D_{X}^{\unicode[STIX]{x1D708}}=2\unicode[STIX]{x1D708}\!\int \!k^{2}X(\boldsymbol{k})\,\text{d}\boldsymbol{k}$, and the friction dissipation rate by $D_{X}^{\unicode[STIX]{x1D707}}=2\unicode[STIX]{x1D707}\!\int \!k^{-4}X(\boldsymbol{k})\,\text{d}\boldsymbol{k}$. In figure 1 we see that $H_{tot}$ varies on a time scale much smaller than $E_{tot}$, suggesting a turbulence governed by helicity rather than energy. On average, we have $D_{E}^{\unicode[STIX]{x1D708}}+D_{E}^{\unicode[STIX]{x1D707}}=0.2$, which corresponds to the total injection rate of energy $\unicode[STIX]{x1D700}_{E}=\int \!\unicode[STIX]{x1D700}_{E}(\boldsymbol{k})\,\text{d}\boldsymbol{k}$. Both $D_{E}^{\unicode[STIX]{x1D707}}$ and $D_{H}^{\unicode[STIX]{x1D707}}$ are mostly identical. This is because friction is dominated by the largest scale $k=1$ where turbulence is close to maximum helicity, implying $H(k=1)\approx kE(k=1)$. In the rest of the paper, the spectra and fluxes are obtained by averaging over frames spread over 50 to 100 time units. For $\tilde{\unicode[STIX]{x1D700}}_{H}=25$, this time window is represented in figure 1 by the thick part of the curves.

An estimate of the Taylor-microscale Reynolds number, $R_{\unicode[STIX]{x1D706}}=u^{\prime }\unicode[STIX]{x1D706}/\unicode[STIX]{x1D708}$, can be obtained with $u^{\prime }=\sqrt{2\bar{E}_{tot}/3}$ and $\unicode[STIX]{x1D706}=\sqrt{15\unicode[STIX]{x1D708}{u^{\prime }}^{2}/\unicode[STIX]{x1D700}_{E}}$ (Pope Reference Pope2000). From figure 1 the mean kinetic energy is $\bar{E}_{tot}\approx 1.35$, which gives $u^{\prime }\approx 0.95$ and $\unicode[STIX]{x1D706}\approx 0.32$, thus leading to $R_{\unicode[STIX]{x1D706}}\approx 200$, which compares well with other numerical simulations of the same resolution (Okamoto et al. Reference Okamoto, Yoshimatsu, Schneider, Farge and Kaneda2007).

3.2 Energy and helicity spectra and fluxes

Taking the dot product of (2.2) with $\boldsymbol{x}(\boldsymbol{k})^{\ast }$, we obtain the equations for modal energy and helicity,

(3.1)$$\begin{eqnarray}\displaystyle \unicode[STIX]{x2202}_{t}X(\boldsymbol{k}) & = & \displaystyle -\text{Re}\{\boldsymbol{x}(\boldsymbol{k})^{\ast }\boldsymbol{\cdot }{\mathcal{F}}[(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}](\boldsymbol{k})\}\nonumber\\ \displaystyle & & \displaystyle -\,2(\unicode[STIX]{x1D708}k^{2}+\unicode[STIX]{x1D707}k^{-4})X(\boldsymbol{k})+\unicode[STIX]{x1D700}_{X}(\boldsymbol{k}),\end{eqnarray}$$

where $(X,\boldsymbol{x})=(E,\boldsymbol{u})$ or $(X,\boldsymbol{x})=(H,\boldsymbol{w})$, ${\mathcal{F}}$ standing for Fourier transform, and $\unicode[STIX]{x1D700}_{X}(\boldsymbol{k})$ satisfying (2.3). Rewriting (3.1) for $\boldsymbol{k}^{\prime }$, and taking the sum over $\boldsymbol{k}^{\prime }$ such that $k<|\boldsymbol{k}^{\prime }|\leqslant k+\text{d}k$ (Verma Reference Verma2019), leads to

