2.1. Basic equations for energy and helicity spectra

The Navier–Stokes equations for an incompressible fluid in Fourier space read

(2.1)\begin{equation} \left(\frac{\partial}{\partial t} + \nu k^2 \right) \tilde{u}_i (\boldsymbol{k},t) ={-} \mathrm{i}M_{ij\ell} (\boldsymbol{k}) \int \mathrm{d}^3\, p \int \mathrm{d}^3 q \,\delta (\boldsymbol{k} - \boldsymbol{p} - \boldsymbol{q}) \tilde{u}_j (\boldsymbol{p},t) \tilde{u}_\ell (\boldsymbol{q},t), \end{equation}

where $\tilde {\boldsymbol {u}} (\boldsymbol {k},t)$ is the Fourier coefficient of the velocity field that satisfies the solenoidal condition $\boldsymbol {k} \cdot \tilde {\boldsymbol {u}} (\boldsymbol {k},t) = 0$, $M_{ij\ell } (\boldsymbol {k}) = [k_j P_{i\ell } (\boldsymbol {k}) + k_\ell P_{ij} (\boldsymbol {k})]/2$ and $P_{ij} (\boldsymbol {k}) = \delta _{ij} -k_i k_j/k^2$. The Fourier transformation of a variable $f(\boldsymbol {x},t)$ and its inverse transformation are defined by

(2.2a,b)\begin{equation} f (\boldsymbol{x},t) = \int \mathrm{d}^3 k \, \tilde{f} (\boldsymbol{k},t) \,\mathrm{e}^{\mathrm{i} \boldsymbol{k} \cdot \boldsymbol{x}},\quad \tilde{f} (\boldsymbol{k},t) = \frac{1}{(2{\rm \pi})^3} \int \mathrm{d}^3 x \, f (\boldsymbol{x},t)\, \mathrm{e}^{-\mathrm{i} \boldsymbol{k} \cdot \boldsymbol{x}}. \end{equation}

The equations for energy and helicity spectra, $E (k,t)$ and $E^H (k,t)$, yield

(2.3)\begin{gather} \left(\frac{\partial}{\partial t} + 2 \nu k^2 \right) E (k,t) = \frac{1}{2} \int _0^\infty \mathrm{d} p\, \int_0^\infty \mathrm{d}q \, S (k,p,q,t), \end{gather}

(2.4)\begin{gather}\left(\frac{\partial}{\partial t} + 2 \nu k^2 \right) E^H (k,t) = \frac{1}{2} \int _0^\infty \mathrm{d} p \,\int_0^\infty \mathrm{d}q \, S^H (k,p,q,t), \end{gather}

where

(2.5)\begin{gather} S (k,p,q,t) ={-} 16{\rm \pi}^2 kpq \varDelta_{kpq} M_{ij\ell} (\boldsymbol{k}) \textrm{Im} [\langle \tilde{u}_i (\boldsymbol{k},t) \tilde{u}_j (-\boldsymbol{p},t) \tilde{u}_\ell (-\boldsymbol{q},t) \rangle], \end{gather}

(2.6)\begin{gather}S^H (k,p,q,t) ={-} 16{\rm \pi}^2 kpq \varDelta_{kpq} \left(\epsilon_{ijm} k_\ell + \epsilon_{i\ell m} k_j\right) k_m \textrm{Re} [ \langle \tilde{u}_i (\boldsymbol{k},t) \tilde{u}_j (-\boldsymbol{p},t) \tilde{u}_\ell (-\boldsymbol{q},t) \rangle], \end{gather}

and $\varDelta _{kpq}$ is unity only when $k$, $p$ and $q$ can form the sides of a triangle. The conservation of energy and helicity are guaranteed by the detailed balance (see e.g. Waleffe Reference Waleffe1992)

(2.7)\begin{gather} S (k,p,q,t) + S (p,q,k,t) + S (q,k,p,t) = 0, \end{gather}

(2.8)\begin{gather}S^H (k,p,q,t) + S^H (p,q,k,t) + S^H (q,k,p,t) = 0. \end{gather}

Therefore, we have

(2.9a,b)\begin{equation} \int _0^\infty \mathrm{d}k \int _0^\infty \mathrm{d}p \int_0^\infty \mathrm{d}q \, S (k,p,q,t) = 0,\quad \int _0^\infty \mathrm{d}k \int _0^\infty \mathrm{d}p \int_0^\infty \mathrm{d}q \, S^H (k,p,q,t) = 0. \end{equation}

Interscale energy flux $\varPi (k,t)$ and helicity flux $\varPi ^H (k,t)$ are defined by

(2.10)\begin{align} \varPi (k,t) &= \tfrac{1}{2} \int_k^\infty \mathrm{d} k' \int _0^\infty \mathrm{d} p' \int_0^\infty \mathrm{d} q' S (k',p',q',t) \nonumber\\ &={-} \tfrac{1}{2} \int_0^k \mathrm{d} k' \int _0^\infty \mathrm{d} p' \int_0^\infty \mathrm{d}q' \,S (k',p',q',t), \end{align}

(2.11)\begin{align} \varPi^H (k,t) &= \tfrac{1}{2} \int_k^\infty \mathrm{d} k' \int _0^\infty \mathrm{d} p' \int_0^\infty \mathrm{d}q' \, S^H (k',p',q',t) \nonumber\\ &={-} \tfrac{1}{2} \int_0^k \mathrm{d} k' \int _0^\infty \mathrm{d} p' \int_0^\infty \mathrm{d}q' \,S^H (k',p',q',t), \end{align}

where we utilise (2.9a,b).

Considering the maximally helical or homochiral condition,

(2.12)\begin{equation} E^H (k,t) = 2k E(k,t). \end{equation}

This condition holds for the Beltrami velocity field, which is defined by $\mathrm {i} \boldsymbol {k} \times \tilde {\boldsymbol {u}} (\boldsymbol {k},t) = k \tilde {\boldsymbol {u}} (\boldsymbol {k},t)$. For the Beltrami velocity field, $S^H(k,p,q,t)$ satisfies

(2.13)\begin{equation} S^H(k,p,q,t) = 2 k S(k,p,q,t), \end{equation}

which denotes a maximally helical condition for the third moment. Using (2.13), the detailed balance for helicity (2.8) yields

(2.14)\begin{equation} k S (k,p,q,t) + p S (p,q,k,t) + q S (q,k,p,t) = 0. \end{equation}

Similarly, the Navier–Stokes equations in two dimensions provide the equation $k^2 S (k,p,q,t) + p^2 S (p,q,k,t) + q^2 S (q,k,p,t) = 0$, which guarantees the conservation of enstrophy. Waleffe (Reference Waleffe1992) analysed the statistical properties of the Navier–Stokes equations by employing the scale-similar structure and the two detailed balance (2.7) and (2.14), which verified that the nonlinear interaction of the velocity fields comprising the same sign helical modes can trigger an inverse transfer of energy. However, it is demonstrated that the turbulent field successively restores the mirror symmetry at small scales via the cascade process (Kraichnan Reference Kraichnan1973; Chen et al. Reference Chen, Chen and Eyink2003a). Hence, the maximally helical condition (2.12) is not persistent in the dynamics of the Navier–Stokes equations. The relationship between this tendency and the property of the closure equations is discussed in Appendix A.

2.2. Properties of closure equations

In this study, we adopt the LRA (Kaneda Reference Kaneda1981) as a closure approximation. For details on the closure, see Kaneda (Reference Kaneda1981, Reference Kaneda2007). A unique feature of LRA is in introducing a mapping function from Eulerian to Lagrangian velocities,

(2.15)\begin{equation} \boldsymbol{v} (\boldsymbol{x}, s\,|\, t) = \int \mathrm{d}^3x' \,\psi (\boldsymbol{x}',t\,|\, \boldsymbol{x},s) \boldsymbol{u} (\boldsymbol{x}', t), \end{equation}

where $\boldsymbol {v} (\boldsymbol {x}, s\,|\, t)$ denotes the Lagrangian velocity and $\psi (\boldsymbol {x}',t\,|\, \boldsymbol {x},s)$ denotes the Lagrangian position function that obeys

(2.16)\begin{equation} \frac{\partial}{\partial t} \psi (\boldsymbol{x}, t\,|\, \boldsymbol{x}', s) + u_i (\boldsymbol{x},t) \frac{\partial}{\partial x_i} \psi (\boldsymbol{x}, t\,|\, \boldsymbol{x}', s) = 0, \end{equation}

with $\psi (\boldsymbol {x},s\,|\, \boldsymbol {x}',s) = \delta (\boldsymbol {x} - \boldsymbol {x}')$. Namely, $\boldsymbol {v} (\boldsymbol {x}, s\,|\, t)$ denotes the velocity of the fluid element at time $t$ which was at $\boldsymbol {x}$ at time $s$. Notably, the Lagrangian velocity satisfies

(2.17)\begin{equation} \boldsymbol{v} (\boldsymbol{x}, s\,|\, s) = \boldsymbol{u} (\boldsymbol{x}, s). \end{equation}

For the representative variables, the LRA adopts the Lagrangian two-time velocity correlation $Q_{ij} (\boldsymbol {k}, t,s)$ and the mean Lagrangian response function $G_{ij} (\boldsymbol {k},t,s)$. They are defined by

(2.18)\begin{gather} Q_{ij} (\boldsymbol{k}, t, s) \delta (\boldsymbol{k} + \boldsymbol{k}') = P_{ia} (\boldsymbol{k}) \left\langle \tilde{v}_a (\boldsymbol{k}, s\,|\,t) \tilde{u}_j (\boldsymbol{k}',s) \right \rangle, \end{gather}

(2.19)\begin{gather}G_{ij} (\boldsymbol{k}, t, s) \delta (\boldsymbol{k} - \boldsymbol{k}') = P_{ia} (\boldsymbol{k}) P_{jb} (\boldsymbol{k}) \left\langle \frac{\delta \tilde{v}_a (\boldsymbol{k}, s\,|\, t)}{\delta \tilde{f}_b (\boldsymbol{k}',s)} \right \rangle, \end{gather}

where $\delta \tilde {\boldsymbol {f}} (\boldsymbol {k},t)$ denotes an infinitesimal forcing driving an infinitesimal perturbation in the Eulerian velocity field $\delta \tilde {\boldsymbol {u}} (\boldsymbol {k},t)$; $\delta \tilde {\boldsymbol {u}} (\boldsymbol {k},t)$ obeys

