2. Numerical experiments

The numerical data are generated by solving the isothermal compressible ideal single-fluid MHD equations, in the presence of both a mechanical and an electromotive driving. They read, in conservative form,

(2.1)$$\begin{gather} \partial_t \rho ={-} \boldsymbol{\nabla} \boldsymbol{\cdot} (\rho {\boldsymbol{v}}), \end{gather}$$

(2.2)$$\begin{gather}\partial_t (\rho {\boldsymbol{v}}) ={-} \boldsymbol{\nabla} \boldsymbol{\cdot} ( \rho {\boldsymbol{v}} {\boldsymbol{v}}{^\textrm{T} }+ (\rho c_s^2 + \tfrac{1}{2}|{\boldsymbol{b}}|^2){\boldsymbol{\mathsf{I}}} - {\boldsymbol{b}} {\boldsymbol{b}}{^\textrm{T} })+\rho\boldsymbol{f}_V, \end{gather}$$

(2.3)$$\begin{gather}\partial_t {\boldsymbol{b}} = \boldsymbol{\nabla} \times ({\boldsymbol{v}} \times {\boldsymbol{b}})+\boldsymbol{f}_M, \end{gather}$$

(2.4)$$\begin{gather}\boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{b}}=0, \end{gather}$$

where $\rho$ is the mass density, ${\boldsymbol {v}}$ the velocity and ${\boldsymbol {b}}$ the magnetic field. The $3\times 3$ identity matrix is denoted by ${\boldsymbol{\mathsf{I}}}$. In the isothermal case, the sound speed $c_s$ is constant, giving the (thermal) pressure $p=\rho c_s^2$. Energetically, this simple thermodynamic approximation introduces a new reservoir of specific potential energy, $u=c_s^2\ln (\rho /\rho _0)$ (with a reference density $\rho _0$), which couples via velocity field dilatation and density fluctuations with the kinetic energy. Kinetic and magnetic energy are linked via incompressible transverse shear Alfvénic fluctuations and via magnetic pressure generated by longitudinal magnetosonic fluctuations (see § 3). This physical approximation can thus be regarded as the structurally next step when going beyond the simplifying assumption of strict incompressibility.

The driving terms $\boldsymbol {f}_V$ and $\boldsymbol {f}_M$ inject kinetic energy at large scales and small-scale magnetic helical fluctuations, respectively, and are described below. The equations are solved using a fourth-order shock-capturing finite-volume solver, which makes use of the constrained-transport approach to ensure the solenoidality of the magnetic field. It is described in detail in Verma et al. (Reference Verma, Teissier, Henze and Müller2019) and Teissier (Reference Teissier2020). The main reconstruction method used is a fourth-order central weighted essentially non-oscillatory (CWENO) method (Levy, Puppo & Russo Reference Levy, Puppo and Russo1999), with a passage through point values in order to keep fourth-order accuracy (McCorquodale & Colella Reference McCorquodale and Colella2011; Buchmüller & Helzel Reference Buchmüller and Helzel2014). The Riemann problems at the cell interfaces are solved using the Rusanov approximation (Rusanov Reference Rusanov1961), and a multidimensional version of the same is used for the line-integrated electric field in the constrained-transport framework (Balsara Reference Balsara2010), which inherently maintains the solenoidality of the magnetic field up to machine precision (Evans & Hawley Reference Evans and Hawley1988). The time stepping is done through a strong stability-preserving Runge–Kutta (SSPRK) method, namely the one described by pseudocode 3 in Ketcheson (Reference Ketcheson2008), with a time step limited by the Courant–Friedrichs–Lewy criterion with a Courant number $C_{CFL}=1.5$. In order to prevent the appearance of negative mass densities as a result of numerical inaccuracies at strong discontinuities and shocks prevalent in high-Mach-number flows, a local reduction of the scheme's order is used in the vicinity of discontinuities, a technique often referred to as ‘flattening’ or ‘fallback approach’ (Colella & Woodward Reference Colella and Woodward1984; Balsara Reference Balsara2012; Núñez-de la Rosa & Munz Reference Núñez-de la Rosa and Munz2016). The increased accuracy of this higher-order numerical model permits one to obtain results comparable to a standard second-order-accurate numerical model at a significantly reduced resolution of the numerical grid (Verma et al. Reference Verma, Teissier, Henze and Müller2019).