(3.2)$$\begin{eqnarray}\unicode[STIX]{x2202}_{t}X(k)=-\unicode[STIX]{x2202}_{k}\unicode[STIX]{x1D6F1}_{X}(k)-D_{X}(k)+\unicode[STIX]{x1D700}_{X}(k),\end{eqnarray}$$

with

(3.3)$$\begin{eqnarray}\displaystyle & \displaystyle X(k)\,\text{d}k=\mathop{\sum }_{k<|\boldsymbol{k}^{\prime }|\leqslant k+\text{d}k}X(\boldsymbol{k}^{\prime }), & \displaystyle\end{eqnarray}$$

(3.4)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F1}_{X}(k)=\mathop{\sum }_{|\boldsymbol{k}^{\prime }|\leqslant k}\text{Re}\{\boldsymbol{x}(\boldsymbol{k}^{\prime })^{\ast }\boldsymbol{\cdot }{\mathcal{F}}[(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}](\boldsymbol{k}^{\prime })\}, & \displaystyle\end{eqnarray}$$

(3.5)$$\begin{eqnarray}\displaystyle & \displaystyle D_{X}(k)\,\text{d}k=\mathop{\sum }_{k<|\boldsymbol{k}^{\prime }|\leqslant k+\text{d}k}2(\unicode[STIX]{x1D708}{k^{\prime }}^{2}+\unicode[STIX]{x1D707}{k^{\prime }}^{-4})X(\boldsymbol{k}^{\prime }), & \displaystyle\end{eqnarray}$$

(3.6)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D700}_{X}(k)\,\text{d}k=\mathop{\sum }_{k<|\boldsymbol{k}^{\prime }|\leqslant k+\text{d}k}\unicode[STIX]{x1D700}_{X}(\boldsymbol{k}^{\prime }). & \displaystyle\end{eqnarray}$$

The energy spectral density $E(k)$ and flux $\unicode[STIX]{x1D6F1}_{E}(k)$ in figure 2, and the helicity spectral density $H(k)$ and flux $\unicode[STIX]{x1D6F1}_{H}(k)$ in figure 3, are plotted for $\tilde{\unicode[STIX]{x1D700}}_{H}\in S$, with $S=\{1;5;7;8;10;13;20;25\}$. For $\tilde{\unicode[STIX]{x1D700}}_{H}=1$, the fluxes of energy and helicity are non-zero only for $k\geqslant 10$. As both fluxes are positive, they correspond to forward cascades of energy and helicity, as expected in three-dimensional turbulence. In figure 3, the shape of the helicity flux obtained for $\tilde{\unicode[STIX]{x1D700}}_{H}=1$, with a maximum at $k\approx 20$, is similar to the one obtained by Kessar et al. (Reference Kessar, Plunian, Stepanov and Balarac2015) in which helicity was injected only at scales smaller than the energy injection scale.

With the increase of $\tilde{\unicode[STIX]{x1D700}}_{H}$ from 1 to 25, the energy flux is shifted downwards. It then becomes negative in the infrared domain $k\leqslant 9$, leading to an inverse cascade of energy. With the increase of $\tilde{\unicode[STIX]{x1D700}}_{H}$, the scale at which the helicity flux is maximum is also shifted, towards the energy injection scale. For $k\geqslant 10$ the helicity flux is positive, corresponding to a forward cascade of helicity. Such a dual cascade is consistent with the homochiral phenomenology. For $\tilde{\unicode[STIX]{x1D700}}_{H}=25$, in the infrared domain $k\leqslant 9$, the energy spectrum obeys a $k^{-5/3}$ scaling law. For $k\geqslant 10$ the energy spectral slope is approximately $E(k)\propto k^{-4.3}$, which is steeper than the $E(k)\propto k^{-7/3}$ predicted by the homochiral phenomenology. Most probably this is due to the fact that, at scales smaller than the energy injection scales, the viscous dissipation cannot be neglected, implying that an inertial range can hardly be identified.