(2.20)\begin{align} \left(\frac{\partial}{\partial t} + \nu k^2\right) \delta \tilde{u}_i (k,t) &={-} 2 \mathrm{i} M_{ij\ell} (\boldsymbol{k}) \int \mathrm{d}^3p \int \mathrm{d}^3 q \, \delta (\boldsymbol{k} - \boldsymbol{p} - \boldsymbol{q})\tilde{u}_j (\boldsymbol{p},t) \delta \tilde{u}_\ell (\boldsymbol{q},t) \nonumber\\ &\quad + P_{ij} (\boldsymbol{k}) \delta \tilde{f}_j (\boldsymbol{k},t). \end{align}

The initial condition for the response function $G_{ij} (\boldsymbol {k},s,s)$ is determined as follows. Based on (2.17) and (2.19), we have

(2.21)\begin{equation} G_{ij} (\boldsymbol{k}, s, s) \delta (\boldsymbol{k} - \boldsymbol{k}') = P_{ia} (\boldsymbol{k}) P_{jb} (\boldsymbol{k}) \left\langle \tilde{G}^\mathrm{E}_{ab} (\boldsymbol{k},s\,|\,\boldsymbol{k}',s) \right \rangle, \end{equation}

where

(2.22)\begin{equation} \tilde{G}^\mathrm{E}_{ij} (\boldsymbol{k},t\,|\,\boldsymbol{k}',s) = \frac{\delta \tilde{u}_i (\boldsymbol{k}, t)}{\delta \tilde{f}_j (\boldsymbol{k}',s)} \end{equation}

and thus,

(2.23)\begin{equation} \delta \tilde{u}_i (\boldsymbol{k}, t) = \int_{t_0}^t \mathrm{d}s \int \mathrm{d}^3k'\, \tilde{G}^\mathrm{E}_{ij} (\boldsymbol{k},t\,|\,\boldsymbol{k}',s) \delta \tilde{f}_j (\boldsymbol{k}',s), \end{equation}

where $t_0$ is the initial time. $\tilde {G}^\mathrm {E}_{ij} (\boldsymbol {k},t\,|\,\boldsymbol {k}',s)$ denotes the Eulerian response function that obeys

(2.24)\begin{align} \left(\frac{\partial}{\partial t} + \nu k^2 \right) \tilde{G}^\mathrm{E}_{ij} (\boldsymbol{k},t\,|\,\boldsymbol{k}',s) &={-} 2 \mathrm{i} M_{i\ell m} (\boldsymbol{k}) \int \mathrm{d}^3 p \int \mathrm{d}^3 q \, \delta (\boldsymbol{k} - \boldsymbol{p} - \boldsymbol{q}) \nonumber\\ &\quad \times \tilde{u}_\ell (\boldsymbol{p},t) \tilde{G}^\mathrm{E}_{mj} (\boldsymbol{q},t\,|\,\boldsymbol{k}',s). \end{align}

The time derivative of the right-hand side of (2.23) must correspond to (2.20), which requires

(2.25)\begin{equation} \tilde{G}^\mathrm{E}_{ij} (\boldsymbol{k},t\,|\,\boldsymbol{k}',t) = P_{ij} (\boldsymbol{k}) \delta (\boldsymbol{k} - \boldsymbol{k}'). \end{equation}

Accordingly, the initial condition for the mean Lagrangian response function yields

(2.26)\begin{equation} G_{ij} (\boldsymbol{k}, s, s) = P_{ij} (\boldsymbol{k}). \end{equation}

Notably, (2.21) should hold even when we adopt another type of response function as the representative. Hence, the initial condition (2.26) is general for the closure theories employing the response function.

For the LRA, the closure equations yield (Kaneda Reference Kaneda1981)

(2.27)\begin{gather} \left(\frac{\partial}{\partial t} + 2\nu k^2 \right) Q_{ij} (\boldsymbol{k}, t,t) = T_{ij} (\boldsymbol{k},t) + T_{ji} (-\boldsymbol{k},t), \end{gather}

(2.28)\begin{gather}\left(\frac{\partial}{\partial t} + \nu k^2 \right) Q_{ij} (\boldsymbol{k}, t,s) ={-} \eta_{ia} (\boldsymbol{k},t,s) Q_{aj} (\boldsymbol{k}, t,s) \quad (t > s), \end{gather}

(2.29)\begin{gather}\left(\frac{\partial}{\partial t} + \nu k^2 \right) G_{ij} (\boldsymbol{k}, t,s) ={-} \eta_{ia} (\boldsymbol{k},t,s) G_{aj} (\boldsymbol{k}, t,s) \quad (t > s), \end{gather}

where

(2.30)\begin{align} T_{ij} (\boldsymbol{k},t) &= M_{i\ell m} (\boldsymbol{k}) \int \mathrm{d}^3 p \int \mathrm{d}^3 q \, \delta(\boldsymbol{k}- \boldsymbol{p} - \boldsymbol{q}) \int_{t_0}^t \mathrm{d} s \nonumber\\ &\quad \times \left[ - 4M_{abc} (\boldsymbol{p}) G_{ma} (\boldsymbol{p},t,s) Q_{\ell b} (\boldsymbol{q},t,s) Q_{jc} (-\boldsymbol{k},t,s) \right. \nonumber\\ &\quad + \left. 2M_{abc} (\boldsymbol{k}) G_{ja} (-\boldsymbol{k},t,s) Q_{mc} (\boldsymbol{p},t,s) Q_{\ell b} (\boldsymbol{q},t,s)\right], \end{align}

(2.31)\begin{align} \eta_{ij} (\boldsymbol{k},t,s) &= 2 \int \mathrm{d}^3 p \int \mathrm{d}^3 q \, \delta(\boldsymbol{k} - \boldsymbol{p} - \boldsymbol{q}) P_{ib} (\boldsymbol{k}) \frac{q_j q_b q_\ell q_m}{q^2} \int_{s}^t \mathrm{d} s' \,Q_{\ell m} (-\boldsymbol{p},t,s'). \end{align}

For a homogeneous isotropic and non-mirror symmetric case, second-order tensor variables read

(2.32)\begin{gather} Q_{ij} (\boldsymbol{k},t,s) = \frac{1}{2} P_{ij} (\boldsymbol{k}) Q (k,t,s) - \frac{\mathrm{i}}{2} \epsilon_{ij\ell} \frac{k_\ell}{k^2} Q^H (k,t,s), \end{gather}

(2.33)\begin{gather}G_{ij} (\boldsymbol{k},t,s) = P_{ij} (\boldsymbol{k}) G (k,t,s) - \mathrm{i} \epsilon_{ij\ell} \frac{k_\ell}{k} G^H (k,t,s), \end{gather}

where $Q(k,t,s)$ and $Q^H(k,t,s)$ are related to the energy and helicity spectra, respectively, and are expressed as

(2.34a,b)\begin{equation} Q(k,t,t) = \frac{E(k,t)}{2{\rm \pi} k^2},\quad Q^H(k,t,t) = \frac{E^H(k,t)}{4{\rm \pi} k^2}, \end{equation}

where $Q(k,t,t)$ and $Q^H (k,t,t)$ denote the spectral densities of energy and helicity, respectively.

The trace parts of (2.27)–(2.29) yield the equations for $Q(k,t,t)$, $Q(k,t,s)$ and $G(k,t,s)$, respectively. Meanwhile, multiplying (2.27)–(2.29) by $\mathrm {i} \epsilon _{ijn} k_n$ yields the equations for $Q^H(k,t,t)$, $Q^H(k,t,s)$ and $G^H(k,t,s)$, respectively. The resulting equations yield

(2.35)\begin{align} & \left(\frac{\partial}{\partial t} + 2\nu k^2 \right) Q (k,t,t) = 2 {\rm \pi}\int_0^\infty \mathrm{d}p \int_0^\infty \mathrm{d}q \,\varDelta_{kpq} \int_{t_0}^t \mathrm{d}s \nonumber\\ &\quad \times\left\{kpq b_{kpq} \left[ G(k,t,s) Q (p,t,s) - G(p,t,s) Q (k,t,s)\right] Q (q,t,s) \vphantom{\left[ \frac{p^2}{k} G^H (k,t,s) Q (p,t,s) - p G^H (p,t,s) Q (k,t,s) \right] }\right. \nonumber\\ &\quad -\frac{pq}{k} c_{kpq} \left[ G(k,t,s) Q^H (p,t,s) - G(p,t,s) Q^H (k,t,s) \right] Q^H (q,t,s) \nonumber\\ &\quad + \frac{pq}{k} \left[ b_{kpq} k G^H (k,t,s) Q^H (p,t,s) - c_{kpq} p G^H (p,t,s) Q^H (k,t,s) \right] Q (q,t,s) \nonumber\\ &\quad - \left. \frac{pq}{k} c_{kpq} \left[ \frac{p^2}{k} G^H (k,t,s) Q (p,t,s) - p G^H (p,t,s) Q (k,t,s) \right] Q^H (q,t,s) \right\}, \end{align}

(2.36)\begin{align} & \left( \frac{\partial}{\partial t} + 2\nu k^2 \right) Q^H (k,t,t) = 2 {\rm \pi}\int_0^\infty \mathrm{d}p \int_0^\infty \mathrm{d}q \,\varDelta_{kpq} \int_{t_0}^t \mathrm{d}s \nonumber\\ &\quad \times \left\{ kpq b_{kpq} \left[ G(k,t,s) Q^H (p,t,s) - G(p,t,s) Q^H (k,t,s) \right] Q (q,t,s) \vphantom{\left[ \frac{p^2}{k} G^H (k,t,s) Q (p,t,s) - p G^H (p,t,s) Q (k,t,s) \right] } \right. \nonumber\\ &\quad - \frac{pq}{k} c_{kpq} \left[ p^2 G(k,t,s) Q (p,t,s) - k^2G(p,t,s) Q (k,t,s) \right] Q^H (q,t,s) \nonumber\\ &\quad + kpq \left[ b_{kpq} k G^H (k,t,s) Q (p,t,s) - c_{kpq} p G^H (p,t,s) Q (k,t,s) \right] Q (q,t,s) \nonumber\\ &\quad - \left.\vphantom{\left[ \frac{p^2}{k} G^H (k,t,s) Q (p,t,s) - p G^H (p,t,s) Q (k,t,s) \right] } \frac{pq}{k} c_{kpq} \left[ k G^H (k,t,s) Q^H (p,t,s) - p G^H (p,t,s) Q^H (k,t,s) \right] Q^H (q,t,s) \right\}, \end{align}