The numerical experiments are performed as follows. Hydrodynamic turbulent steady states are first generated from a fluid at rest (${\boldsymbol {b}}={\boldsymbol {v}}=0$, $\rho =\rho _0=1$) in $512^3$ cubic simulation boxes of size $L=1$ with triply periodic boundary conditions by injecting large-scale kinetic energy through an acceleration field $\boldsymbol {f}_V$. The isothermal sound speed is $c_s=0.1$. The mechanical driving is carried out similarly to the experiments in Federrath et al. (Reference Federrath, Roman-Duval, Klessen, Schmidt and Mac Low2010) and Federrath (Reference Federrath2013). It is governed by an Ornstein–Uhlenbeck process, which injects either purely solenoidal or purely compressive energy at the wavenumber shells $1 \leq K \leq 2$ (with the shell $K \in \mathbb {N}$ defined by the wavevectors $\boldsymbol {k}$ such that $K \leq |\boldsymbol {k}|/\kappa < K+1$, with $\kappa =2{\rm \pi} /L$ the smallest wavenumber in the system). The energy injection rate ${\epsilon ^{K}_{inj}}$ governs the turbulent r.m.s. Mach number ${\mathcal {M}}$ of the statistical steady state obtained when numerical dissipation balances the injected energy. The forcing autocorrelation time is roughly the turbulent turnover time $t_{\mathcal {T}}=L/(2c_s{\mathcal {M}})$. The weak mean velocity field that appears as a result of the forcing is removed at each iteration. When the steady state is reached, a delta-correlated electromotive driving $\boldsymbol {f}_M$ is switched on at a particular instant, which injects fully positive helical magnetic fluctuations at small scales $K \in [48,52]$ with a defined magnetic energy injection rate ${\epsilon ^{M}_{inj}}$. This means that only one sign of magnetic helicity is injected (i.e. the electromotive forcing is only along one helical eigenvector, see § 5), so that, at each wavevector $\boldsymbol {k}$ where the driving injects both magnetic energy $E^M_{\boldsymbol {k},inj}$ and magnetic helicity $H^M_{\boldsymbol {k},inj}$, the so-called ‘realisability condition’ is saturated, $H^M_{\boldsymbol {k},inj}=2E^M_{\boldsymbol {k},inj}/|\boldsymbol {k}|$, and one can estimate the global magnetic helicity injection rate through ${\epsilon ^{{{\mathcal {H}}^M}}_{inj}}=(2{\epsilon ^{M}_{inj}}h_f)/(2{\rm \pi} K^M_f/L)$, with $h_f=+1$ the helical fraction of the injected fluctuations and $K^M_f=50$ the shell around which magnetic helicity is injected. As a consequence, only one sign of magnetic helicity (the positive one) dominates the system at all scales. This simplification allows one to limit the complexity of the present study to a practical level. The mechanical (large-scale) and the electromotive (small-scale) drivings are done at different spatial scales in order to observe the effects of compressibility. Indeed, these enter the magnetic helicity dynamics via a term $\sim \boldsymbol {\nabla } \boldsymbol {\cdot } {\boldsymbol {v}}$, so that if both the kinetic and magnetic fields are driven at small scales (cf. Müller et al. Reference Müller, Malapaka and Busse2012), the velocity field at larger scales would be solely excited by an inverse transfer of the magnetic field and hence far too weak to have observable effects.

In the present paper, shell-to-shell helical analysis of the least and most compressible runs presented in Teissier & Müller (Reference Teissier and Müller2021) are considered, which are labelled by ‘M01s4’, ‘M11s’, ‘M1c’ and ‘M8c’. An outline of that work is given at the beginning of § 6. The number in the label stands for the approximate r.m.s. Mach number during the hydrodynamic turbulent steady state, which is ${\mathcal {M}} \approx 0.116, 11.1, 0.797$ and $7.87$, respectively, and the letter for the forcing type, either purely solenoidal or compressive. The magnetic-to-kinetic energy injection rate ${\epsilon ^{M}_{inj}}/{\epsilon ^{K}_{inj}}$ is taken as unity for all the runs except the M01s4 one, for which ${\epsilon ^{M}_{inj}}/{\epsilon ^{K}_{inj}}=4$, a value that has been observed to result in faster relaxation of spectral dynamics as compared to unity.