Figure 2.

Energy spectra and fluxes (inset). The energy injection rate $\unicode[STIX]{x1D700}_{E}=0.2$ is applied at $k_{E}\in ]\!9,10\!]$. The helicity injection rate $\tilde{\unicode[STIX]{x1D700}}_{H}\in \{1;5;7;8;10;13;20;25\}$ is applied at $k_{H}\in [1,10^{2}]$.

Figure 3.

Helicity spectra and fluxes (inset) for the same parameters as in figure 2.

Figure 4.

Deviation to maximum chirality, $1-H^{r}(k)$, for the same parameters as in figure 2.

A measure of chirality is given by the relative helicity $H^{r}(k)=H(k)/kE(k)$, which satisfies the realizability condition $0\leqslant |H^{r}(k)|\leqslant 1$. A non-helical turbulence corresponds to $|H^{r}(k)|=0$. A maximum helical turbulence corresponds to $|H^{r}(k)|=1$, which can be obtained only in the homochiral framework. In figure 4, $1-H^{r}(k)$ is plotted for $\tilde{\unicode[STIX]{x1D700}}_{H}\in S$. We find that increasing $\tilde{\unicode[STIX]{x1D700}}_{H}$ leads to lower values of $1-H^{r}(k)$ in broader ranges of scales. The turbulent state is then closer to maximum chirality. We note that the lower bound of $10^{-3}$ for $1-H^{r}(k)$ is prescribed by a restriction on $\unicode[STIX]{x1D700}_{H}$ in order to avoid a negative value under the square root in (2.11).

4.1 Energy fluxes

The energy equation for each helical mode is obtained by taking the real part of the dot product of (2.2) with $\boldsymbol{u}^{\pm }(\boldsymbol{k})^{\ast }$. Then the energy equation is given by (3.1) with $(X,\boldsymbol{x})=(E^{\pm },\boldsymbol{u}^{\pm })$ and $2\unicode[STIX]{x1D700}_{E^{\pm }}(\boldsymbol{k})=\unicode[STIX]{x1D700}_{E}(\boldsymbol{k})\pm \unicode[STIX]{x1D700}_{H}(\boldsymbol{k})/k$. The equation satisfied by the energy spectral density of each helical mode is given by (3.2) with (3.3)–(3.6). Of course, the flux of total energy satisfies

(4.1)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{E}(k)=\unicode[STIX]{x1D6F1}_{E^{+}}(k)+\unicode[STIX]{x1D6F1}_{E^{-}}(k).\end{eqnarray}$$

In addition, each flux $\unicode[STIX]{x1D6F1}_{E^{\pm }}(k)$ can be decomposed into

(4.2)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F1}_{E^{+}}(k)=\unicode[STIX]{x1D6F1}_{+}^{+<}(k)+\unicode[STIX]{x1D6F1}_{-}^{+<}(k), & \displaystyle\end{eqnarray}$$

(4.3)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D6F1}_{E^{-}}(k)=\unicode[STIX]{x1D6F1}_{+}^{-<}(k)+\unicode[STIX]{x1D6F1}_{-}^{-<}(k), & \displaystyle\end{eqnarray}$$

where $\unicode[STIX]{x1D6F1}_{b}^{a<}(k)$, with $a,b\equiv \pm$, denotes the energy flux from $\boldsymbol{u}^{a<}(\boldsymbol{k})$, meaning that $\boldsymbol{u}^{a}(\boldsymbol{k})$ is taken at wavenumbers inside a sphere of radius $k$ (the $k$-sphere), to $\boldsymbol{u}^{b}(\boldsymbol{k})$ taken at all wavenumbers. It is defined by