(2.37)\begin{gather} \left[ \frac{\partial}{\partial t} + \nu k^2 + \eta(k,t,s) \right] Q (k, t,s) = 0 \quad (t > s), \end{gather}

(2.38)\begin{gather}\left[ \frac{\partial}{\partial t} + \nu k^2 + \eta(k,t,s) \right] Q^H (k, t,s) = 0\quad (t > s), \end{gather}

(2.39)\begin{gather}\left[ \frac{\partial}{\partial t} + \nu k^2 + \eta(k,t,s) \right] G (k, t,s) = 0 \quad (t > s), \end{gather}

(2.40)\begin{gather}\left[ \frac{\partial}{\partial t} + \nu k^2 + \eta(k,t,s) \right] G^H (k, t,s) = 0\quad (t > s), \end{gather}

where

(2.41)\begin{gather} \eta (k,t,s) = k \int_0^\infty \mathrm{d}q \ q^3 J \left(\frac{q}{k} \right) \int_s^t \mathrm{d}s'\, Q (q,t,s'), \end{gather}

(2.42)\begin{gather}J(x) = \frac{\rm \pi}{2a^4} \left[ (a^2-1)^2 \ln \left( \frac{1+a}{|1-a|} \right) - 2a + \frac{10}{3} a^3 \right], \quad a = \frac{2x}{1+x^2} \end{gather}

and

(2.43a–e)\begin{equation} \left.\begin{gathered} b_{kpq} = \frac{p}{k} (xy + z^3), \quad c_{kpq} = \frac{k}{q} z (x+yz) = \frac{k}{p} z (1-y^2), \nonumber \\ x = \frac{p^2+q^2-k^2}{2pq}, \quad y = \frac{q^2+k^2-p^2}{2qk}, \quad z = \frac{k^2+p^2-q^2}{2kp}. \end{gathered}\right\} \end{equation}

Details on the calculations are provided in Appendix B. Hence, for the closure equations, $S(k,p,q,t)$ and $S^H (k,p,q,t)$ in (2.3)–(2.6) read

(2.44)\begin{align} & S (k,p,q,t) = 4 {\rm \pi}^2 \varDelta_{kpq} \int_{t_0}^t \mathrm{d}s \, kpq \nonumber\\ &\quad \times \left\{ k^2 \left[ (b_{kpq} + b_{kqp}) G(k,t,s) Q (p,t,s) Q (q,t,s) \right. \vphantom{\frac{q^2}{k}} \right. \nonumber\\ &\quad - \left. b_{kpq} G(p,t,s) Q (k,t,s) Q (q,t,s) - b_{kqp} G(q,t,s) Q (k,t,s) Q (p,t,s)\right] \nonumber\\ &\quad - \left[ (c_{kpq} + c_{kqp}) G(k,t,s) Q^H (p,t,s) Q^H (q,t,s) \right. \nonumber\\ &\quad - \left. c_{kpq} G(p,t,s) Q^H (k,t,s) Q^H (q,t,s) - c_{kqp} G(q,t,s) Q^H (k,t,s) Q^H(p,t,s)\right] \nonumber\\ &\quad + k \left[ b_{kpq} G^H (k,t,s) Q^H (p,t,s) Q (q,t,s) + b_{kqp} G^H (k,t,s) Q (p,t,s) Q^H (q,t,s) \vphantom{\left[ \frac{p}{k} G^H (k,t,s) Q (p,t,s) - p G^H (p,t,s) Q (k,t,s) \right]}\right. \nonumber\\ &\quad - \left. \frac{p}{k} c_{kpq} G^H (p,t,s) Q^H (k,t,s) Q (q,t,s) - \frac{q}{k} c_{kqp} G^H (q,t,s) Q^H (k,t,s) Q (p,t,s) \right] \nonumber\\ &\quad - k \left[ \frac{p^2}{k^2} c_{kpq} G^H (k,t,s) Q (p,t,s) Q^H (q,t,s) + \frac{q^2}{k^2} c_{kqp} G^H (k,t,s) Q^H (p,t,s) Q (q,t,s) \right. \nonumber\\ &\quad - \frac{p}{k} c_{kpq} G^H (p,t,s) Q (k,t,s) Q^H (q,t,s) \nonumber\\ &\quad - \left.\left. \frac{q}{k} c_{kqp} G^H (q,t,s) Q (k,t,s) Q^H (p,t,s) \vphantom{\frac{q^2}{k}}\right] \right\}, \end{align}

(2.45)\begin{align} & S^H (k,p,q,t) = 8 {\rm \pi}^2 \varDelta_{kpq} \int_{t_0}^t \mathrm{d}s \, kpq \nonumber\\ &\quad \times \left\{ k^2 \left[ b_{kpq} G(k,t,s) Q^H (p,t,s) Q (q,t,s) + b_{kqp} G(k,t,s) Q (p,t,s) Q^H (q,t,s) \right. \vphantom{\left[ \frac{p^2}{k} G^H (k,t,s) Q (p,t,s) - p G^H (p,t,s) Q (k,t,s) \right] }\right. \nonumber\\ &\quad - \left. b_{kpq} G(p,t,s) Q^H (k,t,s) Q (q,t,s) - b_{kqp} G(q,t,s) Q^H (k,t,s) Q (p,t,s) \right] \nonumber\\ &\quad - k^2 \left[ \frac{p^2}{k^2} c_{kpq} G (k,t,s) Q (p,t,s) Q^H (q,t,s) + \frac{q^2}{k^2} c_{kqp} G (k,t,s) Q^H (p,t,s) Q (q,t,s) \right. \nonumber\\ &\quad - \left.\vphantom{\left[ \frac{p^2}{k} G^H (k,t,s) Q (p,t,s) - p G^H (p,t,s) Q (k,t,s)\right]} c_{kpq} G(p,t,s) Q (k,t,s) Q^H (q,t,s) - c_{kqp} G(q,t,s) Q (k,t,s) Q^H (p,t,s) \right] \nonumber\\ &\quad + k^3 \left[ (b_{kpq} + b_{kqp}) G^H (k,t,s) Q (p,t,s) Q (q,t,s) \vphantom{\left[ \frac{p}{k} G^H (k,t,s) Q (p,t,s) - p G^H (p,t,s) Q (k,t,s) \right] }\right. \nonumber\\ &\quad - \left. \frac{p}{k} c_{kpq} G^H (p,t,s) Q (k,t,s) Q (q,t,s) - \frac{q}{k} c_{kqp} G^H (q,t,s) Q (k,t,s) Q (p,t,s) \right] \nonumber\\ &\quad - k \left[ (c_{kpq} + c_{kqp}) G^H (k,t,s) Q^H (p,t,s) Q^H (q,t,s) \vphantom{\left[ \frac{p}{k} G^H (k,t,s) Q (p,t,s) - p G^H (p,t,s) Q (k,t,s) \right]}\right. \nonumber\\ &\quad - \frac{p}{k} c_{kpq} G^H (p,t,s) Q^H (k,t,s) Q^H (q,t,s) \nonumber\\ &\quad - \left.\vphantom{\left[ \frac{p^2}{k} G^H (k,t,s) Q (p,t,s) - p G^H (p,t,s) Q (k,t,s) \right] } \left. \frac{q}{k} c_{kqp} G^H (q,t,s) Q^H (k,t,s) Q^H (p,t,s) \right]\right\}, \end{align}

where we adopt the symmetry in $p$ and $q$. Equations (2.44) and (2.45) satisfy the detailed balance provided by (2.7) and (2.8), which is presented in Appendix C.

It should be noted that these triple correlations in the closure do not satisfy the maximally helical condition for the third moment given by (2.13) even if the energy and helicity spectra are maximally helical; namely, the closure yields

(2.46)\begin{equation} S^H (k,p,q,t) \neq 2k S (k,p,q,t), \end{equation}

even if $E^H (k,t) = 2k E(k,t)$ at the time $t$ (see also Appendix A). Although this result is not unique to the LRA, it holds for other closures based on the two-point velocity correlations such as the EDQNM, DIA and ALHDIA. The fact that the closure equations are inconsistent with the maximally helical condition is physically plausible. We define the relative helicity as

(2.47)\begin{equation} \rho^H (k,t) = \frac{E^H(k,t)}{2k E(k,t)}. \end{equation}

The relative helicity must be bounded via the realisability condition (1.1), namely

(2.48)\begin{equation} |\rho^H (k,t)| \le 1. \end{equation}

For homogeneous turbulence, it is well known that the relative helicity $\rho ^H(k,t)$ decreases even when it starts from the maximally helical condition $\rho ^H(k,t_0)=1$ (Morinishi, Nakabayashi & Ren Reference Morinishi, Nakabayashi and Ren2001). The closure equations follow this property. Actually, the EDQNM predicts the same tendency (André & Lesieur Reference André and Lesieur1977; Briard & Gomez Reference Briard and Gomez2017). Although some special forcing may realise the maximally helical spectra (Kessar et al. Reference Kessar, Plunian, Stepanov and Balarac2015; Stepanov et al. Reference Stepanov, Golbraikh, Frick and Shestakov2015; Plunian et al. Reference Plunian, Teimurazov, Stepanov and Verma2020), it is beyond the scope of this study because we focus on the scale-similar structures achieved by the nonlinearity of the Navier–Stokes equations. Considering this point, the property (2.13) derived from the Navier–Stokes equations with the Beltrami velocity field possibly holds only for the subensemble satisfying $\mathrm {i} \boldsymbol {k} \times \tilde {\boldsymbol {u}} (\boldsymbol {k},t) = k \tilde {\boldsymbol {u}} (\boldsymbol {k},t)$.