3. Isothermal energetics

In compressible isothermal MHD, energy can be stored in three reservoirs: as kinetic energy ${\mathcal {E}}^K=\frac {1}{2}\rho |{\boldsymbol {v}}|^2$, as magnetic energy ${\mathcal {E}}^M=\frac {1}{2}|{\boldsymbol {b}}|^2$ and as potential energy, in the form of density fluctuations ${\mathcal {E}}^\rho =\rho c_s^2 \ln (\rho /\rho _0)$, with $\rho _0$ the mean density in the system (Kritsuk, Wagner & Norman Reference Kritsuk, Wagner and Norman2013). Their time evolution and the energetic exchanges between them are described in the following.

Starting with the ideal induction equation (2.3), without the forcing term $\partial _t {\boldsymbol {b}}= -({\boldsymbol {v}} \boldsymbol {\cdot } \boldsymbol {\nabla }) {\boldsymbol {b}}+({\boldsymbol {b}} \boldsymbol {\cdot } \boldsymbol {\nabla }) {\boldsymbol {v}}-{\boldsymbol {b}}(\boldsymbol {\nabla } \boldsymbol {\cdot } {\boldsymbol {v}})$, the magnetic energy is governed by

(3.1)\begin{equation} \partial_t {\mathcal{E}}^M=\underbrace{-{\boldsymbol{b}} \boldsymbol{\cdot} ({\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{b}}}_{{\mathcal{B}}_1}+\underbrace{{\boldsymbol{b}}\boldsymbol{\cdot}({\boldsymbol{b}} \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{v}}}_{{\mathcal{B}}_2}-|{\boldsymbol{b}}|^2\boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}. \end{equation}

The terms ${\mathcal {B}}_1$ and ${\mathcal {B}}_2$ can be reformulated as

(3.2)$$\begin{gather} {\mathcal{B}}_1={-}{\boldsymbol{b}} \boldsymbol{\cdot} ({\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{b}}={-}\boldsymbol{\nabla} \boldsymbol{\cdot} (\tfrac12 |{\boldsymbol{b}}|^2{\boldsymbol{v}})+\tfrac12 |{\boldsymbol{b}}|^2 \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}{,} \end{gather}$$

(3.3)$$\begin{gather}{\mathcal{B}}_2={\boldsymbol{b}}\boldsymbol{\cdot}({\boldsymbol{b}} \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{v}}=\boldsymbol{\nabla} \boldsymbol{\cdot} (({\boldsymbol{v}} \boldsymbol{\cdot} {\boldsymbol{b}}){\boldsymbol{b}})-{\boldsymbol{v}}\boldsymbol{\cdot}({\boldsymbol{b}}\boldsymbol{\cdot}\boldsymbol{\nabla}){\boldsymbol{b}}. \end{gather}$$

Plugging these terms into (3.1) leads to

(3.4)\begin{equation} \partial_t {\mathcal{E}}^M={-}\boldsymbol{\nabla} \boldsymbol{\cdot}(\tfrac12 |{\boldsymbol{b}}|^2 {\boldsymbol{v}}-({\boldsymbol{v}} \boldsymbol{\cdot} {\boldsymbol{b}}){\boldsymbol{b}})-{\boldsymbol{v}} \boldsymbol{\cdot} ({\boldsymbol{b}} \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{b}}-\tfrac12 |{\boldsymbol{b}}|^2 \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}. \end{equation}

As compared with the incompressible case, the only new term is $-\frac 12 |{\boldsymbol {b}}|^2\boldsymbol {\nabla } \boldsymbol {\cdot } {\boldsymbol {v}}$. The conservative first term on the right-hand side of (3.4) corresponds to energetic transfers inside the magnetic energy reservoir, whereas the two remaining ones, $-{\boldsymbol {v}} \boldsymbol {\cdot } ({\boldsymbol {b}} \boldsymbol {\cdot } \boldsymbol {\nabla }) {\boldsymbol {b}}$ and $-\frac 12 |{\boldsymbol {b}}|^2\boldsymbol {\nabla } \boldsymbol {\cdot } {\boldsymbol {v}}$, correspond to conversion of energy between the magnetic and kinetic energy reservoirs, through magnetic tension and magnetic pressure, respectively. Their counterparts appear in the kinetic energy equation, as shown below.