(4.4)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{b}^{a<}(k)=\mathop{\sum }_{|\boldsymbol{k}^{\prime }|\leqslant k}\text{Re}\{\boldsymbol{u}^{a}(\boldsymbol{k}^{\prime })^{\ast }\boldsymbol{\cdot }{\mathcal{F}}[(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{b}](\boldsymbol{k}^{\prime })\},\end{eqnarray}$$

where $\boldsymbol{u}^{b}={\mathcal{F}}^{-1}[\boldsymbol{u}^{b}(\boldsymbol{k})]$ is the inverse Fourier transform of $\boldsymbol{u}^{b}(\boldsymbol{k})$ (Verma Reference Verma2004; Kessar et al. Reference Kessar, Plunian, Stepanov and Balarac2015; Plunian, Stepanov & Verma Reference Plunian, Stepanov and Verma2019; Sadhukhan et al. Reference Sadhukhan, Samuel, Plunian, Stepanov, Samtaney and Verma2019). We can further decompose $\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$ as $\unicode[STIX]{x1D6F1}_{+}^{+<}(k)=\unicode[STIX]{x1D6F1}_{+<}^{+<}(k)+\unicode[STIX]{x1D6F1}_{+>}^{+<}(k)$, where $\unicode[STIX]{x1D6F1}_{+<}^{+<}(k)$ denotes the energy flux from $\boldsymbol{u}^{+<}$ to $\boldsymbol{u}^{+<}$, and $\unicode[STIX]{x1D6F1}_{+>}^{+<}(k)$ the energy flux from $\boldsymbol{u}^{+<}$ to $\boldsymbol{u}^{+>}$. Here, again, $\boldsymbol{u}^{+<}$ and $\boldsymbol{u}^{+>}$ denote $\boldsymbol{u}^{+}(\boldsymbol{k})$ taken at wavenumbers, respectively, inside and outside the $k$-sphere (similar notation applies to $\boldsymbol{u}^{-<}$ and $\boldsymbol{u}^{->}$). By definition $\unicode[STIX]{x1D6F1}_{+<}^{+<}(k)=0$, implying that $\unicode[STIX]{x1D6F1}_{+}^{+<}(k)=\unicode[STIX]{x1D6F1}_{+>}^{+<}(k)$ (similarly $\unicode[STIX]{x1D6F1}_{-<}^{-<}(k)=0$ implies $\unicode[STIX]{x1D6F1}_{-}^{-<}(k)=\unicode[STIX]{x1D6F1}_{->}^{-<}(k)$).

An illustration of the four fluxes $\unicode[STIX]{x1D6F1}_{+}^{+<}(k),\unicode[STIX]{x1D6F1}_{-}^{+<}(k),\unicode[STIX]{x1D6F1}_{+}^{-<}(k)$ and $\unicode[STIX]{x1D6F1}_{-}^{-<}(k)$ is given in figure 5. They are plotted in figure 6(a) for $\tilde{\unicode[STIX]{x1D700}}_{H}=25$, together with $\unicode[STIX]{x1D6F1}_{E}(k)$, which is equal to the sum of the four. In the infrared domain $k\in [1,9]$, $\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$ is clearly negative and close to $\unicode[STIX]{x1D6F1}_{E}(k)$, suggesting that it is mostly this flux that is responsible for the inverse energy cascade.

Figure 5.

Illustration of the fluxes among helical modes. The notation $\boldsymbol{u}^{\pm <}$ and $\boldsymbol{u}^{\pm >}$ corresponds to $\boldsymbol{u}^{\pm }(\boldsymbol{k})$ taken at wavenumbers, respectively, inside and outside the $k$-sphere. The non-horizontal arrows denote energy fluxes from one helical mode taken inside the $k$-sphere towards the opposite helical mode taken at all scales.

Figure 6.