For the LRA, $Q(k,t,s)$, $Q^H (k,t,s)$, $G (k,t,s)$ and $G^H (k,t,s)$ obey the same equation as presented in (2.37)–(2.40). The initial condition for the response function (2.26) yields $G(k,s,s) = 1$ and $G^H (k,s,s) = 0$. The latter with (2.40) yields $G^H (k,t,s) = 0$. Therefore, the non-mirror-symmetric part of the response function must disappear for the LRA. Using the mirror-symmetric part of the response function $G (k,t,s)$, the two-time velocity correlations are reduced to

(2.49)\begin{gather} Q(k,t,s) = G(k,t,s) Q(k,s,s) \quad (t \ge s), \end{gather}

(2.50)\begin{gather}Q^H (k,t,s) = G(k,t,s) Q^H (k,s,s) \quad (t \ge s), \end{gather}

which represent the fluctuation-dissipation theorem (FDT) (Marconi et al. Reference Marconi, Puglisi, Rondoni and Vulpiani2008; Matsumoto et al. Reference Matsumoto, Otsuki, Ooshida and Goto2021). It is a feature of LRA that the FDT holds. It should be noted that helicity does not directly influence the response function because $\eta (k,t,s)$, given by (2.41), is solely determined by the mirror-symmetric part of the velocity correlation. Therefore, the effect of the breakage of mirror symmetry disappears from the response function (2.39). Similarly, Kaneda & Gotoh (Reference Kaneda and Gotoh1991) and Rubinstein & Zhou (Reference Rubinstein and Zhou1999) demonstrated that helicity does not affect the Lagrangian two-time velocity correlation. Therefore, we infer that the breakage of mirror symmetry does not affect both the response function and Lagrangian two-time velocity correlation for the LRA.

Using (2.49), (2.50) and $G^H (k,t,s) = 0$, the equations for the spectral densities of energy and helicity (2.35) and (2.36) yield

(2.51)\begin{align} \left( \frac{\partial}{\partial t} + 2\nu k^2 \right) Q (k,t,t) & = 2 {\rm \pi}\int_0^\infty \mathrm{d}p \int_0^\infty \mathrm{d}q \varDelta_{kpq} \int_{t_0}^t \mathrm{d}s \, G(k,t,s) G(p,t,s) G(q,t,s) \nonumber\\ & \quad \times \left\{ kpq b_{kpq} \left[ Q (p,s,s) - Q (k,s,s) \right] Q (q,s,s) \vphantom{\frac{pq}{k}}\right. \nonumber\\ &\quad - \left. \frac{pq}{k} c_{kpq} \left[ Q^H (p,s,s) - Q^H (k,s,s) \right] Q^H (q,s,s)\right\}, \end{align}

(2.52)\begin{align} \left(\frac{\partial}{\partial t} + 2\nu k^2 \right) Q^H (k,t,t) &= 2 {\rm \pi}\int_0^\infty \mathrm{d}p \int_0^\infty \mathrm{d}q \,\varDelta_{kpq} \int_{t_0}^t \mathrm{d}s \, G(k,t,s) G(p,t,s) G(q,t,s) \nonumber\\ &\quad \times \left\{ kpq b_{kpq} \left[ Q^H (p,s,s) - Q^H (k,s,s) \right] Q (q,s,s) \vphantom{\frac{pq}{k} }\right. \nonumber\\ &\quad - \left. \frac{pq}{k} c_{kpq} \left[ p^2 Q (p,s,s) - k^2 Q (k,s,s) \right] Q^H (q,s,s)\right\}. \end{align}

They possess the same form as the EDQNM except for the expression of the relaxation time. The LRA dynamically determines the relaxation time using the response function $G(k,t,s)$, whereas the EDQNM phenomenologically employs the eddy damping time scale (Orszag Reference Orszag1970; Lesieur Reference Lesieur2008).

The numerical simulation of the integro-differential equations provided by (2.39), (2.51) and (2.52) is a primitive approach to investigating the physical properties of the closure equations. However, the LRA involves time integration in the nonlinear interaction term of energy and helicity (2.51) and (2.52). Therefore, we have to perform the triple integration of $p$, $q$ and $s$ for the numerical simulation of LRA, whereas the EDQNM reduces this into the double integral of $p$ and $q$. Consequently, the computation time of LRA increases proportionally to $N_t$ compared with that of EDQNM, where $N_t$ denotes the number of the numerical time step. Therefore, the long-time integration of the high Reynolds number simulation of LRA remains a challenge. For the practical simulations of LRA, reductions such as Markovianisation (Gotoh, Kaneda & Bekki Reference Gotoh, Kaneda and Bekki1988) and the single-time expression of the response function with a short time expansion (Kitamura Reference Kitamura2020) may be effective. It is noteworthy that numerical simulations of closure equations are beneficial when we explore the behaviour of turbulence around the integral and viscous scales. Probably, differences between EDQNM type models and LRA are even more important around these scales than in the inertial range. However, in this study, we focus on the scale similarity of the LRA equations as a first step in understanding the physical properties of the theoretical closure approximation of turbulence in a non-mirror symmetric case. In future work, we will perform the numerical simulations of LRA to explore the behaviour of turbulence outside the similarity range. As will be discussed later, the scale similarity for the simultaneous energy and helicity cascades requires significantly more extensive scale separation than the mirror-symmetric case. In this study, we assume long range similarity scalings and investigate the analytical properties of the LRA for the non-mirror-symmetric case, instead of numerically solving the closure equations.

3.1. Scale similarity assumptions

Here, we assume the extended scale-similar spectra for energy and helicity (Golbraikh & Moiseev Reference Golbraikh and Moiseev2002; Golbraikh Reference Golbraikh2006; Kessar et al. Reference Kessar, Plunian, Stepanov and Balarac2015),

(3.1a,b)\begin{equation} E(k) = C_K^{(n)} \varepsilon^{7/3-n} (\varepsilon^H)^{{-}5/3+n} k^{{-}n}, \quad E^H(k) = C_H^{(m)} \varepsilon^{4/3-m} (\varepsilon^H)^{{-}2/3+m} k^{{-}m}, \end{equation}

where $C_K^{(n)}$ and $C_H^{(m)}$ are constants expected to be universal. In addition, the scale-similar time scale reads

(3.2)\begin{equation} \omega_k^{{-}1} = \varepsilon^{1/3-\ell} (\varepsilon^H)^{{-}2/3+\ell} k^{-\ell}. \end{equation}

Considering the simultaneous cascades of energy and helicity, an additional internal wavenumber or length scale exists (Moiseev & Chkhetiani Reference Moiseev and Chkhetiani1996):

(3.3)\begin{equation} k^H = \varepsilon^H/ \varepsilon. \end{equation}

According to K41 (Kolmogorov Reference Kolmogorov1941b), we assume that the field is approximately statistically steady in the inertial range. In such a case, the spectra and response function are reduced to

(3.4a–c)\begin{equation} Q(k,t,t) = Q(k), \quad Q^H(k,t,t) = Q^H(k),\quad G(k,t,s) = G(k,t-s) = G(\omega_k(t-s)). \end{equation}

One may consider that the scale-similar spectra (3.1a,b) should have the lower and upper bounds $k_{lower}$ and $k_{upper}$, and the response function equation and energy and helicity fluxes are affected from outside the scale-similar range such as the energy-containing and viscous ranges. In the following analyses, the wavenumber variables of integration are normalised by the wavenumber $k$ that lies in the similarity range. In such a case, the integral region accompanied by the similarity laws (3.1a,b), (3.2) and (3.4a–c) extends to the whole wavenumber range in the limit $k_{lower}/k \to 0$ and $k_{upper}/k \to \infty$. If the contributions from outside the similarity range vanish in the limit, one can justify this similarity analysis that the response function equation and energy and helicity fluxes are determined solely by the contributions from the similarity range. Details for the condition for the contributions from outside the similarity range to vanish are given in the supplementary material available at https://doi.org/10.1017/jfm.2021.708. The following scale-similarity analysis considers the situation that the scale-similar spectra extend to a sufficiently wide range that allows for the assumption $k_{lower}/k \ll 1$ and $k_{upper}/k \gg 1$. Besides, if distant interactions such expressed by the contributions from the wavenumber $\sim k_{lower}$ or $k_{upper}$ are not negligible, the wavenumber integral can diverge and this similarity analysis would lose its meaning. We will discuss the conditions for these integrals to converge later.

The realisability condition (1.1) or (2.48) requires

(3.5)\begin{equation} |\rho^H (k)| = \left| \frac{C_H^{(m)}}{2C_K^{(n)}} \left( \frac{\varepsilon^H}{k \varepsilon} \right)^{1-n+m} \right| = \left| \frac{C_H^{(m)}}{2C_K^{(n)}} \left(\frac{k^H}{k} \right)^{1-n+m} \right| \le 1. \end{equation}

The maximally helical condition corresponds to $1-n+m=0$ and $C_H^{(m)} = 2C_K^{(n)}$. However, the maximally helical condition is possibly incompatible with the closure equations in the inertial range in the sense that the closure equations yield (2.46), whereas the Navier–Stokes equations yield (2.13) in this condition. Besides, the relative helicity usually decreases due to the nonlinear interaction (Kraichnan Reference Kraichnan1973; Chen et al. Reference Chen, Chen and Eyink2003a). In several fundamental turbulent flows, the cascade directions of energy and helicity in the simultaneous cascade range (1.3a,b) are both forward (Borue & Orszag Reference Borue and Orszag1997; Chen et al. Reference Chen, Chen and Eyink2003a,Reference Chen, Chen, Eyink and Holmb; Mininni et al. Reference Mininni, Alexakis and Pouquet2006; Baerenzung et al. Reference Baerenzung, Politano, Ponty and Pouquet2008; Deusebio & Lindborg Reference Deusebio and Lindborg2014; Sahoo et al. Reference Sahoo, Bonaccorso and Biferale2015; Alexakis Reference Alexakis2017). Therefore, we consider the scale-similar structures at small scales and assume

(3.6)\begin{equation} k^H < k. \end{equation}

The condition provided by (3.6) is consistent with the concept of the locally isotropic turbulence presented by K41 , which considers small-scale structures. For the $k^H > k$ case, we provide a brief discussion in § 4.4. The realisability (3.5) accompanied by (3.6) requires the exponent of the relative helicity on $k^H/k$ to be positive,

(3.7)\begin{equation} 1-n+m >0. \end{equation}

In this case, a lower bound of the wavenumber that involves $k^H$ exists for the helicity spectrum to satisfy the realisability condition of (1.1) or (3.5). Furthermore, we have to assume that $k_{lower}/k_L$ and $k_L/k^H$ remain finite for the scale-similarity analysis to be reasonable, where $k_L$ denotes the inverse of the integral length scale (see supplementary material). The ratio $k_{lower}/k_L$ is expected to increase if the helicity is injected at large scales because the helicity hinders the energy transfer to small scales (André & Lesieur Reference André and Lesieur1977; Morinishi et al. Reference Morinishi, Nakabayashi and Ren2001; Kessar et al. Reference Kessar, Plunian, Stepanov and Balarac2015; Stepanov et al. Reference Stepanov, Golbraikh, Frick and Shestakov2015). Nevertheless, several numerical simulations of homogeneous turbulence suggest that the relative helicity $\rho ^H (k)$ rapidly decreases almost proportional to $k^{-1}$ as it goes away from the integral scale $k_L$ (André & Lesieur Reference André and Lesieur1977; Borue & Orszag Reference Borue and Orszag1997; Mininni et al. Reference Mininni, Alexakis and Pouquet2006; Baerenzung et al. Reference Baerenzung, Politano, Ponty and Pouquet2008), which suggests that $k_{lower}/k_L$ will remain finite. Besides, several numerical simulations suggest that $k_L/k^H \simeq O(1)$ (Borue & Orszag Reference Borue and Orszag1997; Baerenzung et al. Reference Baerenzung, Politano, Ponty and Pouquet2008; Mininni & Pouquet Reference Mininni and Pouquet2009; Deusebio & Lindborg Reference Deusebio and Lindborg2014). However, further verification is needed to evaluate the ratios $k_{lower}/k_L$ and $k_L/k^H$ in more general helical turbulent flows. In this study, we assume that $k_{lower}/k_L$ and $k_L/k^H$ remain finite and do not explicitly consider the lower bound of the scale-similar range involving $k_{lower}$ or $k^H$.