Using the time evolution of the mass density and the velocity field,

(3.5)$$\begin{gather} \partial_t \rho ={-} \boldsymbol{\nabla} \boldsymbol{\cdot}(\rho {\boldsymbol{v}}), \end{gather}$$

(3.6)$$\begin{gather}\partial_t {\boldsymbol{v}} ={-} ({\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{\nabla}){\boldsymbol{v}}+\tfrac{1}{\rho}\boldsymbol{\nabla} \boldsymbol{\cdot} \left({\boldsymbol{b}}{\boldsymbol{b}}{^\textrm{T}}-\left(\rho c_s^2+\tfrac12 |{\boldsymbol{b}}|^2\right){\boldsymbol{\mathsf{I}}}\right), \end{gather}$$

yields the time evolution of the kinetic energy:

(3.7)\begin{align} \partial_t {\mathcal{E}}^K &=\rho {\boldsymbol{v}} \boldsymbol{\cdot} \partial_t {\boldsymbol{v}}+\tfrac12 |{\boldsymbol{v}}|^2\partial_t \rho, \end{align}

(3.8)\begin{align} &=\underbrace{-\rho {\boldsymbol{v}} \boldsymbol{\cdot} ({\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{\nabla}){\boldsymbol{v}}}_{{\mathcal{V}}_1}+{\boldsymbol{v}} \boldsymbol{\cdot} ({\boldsymbol{b}} \boldsymbol{\cdot}\boldsymbol{\nabla}){\boldsymbol{b}}\underbrace{-\tfrac12 {\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{\nabla}(|{\boldsymbol{b}}|^2)}_{{\mathcal{V}}_2}\underbrace{-{\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{\nabla}(\rho c_s^2)}_{{\mathcal{V}}_3} +\tfrac12 |{\boldsymbol{v}}|^2\partial_t \rho. \end{align}

The terms ${\mathcal {V}}_i$ can be reformulated as

(3.9)$$\begin{gather} {\mathcal{V}}_1 ={-}\rho {\boldsymbol{v}} \boldsymbol{\cdot} ({\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{\nabla}){\boldsymbol{v}}={-}\boldsymbol{\nabla} \boldsymbol{\cdot} (\tfrac12 \rho |{\boldsymbol{v}}|^2 {\boldsymbol{v}})-\tfrac12 |{\boldsymbol{v}}|^2 \partial_t \rho. \end{gather}$$

(3.10)$$\begin{gather}{\mathcal{V}}_2 ={-}\tfrac12 {\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{\nabla}(|{\boldsymbol{b}}|^2)={-}\boldsymbol{\nabla} \boldsymbol{\cdot} (\tfrac12 |{\boldsymbol{b}}|^2{\boldsymbol{v}})+\tfrac12 |{\boldsymbol{b}}|^2 \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}, \end{gather}$$

(3.11)$$\begin{gather}{\mathcal{V}}_3 ={-}{\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{\nabla}(\rho c_s^2)={-}\boldsymbol{\nabla} \boldsymbol{\cdot} (\rho c_s^2{\boldsymbol{v}})+\rho c_s^2\boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}. \end{gather}$$

Plugging them into (3.8), one obtains

(3.12)\begin{equation} \partial_t {\mathcal{E}}^K={-}\boldsymbol{\nabla} \boldsymbol{\cdot} (\tfrac12 \rho |{\boldsymbol{v}}|^2 {\boldsymbol{v}}+p^*{\boldsymbol{v}})+{\boldsymbol{v}} \boldsymbol{\cdot} ({\boldsymbol{b}} \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{b}}+\tfrac12 |{\boldsymbol{b}}|^2 \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}+\rho c_s^2 \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}, \end{equation}

with $p^*=\rho c_s^2+\frac 12 |{\boldsymbol {b}}|^2$ the sum of thermal and magnetic pressure. In this equation (3.12), the same $-{\boldsymbol {v}} \boldsymbol {\cdot } ({\boldsymbol {b}} \boldsymbol {\cdot } \boldsymbol {\nabla }) {\boldsymbol {b}}$ and $-\frac 12 |{\boldsymbol {b}}|^2\boldsymbol {\nabla } \boldsymbol {\cdot } {\boldsymbol {v}}$ terms as in (3.4) appear as well, but with an opposite sign, confirming that they correspond to energetic exchanges between the magnetic and kinetic energy reservoirs. Compared to incompressible MHD, where energy conversion can only take place through magnetic stretching i.e. shear Alfvén fluctuations, isothermal MHD allows energetic conversion through two additional channels: kinetic $\leftrightarrow$ magnetic through magnetic pressure (fast and slow magnetosonic fluctuations) and kinetic $\leftrightarrow$ potential through the velocity field's dilatational part $\boldsymbol {\nabla } \boldsymbol {\cdot } {\boldsymbol {v}}$, i.e. work done by the interaction of the velocity field with thermal pressure. Indeed, the last term on the right-hand side of (3.12) can be found with an opposite sign in the potential energy ${\mathcal {E}}^\rho =\rho c_s^2 \ln (\rho /\rho _0)$ time evolution:

(3.13)\begin{align} \partial_t {\mathcal{E}}^\rho &= c_s^2((\partial_t \rho) \ln(\rho/\rho_0)+\rho \partial_t \rho/\rho), \end{align}

(3.14)\begin{align} &= c_s^2 -\boldsymbol{\nabla} \boldsymbol{\cdot} (\rho {\boldsymbol{v}}) (1+\ln(\rho/\rho_0)), \end{align}

(3.15)\begin{align} &={-}\boldsymbol{\nabla} \boldsymbol{\cdot} ({\mathcal{E}}^\rho {\boldsymbol{v}}) - \rho c_s^2 \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}. \end{align}

To summarize, the energy conversion terms between reservoirs can be underlined by performing a volume average of (3.4), (3.12) and (3.15) over a closed or triply-periodic domain:

(3.16)$$\begin{gather} \partial_t \langle {\mathcal{E}}^K \rangle = \langle {\boldsymbol{v}} \boldsymbol{\cdot}({\boldsymbol{b}} \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{b}} \rangle+\langle \tfrac12 |{\boldsymbol{b}}|^2 \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}} \rangle+\langle \rho c_s^2 \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}} \rangle, \end{gather}$$

(3.17)$$\begin{gather}\partial_t \langle {\mathcal{E}}^M \rangle ={-}\langle {\boldsymbol{v}} \boldsymbol{\cdot}({\boldsymbol{b}} \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{b}} \rangle-\langle \tfrac12 |{\boldsymbol{b}}|^2 \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}} \rangle, \end{gather}$$

(3.18)$$\begin{gather}\partial_t \langle {\mathcal{E}}^\rho \rangle ={-}\langle \rho c_s^2 \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}} \rangle. \end{gather}$$

Conservative cross-scale nonlinear fluxes for each energy reservoir are present as well in the form of divergence terms in (3.4), (3.12) and (3.15).

In summary, the influence of isothermal compressibility has two dynamic consequences: (i) the emergence of a potential energy reservoir that is linked with kinetic energy via the interaction of thermal pressure and the velocity field; and (ii) the emergence of magnetosonic waves, which, in addition to inherently incompressible shear Alfvén waves (with perturbations orthogonal in Fourier space to the plane spanned by the wavevector and the large-scale magnetic field), open a second channel of kinetic $\leftrightarrow$ magnetic energy conversion via the interaction of velocity and magnetic pressure. In addition, the compressive velocity component may also modify the magnetic tension exchange. All these effects have in common that they appear as dependences on the velocity dilatation.

This brief overview of isothermal MHD energy dynamics is given to aid the subsequent interpretation of magnetic helicity transfer, which is the main focus of the present work.

4. Shell-to-shell transfers

The formalism used in the incompressible case in Alexakis et al. (Reference Alexakis, Mininni and Pouquet2006) for magnetic helicity and energetic shell-to-shell transfers is reviewed and extended here for compressible MHD. For a field $\boldsymbol {f}$, the field obtained by keeping only the wavenumbers in a certain shell $K \in \mathbb {N}$, defined by the wavevectors $\boldsymbol {k}$ such that $K \leq |\boldsymbol {k}|/\kappa < K+1$ (with $\kappa =2{\rm \pi} /L$ the smallest wavenumber in the system) is labelled by $\boldsymbol {f}_K$.