Helical mode fluxes for $\unicode[STIX]{x1D700}_{E}=0.2$ and $\tilde{\unicode[STIX]{x1D700}}_{H}=25$. In (a), the four fluxes $\unicode[STIX]{x1D6F1}_{+}^{+<}(k),\unicode[STIX]{x1D6F1}_{-}^{+<}(k),\unicode[STIX]{x1D6F1}_{+}^{-<}(k)$ and $\unicode[STIX]{x1D6F1}_{-}^{-<}(k)$, illustrated in figure 5, are plotted together with the total energy flux $\unicode[STIX]{x1D6F1}_{E}(k)$ already plotted in figure 2. In (b), the flux $\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$ is split into the homochiral flux $^{+}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$ and the heterochiral flux $^{-}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$.

It is thus interesting to further decompose the flux $\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$ as $\unicode[STIX]{x1D6F1}_{+}^{+<}(k)=^{+}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)+^{-}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$ with

(4.5)$$\begin{eqnarray}\displaystyle \text{}^{\pm }\unicode[STIX]{x1D6F1}_{+}^{+<}(k)=\mathop{\sum }_{|\boldsymbol{k}^{\prime }|\leqslant k}\text{Re}\{\boldsymbol{u}^{+}(\boldsymbol{k}^{\prime })^{\ast }\boldsymbol{\cdot }{\mathcal{F}}[(\boldsymbol{u}^{\pm }\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{+}](\boldsymbol{k}^{\prime })\}, & & \displaystyle\end{eqnarray}$$

where $^{\pm }\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$ is interpreted as the energy flux from $\boldsymbol{u}^{+<}$ to $\boldsymbol{u}^{+}$, with $\boldsymbol{u}^{\pm }$ acting as a mediator.

Figure 7.

Ratio of the energy flux due to homochiral triads to the energy flux due to heterochiral triads, versus $k$ and for the same parameters as in figure 2.

The three fluxes $\unicode[STIX]{x1D6F1}_{+}^{+<}(k),^{+}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$ and $^{-}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$ are plotted in figure 6(b). In the infrared domain $k\in [1,9]$, $^{+}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)\leqslant 0$ and $^{-}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)\geqslant 0$, suggesting that the homochiral triadic interactions $(+,+,+)$ are mostly responsible for the inverse energy cascade. At scales smaller than the energy injection scales, $k\geqslant 10$, we observe that $^{+}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)\geqslant 0$, which is in striking difference with previous simulations in which energy was injected at large scales (Kessar et al. Reference Kessar, Plunian, Stepanov and Balarac2015). In addition $|^{+}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)|\geqslant |^{-}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)|$, which contradicts the assumptions of the phenomenological model by Stepanov et al. (Reference Stepanov, Golbraikh, Frick and Shestakov2015).

From figure 2 we observe that the inverse cascade is continuously greater for increasing values of $\tilde{\unicode[STIX]{x1D700}}_{H}$. This is in contrast with Sahoo et al. (Reference Sahoo, Alexakis and Biferale2017) in which a sharp transition between the cases with and without inverse cascade is reported. There are at least two main reasons to explain this discrepancy. First, in Sahoo et al. (Reference Sahoo, Alexakis and Biferale2017) such a sharp transition is only expected within the limit of high Reynolds numbers, which we are not studying here. Second, in Sahoo et al. (Reference Sahoo, Alexakis and Biferale2017), the ratio of heterochiral to homochiral triadic interactions is imposed, and it is imposed in the same way at all scales. This was done by solving a modified version of the NSE including a prescribed weighting between the homochiral and heterochiral nonlinear terms. Here, as we solve the NSE (2.2), the importance of homochiral versus heterochiral triadic interactions is not prescribed, and remains scale-dependent.