3.2. Response function

Substituting (3.1a,b), (3.2) and (3.4a–c) into the response function (2.39) in an inviscid condition, we have

(3.8)\begin{align} \frac{\partial}{\partial t} G (\omega_k (t-s)) &={-} k \int_0^\infty \mathrm{d}q \, q^3 J (q/k) \int_s^t \mathrm{d}s' \,G ( (q/k)^\ell \omega_k (t-s')) \nonumber\\ &\quad \times \frac{C_K^{(n)}}{2{\rm \pi}} \varepsilon^{7/3-n} (\varepsilon^H)^{{-}5/3+n} q^{{-}n-2} G (\omega_k (t-s)). \end{align}

Setting $\gamma \tau = \omega _k (t-s)$, $\gamma \sigma = \omega _k (t-s')$ and $\gamma = (2{\rm \pi} /C_K^{(n)})^{1/2}$, we obtain

(3.9)\begin{equation} \frac{\partial}{\partial \tau} \bar{G} (\tau) ={-} (k^H/k)^{{-}3+n+2\ell} \int_0^\infty \mathrm{d}w \, w^{{-}n+1} J (w) \int_0^\tau \mathrm{d}\sigma \,\bar{G} ( w^\ell \sigma) \bar{G} (\tau), \end{equation}

where $\bar {G} (\tau ) = G(\gamma \tau )$ and $w = q/k$. For (3.9) to be explicitly independent of the wavenumber $k$, the exponent of $k^H/k$ must be zero, namely

(3.10)\begin{equation} -3+n+2\ell = 0. \end{equation}

Accordingly, we have

(3.11)\begin{equation} \frac{\partial}{\partial \tau} \bar{G} (\tau) ={-} \int_0^\infty \mathrm{d} w \, w^{2\ell -2} J (w) \int_0^\tau \mathrm{d}\sigma \,\bar{G} ( w^\ell \sigma) \bar{G} (\tau). \end{equation}

Here, $\ell$ should be determined by another constraint, for example, the constant energy flux. If the $w$ integral in (3.11) diverges, this scale-similarity analysis would lose its meaning. For the DIA, the wavenumber integral of the Eulerian response function equation diverges at low wavenumber regions when we assume the simplified inertial range response function as well as the Kolmogorov energy spectrum (Leslie Reference Leslie1971). Note that $\ell < 0$ indicates that the small-scale velocity fluctuations exhibit longer time scales than the large scales, which may be unphysical. Hence, we consider $\ell \ge 0$. We can confirm that $J(x) \le (16{\rm \pi} /15)x$ for $x \le 1$ and its asymptote at $x\ll 1$ yields $J(x) = (16{\rm \pi} /15)x + O(x^3)$ (Kaneda Reference Kaneda1986). Therefore, for $\ell =0$, the $w$ integral in (3.11) diverges at $w \ll 1$. We decompose the $w$ integration in (3.11) into $w \le 1$ and $w \ge 1$. For $\ell > 0$ and $w \le 1$, we have

(3.12)\begin{align} & \int_0^1 \mathrm{d} w \, w^{2\ell - 2} J (w) \int_0^\tau \mathrm{d}\sigma \, \bar{G} (w^\ell\sigma) \nonumber\\ &\quad \le \int_0^1 \mathrm{d} w \, w^{2\ell - 2} \times \frac{16{\rm \pi}}{15} w \times \int_0^\tau \mathrm{d}\sigma \times 1 = \frac{16{\rm \pi}}{15} \int_0^1 \mathrm{d}w \, w^{2\ell - 1} \tau, \end{align}

because $\bar {G}(\tau ) \le 1$. Therefore, the left-hand side of (3.12) converges for a finite $\tau$ when $\ell > 0$. By putting $\zeta = w^\ell \sigma$, (3.11) yields

(3.13)\begin{equation} \frac{\partial}{\partial \tau} \bar{G} (\tau) ={-} \int_0^\infty \mathrm{d} w \, w^{\ell -2} J (w) \int_0^{w^\ell \tau} \mathrm{d} \zeta \,\bar{G} (\zeta) \bar{G} (\tau). \end{equation}

Then, for $w \ge 1$ with $\ell > 0$, we have

(3.14)\begin{align} & \int_1^\infty w^{\ell - 2} J (w) \int_0^{w^\ell \tau} \mathrm{d} \zeta \, \bar{G} (\zeta) \nonumber\\ &\quad \le \int_1^\infty w^{\ell - 2} \times \frac{16{\rm \pi}}{15} w^{{-}1} \times \int_0^\infty \mathrm{d} \zeta \,\bar{G} (\zeta) = \int_1^\infty \frac{16{\rm \pi}}{15} w^{\ell - 3} \int_0^\infty \mathrm{d} \zeta \,\bar{G} (\zeta), \end{align}

where we utilise $J(x) = J(1/x) \le (16{\rm \pi} /15)/x$ for $x \ge 1$ and $\int _0^{w^\ell \tau } \mathrm {d} \zeta \,\bar {G} (\zeta ) \le \int _0^\infty \mathrm {d} \zeta \,\bar {G} (\zeta )$. Because $\int _0^\infty \mathrm {d} \zeta \,\bar {G} (\zeta )$ is finite for a physically reasonable $\bar {G}(\tau )$, the left-hand side of (3.14) converges when $\ell < 2$. Consequently, the condition for the $w$ integral in (3.11) to converge yields $0 < \ell < 2$.

3.3. Energy flux

The most significant relation in turbulence statistics may be Kolmogorov's four-fifths law (Kolmogorov Reference Kolmogorov1941a; Frisch Reference Frisch1995),

(3.15)\begin{equation} \left\langle [{\rm \Delta} u_L (\boldsymbol{x},\boldsymbol{r},t) ]^3 \right \rangle ={-} \frac{4}{5} \varepsilon r, \end{equation}

where ${\rm \Delta} u_L (\boldsymbol {x},\boldsymbol {r},t) = {\rm \Delta} \boldsymbol {u} (\boldsymbol {x},\boldsymbol {r},t) \cdot \boldsymbol {r}/r$ and ${\rm \Delta} \boldsymbol {u} (\boldsymbol {x},\boldsymbol {r},t) = \boldsymbol {u} (\boldsymbol {x}+\boldsymbol {r},t) - \boldsymbol {u} (\boldsymbol {x},t)$. Notably, owing to the complete separation of the longitudinal third moment equation from the non-mirror symmetric contributions, the four-fifths law (3.15) holds even when the mirror symmetry is broken and the flow involves non-zero helicity (L'vov et al. Reference L'vov, Podivilov and Procaccia1997; Gomez et al. Reference Gomez, Politano and Pouquet2000; Kurien Reference Kurien2003). As demonstrated by Frisch (Reference Frisch1995), the four-fifths law described in physical space agrees with the constant energy flux in Fourier space, namely $\varPi = \varepsilon$, where $\varPi$ is defined by (2.10). Here, we investigate the scale-similar structure under the constant energy flux.

Substituting (3.1a,b), (3.2) and (3.4a–c) into the energy flux (2.10) yields (see Kraichnan Reference Kraichnan1966; Leslie Reference Leslie1971; Kaneda Reference Kaneda1986)

(3.16)\begin{align} \varPi (k) &= 4 {\rm \pi}^2 \int_k^\infty \mathrm{d}k' \int_0^k \mathrm{d} p' \int_{max (p',k'-p')}^{k'+p'} \mathrm{d}q' \nonumber\\ &\quad \times \int_{t_0}^t \mathrm{d}s\, G(\omega_{k'}(t-s)) G(\omega_{p'}(t-s)) G(\omega_{q'}(t-s)) \nonumber\\ &\quad \times \left\{ k'{}^3p'q' \left[(b_{k'p'q'} + b_{k'q'p'}) \left(\frac{C_K^{(n)}}{2{\rm \pi}} \varepsilon^{7/3-n} (\varepsilon^H)^{{-}5/3+n}\right)^2 p'{}^{{-}n-2} q'{}^{{-}n-2} \right. \right. \nonumber\\ &\quad - b_{k'p'q'} \left( \frac{C_K^{(n)}}{2{\rm \pi}} \varepsilon^{7/3-n} (\varepsilon^H)^{{-}5/3+n} \right)^2 k'{}^{{-}n-2} q'{}^{{-}n-2} \nonumber\\ &\quad - \left. b_{k'q'p'} \left( \frac{C_K^{(n)}}{2{\rm \pi}} \varepsilon^{7/3-n} (\varepsilon^H)^{{-}5/3+n} \right)^2 k'{}^{{-}n-2} p'{}^{{-}n-2} \right] \nonumber\\ &\quad - k'p'q' \left[ (c_{k'p'q'} + c_{k'q'p'}) \left( \frac{C_H^{(m)}}{4{\rm \pi}} \varepsilon^{4/3-m} (\varepsilon^H)^{{-}2/3+m} \right)^2 p'{}^{{-}m-2} q'{}^{{-}m-2} \right. \nonumber\\ &\quad - c_{k'p'q'} \left( \frac{C_H^{(m)}}{4{\rm \pi}} \varepsilon^{4/3-m} (\varepsilon^H)^{{-}2/3+m}\right)^2 k'{}^{{-}m-2} q'{}^{{-}m-2} \nonumber\\ &\quad - \left. \left. c_{k'q'p'} \left( \frac{C_H^{(m)}}{4{\rm \pi}} \varepsilon^{4/3-m} (\varepsilon^H)^{{-}2/3+m} \right)^2 k'{}^{{-}m-2} p'{}^{{-}m-2} \right] \right\}. \end{align}