In the absence of forcing and of dissipative effects, $H^M_K=\langle {\boldsymbol {a}}_K \boldsymbol {\cdot } {\boldsymbol {b}}_K \rangle$, the magnetic helicity present in shell $K$ is governed by (Alexakis et al. Reference Alexakis, Mininni and Pouquet2006; Plunian, Stepanov & Verma Reference Plunian, Stepanov and Verma2019)

(4.1)\begin{equation} \partial_t {{\mathcal{H}}^M_K} = \sum_Q {\mathcal{T}}^{{{\mathcal{H}}^M}}(Q,K)= \sum_Q 2\langle {\boldsymbol{b}}_K \boldsymbol{\cdot} ({\boldsymbol{v}} \times {\boldsymbol{b}}_Q) \rangle, \end{equation}

where ${\mathcal {T}}^{{{\mathcal {H}}^M}}(Q,K)$ corresponds to the transfer rate of magnetic helicity from shell $Q$ to shell $K$. This transfer function, derived in the incompressible case, remains valid without formal modification in compressible MHD because of the magnetic field solenoidality. However, as outlined in § 3, isothermal compressibility changes the energetic dynamics of the MHD system fundamentally and impacts the nonlinear transfer of magnetic helicity via compressible modifications of velocity and the magnetic field. The interpretation as a transfer rate of magnetic helicity between shells is justified through the antisymmetric property ${\mathcal {T}}^{{{\mathcal {H}}^M}}(Q,K)=-{\mathcal {T}}^{{{\mathcal {H}}^M}}(K,Q)$, and because magnetic helicity is a purely magnetic ideal invariant. Hence, the velocity field cannot exchange helicity with the magnetic field and plays only a mediating role for transfers of magnetic helicity between two magnetic modes. The situation regarding energetic transfers is more subtle and left for future work, as outlined below. Please note that in Alexakis et al. (Reference Alexakis, Mininni and Pouquet2006) the inverse convention is used, with $T_h(Q,K)$ the transfer of magnetic helicity received by shell $Q$ from shell $K$. The magnetic helicity is also defined there as ${{\mathcal {H}}^M}=\frac {1}{2} \int {\boldsymbol {a}} \boldsymbol {\cdot } {\boldsymbol {b}} \, \, \textrm {d}V$ so that a factor 2 is not present in the transfer rates. The velocity field can also be decomposed in shells, yielding the transfer function

(4.2)\begin{equation} {\mathcal{T}}^{{{\mathcal{H}}^M}}(Q,P,K)= 2\langle {\boldsymbol{b}}_K \boldsymbol{\cdot} ({\boldsymbol{v}}_P \times {\boldsymbol{b}}_Q) \rangle, \end{equation}

which represents the transfer of magnetic helicity from shell $Q$ to shell $K$ mediated by the velocity field at shell $P$. Summing the function of three variables ${\mathcal {T}}^{{{\mathcal {H}}^M}}(Q,P,K)$ along $Q$ allows one to quantify the importance of the mediating velocity field with respect to magnetic helicity transfers to shell $K$ as a function of the scale $P$:

(4.3)\begin{equation} {{\mathcal{M}}^{{{\mathcal{H}}^M}}(P,K)=\sum_Q {\mathcal{T}}^{{{\mathcal{H}}^M}}(Q,P,K)}. \end{equation}

In this work, the indices $K$ and $Q$ always correspond to magnetic field shells, whereas the index $P$ always corresponds to the velocity field.

Following § 3, a shell decomposition allows one to derive the time evolution of ${\mathcal {E}}^M_K=\langle \frac 12 |{\boldsymbol {b}}_K|^2 \rangle$, the magnetic energy contained in shell $K$,

(4.4)\begin{equation} \partial_t {\mathcal{E}}^M_K = \sum_{Q,P} {\mathcal{T}}^{\mathcal{E}}_{b \to b}(Q,P,K) + \sum_{Q,P} \left({\mathcal{T}}^{\mathcal{E}}_{v \xrightarrow[T]{} b}(P,Q,K)+ {\mathcal{T}}^{\mathcal{E}}_{v \xrightarrow[P]{} b}(P,Q,K)\right), \end{equation}

with the transfer functions

(4.5)$$\begin{gather} {\mathcal{T}}^{\mathcal{E}}_{b \to b}(Q,P,K) = \langle -{\boldsymbol{b}}_K \boldsymbol{\cdot} ({\boldsymbol{v}}_P \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{b}}_Q - \tfrac{1}{2} ({\boldsymbol{b}}_K \boldsymbol{\cdot} {\boldsymbol{b}}_Q) \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}_P \rangle, \end{gather}$$