To contrast the strengths of heterochiral and homochiral triadic interactions, we calculate the ratio of their energy fluxes, which is given by

(4.6)$$\begin{eqnarray}\frac{\unicode[STIX]{x1D6F1}_{Hom}(k)}{\unicode[STIX]{x1D6F1}_{Het}(k)}=\frac{\text{}^{+}\unicode[STIX]{x1D6F1}_{+>}^{+<}(k)+^{-}\unicode[STIX]{x1D6F1}_{->}^{-<}(k)}{\unicode[STIX]{x1D6F1}_{E}(k)-(\text{}^{+}\unicode[STIX]{x1D6F1}_{+>}^{+<}(k)+^{-}\unicode[STIX]{x1D6F1}_{->}^{-<}(k))}.\end{eqnarray}$$

In figure 7, this ratio is plotted versus $k$, for $\tilde{\unicode[STIX]{x1D700}}_{H}\in S$. This ratio is negative at scales larger than the energy injection scales, and positive at scales smaller than the energy injection scales. This is in agreement with the dual inverse energy cascade at large scales and forward helicity cascade at small scales. It is also clearly $k$-dependent. Finally, with the increase of $\tilde{\unicode[STIX]{x1D700}}_{H}$, the ratio $|\unicode[STIX]{x1D6F1}_{Hom}(k)|/|\unicode[STIX]{x1D6F1}_{Het}(k)|$ increases without sharp transition.

Figure 8.

In $(k_{1},k_{2})$ maps, two representations of fluxes from a $k_{1}$ inner sphere to a $k_{2}$ outer sphere $\unicode[STIX]{x1D6F1}_{{>}k_{2}}^{{<}k_{1}}$, for $\unicode[STIX]{x1D700}_{E}=0.2$ and $\tilde{\unicode[STIX]{x1D700}}_{H}=25$. In (a) the fluxes are calculated among positive homochiral triads only. In (b) the fluxes are calculated among all other triads, namely heterochiral and negative homochiral. The range of values taken by the fluxes is $[-15;15]$ in (a) and $[-5;5]$ in (b).

4.2 Non-local energy transfers

In order to investigate the degree of locality of the cascades, we further introduce the energy flux from an inner sphere of radius $k_{1}$ to an outer sphere or radius $k_{2}$, which is defined as

(4.7)$$\begin{eqnarray}\unicode[STIX]{x1D6F1}_{{>}k_{2}}^{{<}k_{1}}=\mathop{\sum }_{|\boldsymbol{k}^{\prime }|\geqslant k_{2}}\text{Re}\{\boldsymbol{u}(\boldsymbol{k}^{\prime })^{\ast }\boldsymbol{\cdot }{\mathcal{F}}[(\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735})\boldsymbol{u}^{{<}k_{1}}](\boldsymbol{k}^{\prime })\},\end{eqnarray}$$

with

(4.8)$$\begin{eqnarray}\boldsymbol{u}^{{<}k_{1}}=\mathop{\sum }_{|\boldsymbol{k}^{\prime }|\leqslant k_{1}}{\mathcal{F}}^{-1}[\boldsymbol{u}(\boldsymbol{k}^{\prime })]\end{eqnarray}$$

and $k_{1}\leqslant k_{2}$. For $k_{1}=k_{2}=k$ the two spheres are adjacent, implying that $\unicode[STIX]{x1D6F1}_{{>}k}^{{<}k}=\unicode[STIX]{x1D6F1}_{E}(k)$. Increasing the value of $k_{2}-k_{1}$ corresponds to moving the two spheres apart, and then to investigate non-local fluxes.