Setting $p' = k' r$, $q' = k' w$, $k' = k/v$, $t_0 \to -\infty$, and $\gamma \tau = \omega _{k'} (t-s)$, we obtain

(3.17)\begin{align} \varPi (k) & = 4 {\rm \pi}^2 \left( \frac{C_K^{(n)}}{2{\rm \pi}} \right)^{3/2} ( k^H/k )^{{-}4+2n+\ell} \varepsilon \int_0^1 \mathrm{d}v \int_0^v \mathrm{d} r \int_{max (r,1-r)}^{1+r} \mathrm{d} w \nonumber\\ &\quad \times \int_0^\infty \mathrm{d} \tau \, \bar{G} (\tau) \bar{G} (r^\ell \tau) \bar{G} (w^\ell \tau) v^{{-}5+2n+\ell} \nonumber\\ &\quad \times \left\{\left[(b_{1rw} + b_{1wr}) r^{{-}n-1} w^{{-}n-1} - b_{1rw} r w^{{-}n-1}- b_{1wr} r^{{-}n-1} w \right] \vphantom{\left( \frac{C_H^{(m)}}{2 C_K^{(n)}} \right)^2}\right. \nonumber\\ &\quad - v^{2(1-n+m)} \left( \frac{C_H^{(m)}}{2 C_K^{(n)}} \right)^2 ( k^H/k )^{2(1-n+m)} \nonumber\\ &\quad \times \left.\vphantom{\left( \frac{C_H^{(m)}}{2 C_K^{(n)}} \right)^2} \left[(c_{1rw} + c_{1wr}) r^{{-}m-1} w^{{-}m-1} - c_{1rw} r w^{{-}m-1}- c_{1wr} r^{{-}m-1} w \right]\right\}. \end{align}

Considering the conditions in (3.6) and (3.7), the first line in $\{ \cdot \}$ should be the leading terms. In such a case, the constant energy flux independent of $k$ requires

(3.18)\begin{equation} -4+2n+\ell = 0. \end{equation}

Hence, we have

(3.19)\begin{align} \varPi (k) &= 4 {\rm \pi}^2 \left( \frac{C_K^{(n)}}{2{\rm \pi}} \right)^{3/2} \varepsilon \int_0^1 \mathrm{d}v \int_0^v \mathrm{d} r \int_{max (r,1-r)}^{1+r} \mathrm{d} w \int_0^\infty \mathrm{d} \tau \, \bar{G} (\tau) \bar{G} (r^{4-2n} \tau) \bar{G} (w^{4-2n} \tau) \nonumber\\ &\quad \times v^{{-}1} \left\{(b_{1rw} + b_{1wr}) r^{{-}n-1} w^{{-}n-1} - b_{1rw} rw^{{-}n-1}- b_{1wr} r^{{-}n-1} w \vphantom{\left( \frac{C_H^{(m)}}{2 C_K^{(n)}} \right)^2}\right. \nonumber\\ &\quad - v^{2(1-n+m)} \left( \frac{C_H^{(m)}}{2 C_K^{(n)}} \right)^2 ( k^H/k )^{2(1-n+m)} \nonumber\\ &\quad \times \left.\vphantom{\left( \frac{C_H^{(m)}}{2 C_K^{(n)}} \right)^2} \left[(c_{1rw} + c_{1wr}) r^{{-}m-1} w^{{-}m-1} - c_{1rw} r w^{{-}m-1}- c_{1wr} r^{{-}m-1} w \right] \right\}. \end{align}

Because we assume (3.7), the integration of (3.19) by parts on $v$ yields

(3.20)\begin{align} \varPi (k) &= 4 {\rm \pi}^2 \left( \frac{C_K^{(n)}}{2{\rm \pi}} \right)^{3/2} \varepsilon \int_0^1 \mathrm{d}v \int_{max (v,1-v)}^{1+v} \mathrm{d} w \int_0^\infty \mathrm{d} \tau \, \bar{G} (\tau) \bar{G} (v^{4-2n} \tau) \bar{G} (w^{4-2n} \tau) \nonumber\\ &\quad \times \left\{ \ln \left( \frac{1}{v} \right) \left[(b_{1vw} + b_{1wv}) v^{{-}n-1} w^{{-}n-1} - b_{1vw} vw^{{-}n-1}- b_{1wv} v^{{-}n-1} w \right] \vphantom{\left( \frac{C_H^{(m)}}{2 C_K^{(n)}} \right)^2}\right. \nonumber\\ &\quad - \frac{1}{2(1-n+m)} (1-v^{2(1-n+m)}) \left( \frac{C_H^{(m)}}{2 C_K^{(n)}} \right)^2 (k^H/k )^{2(1-n+m)} \nonumber\\ &\quad \times \left.\vphantom{\left( \frac{C_H^{(m)}}{2 C_K^{(n)}} \right)^2} \left[(c_{1vw} + c_{1wv}) v^{{-}m-1} w^{{-}m-1}- c_{1vw} v w^{{-}m-1} - c_{1wv} v^{{-}m-1} w \right] \vphantom{\left( \frac{1}{v} \right)}\right\}. \end{align}

Equation (3.20) still depends on $k$ via the two lines from the bottom in $\{ \cdot \}$. Because we consider (3.7), the requirement for $\varPi (k)$ to be independent of $k$ yields $k^H/k \ll 1$. This suggests that the constant energy flux holds only when the relative helicity is negligible, $|\rho ^H (k)| \ll 1$. Otherwise, the energy flux depends on the wavenumber $k$, which conflicts with the constant energy flux or the four-fifths law (3.15). The wavenumber-dependent energy flux also conflicts with the statistical steadiness of the energy spectrum. Using (2.3) and (2.10), the energy spectrum equation in an inviscid range reads

(3.21)\begin{equation} \frac{\partial}{\partial t} E(k,t) ={-} \frac{\partial}{\partial k} \varPi (k,t). \end{equation}

Therefore, if $\varPi$ depends on $k$, $E(k,t)$ must depend on the time $t$, which conflicts with the assumption of the statistical steadiness of the inertial range. Even if the scale-similar spectra are explicitly bounded by $k^H$, contributions from $k \le k^H$ are expected to vanish at $k^H/k \ll 1$, the same as the case of $k_{lower}$ (see supplementary material). If the contributions from $k \le k^H$ are negligible, we need not explicitly consider $k^H$ as a lower bound of the scale-similar helicity spectrum.

As discussed in § 3.2 for the response function, this scale-similarity analysis for the energy flux would also lose its meaning if the wavenumber integral in (3.20) diverges. According to Kraichnan (Reference Kraichnan1966), if the energy transfer is effectively local in wavenumber, the wavenumber integrations converge in a manner that permits the sole contribution of the inertial-range wavenumbers. We readily observe that the integrand of (3.20) is possibly singular at $v=0$ for $n>0$ and $m>0$. The asymptotic analysis at $v \ll 1$ yields (for the asymptotic analysis of geometrical factors refer to, for example, Leslie (Reference Leslie1971) and Gotoh (Reference Gotoh1998))

(3.22)\begin{align} & \int_{1-v}^{1+v} \mathrm{d} w \int_0^\infty \mathrm{d} \tau \, \bar{G} (\tau) \bar{G} (v^{4-2n}\tau) \bar{G} (w^{4-2n} \tau) \nonumber\\ &\quad \times\ln \left( \frac{1}{v} \right) \left[(b_{1vw} + b_{1wv}) v^{{-}n-1} w^{{-}n-1} - b_{1vw} vw^{{-}n-1}- b_{1wv} v^{{-}n-1} w \right] \nonumber\\ &\quad \propto \ln \left( \frac{1}{v} \right) \left[ v^{{-}n+2} + O(v^{{-}n+3}) \right], \end{align}

(3.23)\begin{align} & \int_{1-v}^{1+v} \mathrm{d} w \int_0^\infty \mathrm{d} \tau \, \bar{G} (\tau) \bar{G} (v^{4-2n} \tau) \bar{G} (w^{4-2n} \tau) \nonumber\\ &\quad \times(1-v^{2(1-n+m)}) \left[(c_{1vw} + c_{1wv}) v^{{-}m-1} w^{{-}m-1} - c_{1vw} v w^{{-}m-1}- c_{1wv} v^{{-}m-1} w \right] \nonumber\\ &\quad \propto v^{{-}m+2} + O(v^{{-}m+3}). \end{align}

Therefore, the integral of the energy-related part converges for $n<3$ and that of the helicity-related part converges for $m<3$. When $m<3$ and $k^H/k \ll 1$, (3.20) yields

(3.24)\begin{align} \varPi &= 4 {\rm \pi}^2 \left( \frac{C_K^{(n)}}{2{\rm \pi}} \right)^{3/2} \varepsilon \int_0^1 \mathrm{d}v \int_{max (v,1-v)}^{1+v} \mathrm{d} w \int_0^\infty \mathrm{d}\tau \, \bar{G} (\tau) \bar{G} (v^{4-2n} \tau) \bar{G} (w^{4-2n} \tau) \nonumber\\ &\quad \times \ln \left( \frac{1}{v} \right) \left[(b_{1vw} + b_{1wv}) v^{{-}n-1} w^{{-}n-1} - b_{1vw} vw^{{-}n-1}- b_{1wv} v^{{-}n-1} w \right]. \end{align}

For $\ell = 4-2n < 0$, it may be unphysical as discussed in § 3.2. Considering the condition that the integral in the response function equation converges, $0 < \ell < 2$, the physically meaningful constant energy flux regime holds for $1 < n < 2$ for the LRA. Regarding the LRA, the scale similarity for the response function equation requires (3.10). Combining (3.18) with (3.10) yields

(3.25a,b)\begin{equation} n = \frac{5}{3}, \quad \ell = \frac{2}{3}, \end{equation}

which corresponds with the conventional inertial-range energy spectrum and time scale, namely $E(k) = C_K^{(5/3)} \varepsilon ^{2/3} k^{-5/3}$ and $\omega _k^{-1} = \varepsilon ^{-1/3} k^{-2/3}$. Substituting (3.25a,b) into the energy flux (3.24) yields

(3.26)\begin{align} \varPi &= 4 {\rm \pi}^2 \left( \frac{C_K^{(5/3)}}{2{\rm \pi}} \right)^{3/2} \varepsilon \int_0^1 \mathrm{d}v \int_{max (v,1-v)}^{1+v} \mathrm{d} w \int_0^\infty \mathrm{d} \tau \, \bar{G} (\tau) \bar{G} (v^{2/3} \tau) \bar{G} (w^{2/3} \tau) \nonumber\\ &\quad \times \ln \left( \frac{1}{v} \right) \left[ (b_{1vw} + b_{1wv}) v^{{-}8/3} w^{{-}8/3} - b_{1vw} vw{}^{{-}8/3} - b_{1wv} v^{{-}8/3} w \right]. \end{align}

Similarly, the response function (3.11) yields

(3.27)\begin{equation} \partial_\tau \bar{G} (\tau) ={-} \int_0^\infty \mathrm{d} w \, w^{{-}2/3} J (w) \int_0^\tau \mathrm{d}\sigma \,\bar{G} ( w^{2/3} \sigma) \bar{G} (\tau). \end{equation}

Equations (3.26) and (3.27) are exactly the same as those in the mirror-symmetric case (Kaneda Reference Kaneda1986). Consequently, the scale similarity accompanied by the wavenumber-independent response function and constant energy flux yields the conventional Kolmogorov inertial range. Hence, the closure predicts that the Kolmogorov constant is universal regardless of the existence of helicity. The numerical integration of (3.26) and (3.27) with $\varPi = \varepsilon$ yields (Kaneda Reference Kaneda1986)

(3.28)\begin{equation} C_K^{(5/3)} = C_K = 1.72. \end{equation}

3.4. Helicity flux

The non-mirror-symmetric counterpart of the four-fifths law is expressed as (Chkhetiani Reference Chkhetiani1996; L'vov et al. Reference L'vov, Podivilov and Procaccia1997; Gomez et al. Reference Gomez, Politano and Pouquet2000; Kurien Reference Kurien2003)

(3.29)\begin{equation} \left\langle {\rm \Delta} \boldsymbol{u}_L (\boldsymbol{x},\boldsymbol{r},t) \cdot \boldsymbol{u}_T (\boldsymbol{x}+\boldsymbol{r},t) \times \boldsymbol{u}_T (\boldsymbol{x},t) \right \rangle = \tfrac{1}{15} \varepsilon^H r^2, \end{equation}

where ${\rm \Delta} \boldsymbol {u}_L (\boldsymbol {x},\boldsymbol {r},t) = {\rm \Delta} u_L (\boldsymbol {x},\boldsymbol {r},t) \boldsymbol {r}/r$, $\boldsymbol {u}_T (\boldsymbol {x},t) = \boldsymbol {u} (\boldsymbol {x},t) - [\boldsymbol {u} (\boldsymbol {x},t) \cdot \boldsymbol {r}/r] \boldsymbol {r}/r$. Although different notations are used in several studies, they are all essentially equivalent (Gomez et al. Reference Gomez, Politano and Pouquet2000; Kurien Reference Kurien2003). The notation in this study is based on L'vov et al. (Reference L'vov, Podivilov and Procaccia1997) (the coefficient is $2/15$ in L'vov et al. (Reference L'vov, Podivilov and Procaccia1997) because the definitions of helicity and its dissipation rate are divided by $2$, such that it is referred to as the two-fifteenths law). Notably, the four-fifths law (3.15) and its helical counterpart (3.29) simultaneously hold without contradiction (L'vov et al. Reference L'vov, Podivilov and Procaccia1997; Gomez et al. Reference Gomez, Politano and Pouquet2000; Kurien Reference Kurien2003). As the four-fifths law (3.15) agrees with the constant energy flux, (3.29) corresponds to the constant helicity flux in the wavenumber space (see Appendix D). Kurien, Taylor & Matsumoto (Reference Kurien, Taylor and Matsumoto2004b) confirmed (3.29) in the DNS of homogeneous turbulence injecting helicity.

Using (3.1a,b), (3.2) and (3.4a–c), the helicity flux (2.11) in the scale-similar range yields

(3.30)\begin{align} \varPi^H (k) &= 8 {\rm \pi}^2 \int_k^\infty \mathrm{d}k' \int_0^k \mathrm{d} p' \int_{max (p',k'-p')}^{k'+p'} \mathrm{d}q' \nonumber\\ &\quad \times \int_{t_0}^t \mathrm{d}s\, G(\omega_{k'}(t-s)) G(\omega_{p'}(t-s)) G(\omega_{q'}(t-s)) \nonumber\\ &\quad \times \left\{ k'{}^3p'q' \left[ \left( b_{k'p'q'} - \frac{q'{}^2}{k'{}^2} c_{k'q'p'} \right)\frac{C_K^{(n)}}{2{\rm \pi}} \frac{C_H^{(m)}}{4{\rm \pi}} \right. \right. \nonumber\\ &\quad \times \varepsilon^{11/3-n-m} (\varepsilon^H)^{{-}7/3+n+m} p'{}^{{-}m-2} q'{}^{{-}n-2} \nonumber\\ &\quad +\left( b_{k'q'p'} - \frac{p'{}^2}{k'{}^2} c_{k'p'q'} \right) \frac{C_K^{(n)}}{2{\rm \pi}} \frac{C_H^{(m)}}{4{\rm \pi}} \nonumber\\ &\quad \times \varepsilon^{11/3-n-m} (\varepsilon^H)^{{-}7/3+n+m} p'{}^{{-}n-2} q'{}^{{-}m-2} \nonumber\\ &\quad - b_{k'p'q'}\frac{C_K^{(n)}}{2{\rm \pi}} \frac{C_H^{(m)}}{4{\rm \pi}} \varepsilon^{11/3-n-m} (\varepsilon^H)^{{-}7/3+n+m} k'{}^{{-}m-2} q'{}^{{-}n-2} \nonumber\\ &\quad + c_{k'p'q'}\frac{C_K^{(n)}}{2{\rm \pi}} \frac{C_H^{(m)}}{4{\rm \pi}} \varepsilon^{11/3-n-m} (\varepsilon^H)^{{-}7/3+n+m} k'{}^{{-}n-2} q'{}^{{-}m-2} \nonumber\\ &\quad - b_{k'q'p'}\frac{C_K^{(n)}}{2{\rm \pi}} \frac{C_H^{(m)}}{4{\rm \pi}} \varepsilon^{11/3-n-m} (\varepsilon^H)^{{-}7/3+n+m} k'{}^{{-}m-2} p'{}^{{-}n-2} \nonumber\\ &\quad + \left. \left. c_{k'q'p'} \frac{C_K^{(n)}}{2{\rm \pi}} \frac{C_H^{(m)}}{4{\rm \pi}} \varepsilon^{11/3-n-m} (\varepsilon^H)^{{-}7/3+n+m} k'{}^{{-}n-2} p'{}^{{-}m-2} \right]\right\}. \end{align}

Setting $p' = k' r$, $q' = k' w$, $k' = k/v$, $t_0 \to -\infty$ and $\gamma \tau = \omega _{k'} (t-s)$, we obtain

(3.31)\begin{align} \varPi^H (k) &= 8 {\rm \pi}^2 \left(\frac{C_K^{(n)}}{2{\rm \pi}} \right)^{1/2} \frac{C_H^{(m)}}{4{\rm \pi}} (k^H/k)^{{-}4+n+m+\ell} \varepsilon^H \int_0^1 \mathrm{d}v \int_0^v \mathrm{d} r \int_{max (r,1-r)}^{1+r} \mathrm{d}w \nonumber\\ &\quad \times \int_0^\infty \mathrm{d}\tau \, \bar{G} (\tau) \bar{G} (r^\ell \tau) \bar{G} (w^\ell \tau) v^{{-}5+n+m+\ell} \nonumber\\ &\quad \times \left[ ( b_{1rw} - w^2 c_{1wr} ) r^{{-}m-1} w^{{-}n-1} + (b_{1wr} - r^2 c_{1rw} )r^{{-}n-1} w^{{-}m-1} \right. \nonumber\\ &\quad - \left. b_{1rw} r w^{{-}n-1} + c_{1rw} r w^{{-}m-1} - b_{1wr} r^{{-}n-1} w + c_{1wr} r^{{-}m-1} w\right]. \end{align}

For (3.31) to be independent of $k$, it requires

(3.32)\begin{equation} -4+n+m+\ell = 0. \end{equation}

By integrating (3.31) by parts on $v$, the helicity flux yields

(3.33)\begin{align} \varPi^H &= 8 {\rm \pi}^2 \left(\frac{C_K^{(n)}}{2{\rm \pi}} \right)^{1/2} \frac{C_H^{(m)}}{4{\rm \pi}} \varepsilon^H \int_0^1 \mathrm{d}v \int_{max (v,1-v)}^{1+v} \mathrm{d}w \nonumber\\ &\quad \times \int_0^\infty \mathrm{d}\tau \, \bar{G} (\tau) \bar{G} (v^{4-n-m} \tau) \bar{G} (w^{4-n-m} \tau) \nonumber\\ &\quad \times \ln \left( \frac{1}{v} \right)\left[ ( b_{1vw} - w^2 c_{1wv} ) v^{{-}m-1} w^{{-}n-1} + ( b_{1wv} - v^2 c_{1vw} ) v^{{-}n-1} w^{{-}m-1} \right. \nonumber\\ &\quad - \left. b_{1vw} v w^{{-}n-1} + c_{1vw} v w^{{-}m-1} - b_{1wv} v^{{-}n-1} w + c_{1wv} v^{{-}m-1} w \right]. \end{align}