In figure 8, two $(k_{1},k_{2})$ maps of $\unicode[STIX]{x1D6F1}_{{>}k_{2}}^{{<}k_{1}}$ are given, figure 8(a) giving the flux among positive homochiral triads, figure 8(b) giving the flux among all other triads (heterochiral and negative homochiral). In figure 8(a) the diagonal coincides with the flux $^{+}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$ represented in figure 6(b). In figure 8(b) the diagonal coincides with $\unicode[STIX]{x1D6F1}_{E}(k)-^{+}\unicode[STIX]{x1D6F1}_{+}^{+<}(k)$. In both cases the further away from the diagonal, the more non-local the fluxes are. In figure 8(a) the fluxes are mostly negative, with a minimum at $(k_{1},k_{2})\approx (7,9)$. This shows that the energy cascade is inverse, and due to weakly non-local interactions. In figure 8(b) the fluxes are positive and maximum on the diagonal, implying a forward cascade due to local interactions. In figure 8(b) the fluxes are approximately three times smaller than in figure 8(a), implying that the sum of both is qualitatively similar to figure 8(a). Therefore the inverse cascade obtained in the infrared domain is mainly due to weakly non-local interactions among positive homochiral triads $(+,+,+)$. The sum of the other triadic interactions leads to forward cascade, mainly local. The aforementioned non-local interactions leading to inverse energy cascade are also observed in two-dimensional hydrodynamic turbulence (Gupta et al. Reference Gupta, Jayaraman, Chatterjee, Sadhukhan, Samtaney and Verma2019; Verma Reference Verma2019).

Figure 9.

Energy spectra and fluxes (inset). The energy injection rate $\unicode[STIX]{x1D700}_{E}=0.2$ is applied at $k_{E}\in ]\!9,10\!]$. From bottom to top of the energy spectra, the helicity injection rate $\tilde{\unicode[STIX]{x1D700}}_{H}=25$ is applied at $k_{H}\in [1,10]$, $]\!9,10^{2}\! ]$, $]\!5,20\!]$, $[1,20]$, $[1,10^{2}]$ and $[1,10^{2}]$ again. For the last curve the viscosity is twice as small and the resolution is equal to $1024^{3}$.

4.3 Varying the helicity injection range of scales

In contrast to previous studies (Biferale et al. Reference Biferale, Musacchio and Toschi2012, Reference Biferale, Musacchio and Toschi2013; Sahoo et al. Reference Sahoo, Alexakis and Biferale2017), here we can vary the range of scales in which helicity is injected. We already know that injecting helicity at scales smaller than the energy injection scales leads to a forward cascade of energy (Kessar et al. Reference Kessar, Plunian, Stepanov and Balarac2015). Here this would correspond to injecting helicity in the range $k_{H}\in ]\!9,10^{2}\! ]$. From § 3.2, we also know that injecting helicity in the range $k_{H}\in [1,10^{2}]$ leads to an inverse energy cascade at large scales.

We now investigate additional cases injecting helicity in different scale ranges, $k_{H}\in [1,20]$, $]\!5,20\!]$ and $[1,10]$, keeping all other parameters the same. The results of these three cases are plotted in figure 9, together with the case $k_{H}\in [1,10^{2}]$, already shown in figure 2, and the case $k_{H}\in ]\!9,10^{2}\! ]$.

The inverse cascade is obtained for $k_{H}\in [1,20]$ but not for $k_{H}\in [1,10]$, suggesting that a sufficiently broad range of scales smaller than the energy injection scales is necessary to produce the inverse cascade. This emphasizes the role of non-local transfers identified in § 4.2. For $k_{H}\in ]\!5,20\!]$ the energy flux is negative for $k\in ]\!5,10\!]$ with an inverse cascade limited to these scales. This emphasizes the role of helicity injection at scales larger than the energy injection scale, i.e. the inverse energy cascade stops at scales where helicity is not injected. In summary, to obtain an inverse cascade it is necessary to inject helicity in a sufficiently broad range of scales on both sides of the energy injection range of scales.

Finally, the case $k_{H}\in [1,10^{2}]$ with a twice smaller viscosity $\unicode[STIX]{x1D708}=0.75\times 10^{-3}$ and a resolution $1024^{3}$ is also plotted in figure 9. It corresponds to a mean kinetic energy $\bar{E}_{tot}\approx 2.1$, thus leading to $\unicode[STIX]{x1D706}\approx 0.28$, and a Taylor-microscale Reynolds number $R_{\unicode[STIX]{x1D706}}\approx 440$. The energy spectrum and flux are qualitatively similar to the case shown in figure 2, with again an inverse energy cascade at large scales.