The constant helicity flux condition requires the convergence of the integral of (3.33). Because the maximally helical condition is not consistent with the closure equations, the pure helicity cascade barely occurs, where $\varepsilon ^H$ solely determines the scaling laws. When both $\varepsilon$ and $\varepsilon ^H$ determine the scaling laws, the constant energy and helicity fluxes should be simultaneously satisfied. Consequently, both (3.18) and (3.32) are required, which yields $m=n$. Therefore, regardless of the choice of time scale or value of $\ell$, the simultaneous constant fluxes of energy and helicity yield the same exponent on the wavenumber for their spectra, namely $E(k), E^H (k) \propto k^{-n}$ when $k^H/k \ll 1$. The obtained result is consistent with the spectra associated with the helical time scale (1.4a,b) suggested by Kurien et al. (Reference Kurien, Taylor and Matsumoto2004a). Here, (3.33) reduces to

(3.34)\begin{align} \varPi^H &= 8 {\rm \pi}^2 \left(\frac{C_K^{(n)}}{2{\rm \pi}} \right)^{1/2} \frac{C_H^{(n)}}{4{\rm \pi}} \varepsilon^H \int_0^1 \mathrm{d}v \int_{max (v,1-v)}^{1+v} \mathrm{d}w \nonumber\\ &\quad \times \int_0^\infty \mathrm{d}\tau \, \bar{G} (\tau) \bar{G} (v^{4-2n} \tau) \bar{G} (w^{4-2n} \tau) \nonumber\\ &\quad \times \ln \left( \frac{1}{v} \right)\left[ ( b_{1vw} - w^2 c_{1wv} ) v^{{-}n-1} w^{{-}n-1} + ( b_{1wv} - v^2 c_{1vw} ) v^{{-}n-1} w^{{-}n-1} \right. \nonumber\\ &\quad - \left. b_{1vw} v w^{{-}n-1} + c_{1vw} v w^{{-}n-1} - b_{1wv} v^{{-}n-1} w + c_{1wv} v^{{-}n-1} w \right]. \end{align}

We can confirm that the integral of (3.34) converges for $n=m<3$, similar to that of the energy flux, as demonstrated in § 3.3. Similarly, $n>2$ may be unphysical because the small scales exhibit longer time scales than the large scales in this case. Considering the condition that the integral in the response function equation converges, $0 < \ell < 2$, we have $1 < n=m < 2$. For the LRA, (3.10) determines the exponent for the time scale, which yields

(3.35)\begin{equation} m = \frac{5}{3}. \end{equation}

Hence, the LRA with the constant energy and helicity fluxes yields the conventional simultaneous cascade range spectra (1.3a,b). Substituting $n=m=5/3$ and $\ell = 2/3$ into (3.34) yields

(3.36)\begin{align} \varPi^H &= 8 {\rm \pi}^2 \left(\frac{C_K^{(5/3)}}{2{\rm \pi}} \right)^{1/2} \frac{C_H^{(5/3)}}{4{\rm \pi}} \varepsilon^H \int_0^1 \mathrm{d}v \int_{max (v,1-v)}^{1+v} \mathrm{d}w \int_0^\infty \mathrm{d}\tau \, \bar{G} (\tau) \bar{G} (v^{2/3} \tau) \bar{G} (w^{2/3} \tau) \nonumber\\ &\quad \times \ln \left( \frac{1}{v} \right) \left[( b_{1vw} + b_{1wv} - v^2 c_{1vw} - w^2 c_{1wv} ) v^{{-}8/3} w^{{-}8/3} \right. \nonumber\\ &\quad - \left. ( b_{1vw} - c_{1vw} ) v w^{{-}8/3}- ( b_{1wv} - c_{1wv} ) v^{{-}8/3} w \right]. \end{align}

Here, we analytically have $b_{1vw} + b_{1wv} - v^2 c_{1vw} - w^2 c_{1wv} = 0$. The numerical integration of (3.27) and (3.36) with $\varPi ^H = \varepsilon ^H$ and (3.28) yields

(3.37)\begin{equation} C_H^{(5/3)} = C_H = 2.81. \end{equation}

3.5. Localness of interscale interaction

Kolmogorov's similarity hypothesis assumes the localness of the interscale interaction for the energy cascade process (Kolmogorov Reference Kolmogorov1941b). Several studies have investigated the localness of the interscale interaction based on closure theories (Kraichnan Reference Kraichnan1966, Reference Kraichnan1971; Kaneda Reference Kaneda1986; Domaradzki & Rogallo Reference Domaradzki and Rogallo1990). Similarly, we investigate it for the helicity cascade in the simultaneous energy and helicity cascades range based on the LRA. For the scale-similar range, the energy flux (2.10) can be rewritten as follows (Kraichnan Reference Kraichnan1966; Kaneda Reference Kaneda1986):

(3.38a,b)\begin{equation} \varPi = \int_1^\infty \mathrm{d}\alpha \frac{W(\alpha)}{\alpha},\quad \alpha = \frac{max (k,p,q)}{\min (k,p,q)}, \end{equation}

where

(3.39a–c)\begin{equation} \left.\begin{gathered} W (\alpha) = \frac{\ln \alpha}{\alpha} \int_1^{\alpha_*} \mathrm{d}\beta \frac{1}{\beta^2} S \left(1, \frac{1}{\alpha}, \frac{1}{\beta} \right) + \alpha \int_{\alpha_{**}}^\alpha \mathrm{d}\beta \frac{\ln \beta}{\beta^3} S \left( 1, \frac{1}{\beta}, \frac{\alpha}{\beta} \right), \\ \alpha_* = \left( \left| \frac{1}{\alpha} - \frac{1}{2} \right| + \frac{1}{2} \right)^{{-}1}, \quad \alpha_{**} = max (1, \alpha-1), \end{gathered}\right\}\end{equation}

and we utilise the scale-similar property for the triple correlation,

(3.40)\begin{equation} S(ak,ap,aq) = a^{{-}3} S(k,p,q). \end{equation}

Notably, the scale similarity provided by (3.40) is satisfied when (3.18) and $k^H/k \ll 1$ are satisfied for the closure. Here $W(\alpha )\,\mathrm {d} \alpha /\alpha$ denotes the contribution to $\varPi$ from all triad interactions such that the ratio of the maximum to minimum wavenumber lies between $\alpha$ and $\alpha + \mathrm {d} \alpha$ (Kraichnan Reference Kraichnan1966; Kaneda Reference Kaneda1986). We can also define the filling factor $F(\alpha )$ as (Kraichnan Reference Kraichnan1966; Gotoh Reference Gotoh1998)

(3.41)\begin{equation} F (\alpha) = \int_1^\alpha \,\mathrm{d}\alpha' \frac{W(\alpha')}{\alpha'}. \end{equation}

Similarly, the interscale interaction for the helicity cascade (2.11) using $\alpha$ reads

(3.42)\begin{gather} \varPi^H = \int_1^\infty \mathrm{d}\alpha \frac{W^H (\alpha)}{\alpha}, \end{gather}

(3.43)\begin{gather}W^H (\alpha) = \frac{\ln \alpha}{\alpha} \int_1^{\alpha_*} \mathrm{d}\beta \frac{1}{\beta^2} S^H \left(1, \frac{1}{\alpha}, \frac{1}{\beta} \right) + \alpha \int_{\alpha_{**}}^\alpha \mathrm{d}\beta \frac{\ln \beta}{\beta^3} S^H \left( 1, \frac{1}{\beta}, \frac{\alpha}{\beta} \right), \end{gather}

(3.44)\begin{gather}F^H (\alpha) = \int_1^\alpha \mathrm{d}\alpha' \frac{W^H (\alpha')}{\alpha'}, \end{gather}

where we employ the scale-similar property for the triple correlation in the helicity equation,

(3.45)\begin{equation} S^H (ak,ap,aq) = a^{{-}3} S^H (k,p,q), \end{equation}

which is satisfied when (3.32) is satisfied for the closure.

Figure 1 shows the profiles of $W(\alpha )$ and $W^H (\alpha )$ for the simultaneous energy and helicity cascades range of the LRA. We also plot $W(\alpha )$ for the inverse energy cascade range of the 2-D turbulence based on the LRA (Kaneda Reference Kaneda1987) for reference. In figure 2, we plot the profiles of $F(\alpha )$ and $F^H (\alpha )$. Kraichnan (Reference Kraichnan1966) demonstrated that $W(\alpha ) \propto \alpha ^{-4/3} \ln \alpha$ at $\alpha \gg 1$ in the inertial range for the ALHDIA. The LRA yields the same asymptote for $W(\alpha )$. The asymptote of its helicity counterpart also yields $W^H(\alpha ) \propto \alpha ^{-4/3} \ln \alpha$ at $\alpha \gg 1$ when $n=m=5/3$ and $\ell = 2/3$. The inset of figure 1 shows these asymptotes using a log–log plot. The fact that $W(\alpha )$ and $W^H (\alpha )$ decrease at large $\alpha$ indicates that we can interpret that the interscale interactions for energy and helicity cascades are both local.

Figure 1. Measures of localness of the interscale energy and helicity transfers in the simultaneous cascade range. The red solid and green dashed lines depict the energy and helicity transfers, respectively. The blue dotted line depicts the interscale energy transfer in the inverse energy cascade range of the 2-D turbulence based on the LRA. The inset shows the asymptotes of $W(\alpha )$ and $W^H(\alpha )$ for the LRA of the 3-D case.

Figure 2. Filling factor of the interscale energy and helicity transfers. The line colours and types are the same as in figure 1.

For the LRA of the 3-D case, $W (\alpha )$ peaks at $\alpha = 2$, and $F (\alpha )$ reaches $0.6$ at $\alpha = 4$. Hence, the contributions from the small wavenumber ratio are considered dominant for the 3-D energy cascade. In contrast, the $W^H (\alpha )$ profile is relatively gentle, which indicates that the interscale interaction for the helicity cascade is slightly non-local compared with the energy cascade in the sense that the contributions from $\alpha \ge 4$ to $W^H(\alpha )$ are larger than those to $W(\alpha )$. This result is consistent with the EDQNM performed by André & Lesieur (Reference André and Lesieur1977). Note that the helicity cascade is relatively local compared with the 2-D case (see also Kraichnan Reference Kraichnan1971). This slightly non-local property of the helicity cascade process can pave the way for further improvements in helical shell models (Benzi et al. Reference Benzi, Biferale, Kerr and Trovatore1996). In addition, the slightly non-local property of the helicity cascade suggests that it requires a significantly extensive scale separation to accurately predict the scale-similar range of the helicity spectrum.