(4.6)$$\begin{gather}{\mathcal{T}}^{\mathcal{E}}_{v \xrightarrow[T]{} b}(P,Q,K) = \langle {\boldsymbol{b}}_K \boldsymbol{\cdot} ({\boldsymbol{b}}_Q \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{v}}_P \rangle, \end{gather}$$

(4.7)$$\begin{gather}{\mathcal{T}}^{\mathcal{E}}_{v \xrightarrow[P]{} b}(P,Q,K) = \langle - \tfrac{1}{2} ({\boldsymbol{b}}_K \boldsymbol{\cdot} {\boldsymbol{b}}_Q) \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}_P \rangle. \end{gather}$$

The ${\mathcal {T}}^{\mathcal {E}}_{b \to b}$ term represents transfers inside the magnetic energy reservoir, and the other two between magnetic and kinetic energy reservoirs through magnetic tension (${\mathcal {T}}^{\mathcal {E}}_{v \xrightarrow [T]{} b}$) and magnetic pressure (${\mathcal {T}}^{\mathcal {E}}_{v \xrightarrow [P]{} b}$). They have, respectively, the following counterpart for kinetic energy exchanges:

(4.8)$$\begin{gather} {\mathcal{T}}^{\mathcal{E}}_{b \xrightarrow[T]{} v}(K,Q,P) = \langle {\boldsymbol{v}}_P \boldsymbol{\cdot} ({\boldsymbol{b}}_Q \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{b}}_K \rangle, \end{gather}$$

(4.9)$$\begin{gather}{\mathcal{T}}^{\mathcal{E}}_{b \xrightarrow[P]{} v}(K,Q,P) = \langle -\tfrac12 {\boldsymbol{v}}_P \boldsymbol{\cdot} \boldsymbol{\nabla} ({\boldsymbol{b}}_K \boldsymbol{\cdot} {\boldsymbol{b}}_Q) \rangle. \end{gather}$$

These transfer functions have appropriate antisymmetries so that they could be interpreted as transfers of energy between fields and shells. Indeed, for magnetic $\leftrightarrow$ magnetic transfers, ${\mathcal {T}}^{\mathcal {E}}_{b \to b}(Q,P,K)=-{\mathcal {T}}^{\mathcal {E}}_{b \to b}(K,P,Q)$, and for kinetic $\leftrightarrow$ magnetic transfers ${\mathcal {T}}^{\mathcal {E}}_{b \xrightarrow [T]{} v}(K,Q,P)=-{\mathcal {T}}^{\mathcal {E}}_{v \xrightarrow [T]{} b}(P,Q,K)$ for magnetic stretching, and similarly for the magnetic pressure contribution ${\mathcal {T}}^{\mathcal {E}}_{v \xrightarrow [P]{} b}$. However, as pointed out in Aluie (Reference Aluie2017), by using a coarse-graining approach, magnetic stretching does not result in a cross-scale conversion term between magnetic and kinetic energy reservoirs in incompressible MHD. It has been shown that the Lorentz force has two simultaneous effects: a local-in-scale energy conversion between both reservoirs, and a transfer across scales inside each of the reservoirs. In other words, even though ${\mathcal {T}}^{\mathcal {E}}_{b \xrightarrow [T]{} v}(K,Q,P)=-{\mathcal {T}}^{\mathcal {E}}_{v \xrightarrow [T]{} b}(P,Q,K)$ shows the total effect of the magnetic stretching term, the energy is in fact transferred from ${\boldsymbol {v}}_P$ to ${\boldsymbol {b}}_P$ in a Fourier-local fashion, and, simultaneously, magnetic $\leftrightarrow$ magnetic and kinetic $\leftrightarrow$ kinetic transfers across scales occur. How to split these contributions in incompressible MHD is described in that reference.

In compressible MHD, the situation is more complex, since, in addition to these two effects, which need to be considered separately, transfers across scales could occur directly through the coupling with the mass density field, which is now included in the kinetic energy field. Moreover, the energy contained at a certain length scale $l_P\sim 1/P$ is not $\int \frac 12 |{\boldsymbol {v}}_P|^2$ any longer, but may be defined in several ways: using $\int \frac 12 (\rho ^\alpha {\boldsymbol {v}})_P\boldsymbol {\cdot }(\rho ^{1-\alpha }{\boldsymbol {v}})_P$ for any $\alpha \in [0,1/2]$, typical choices being $\alpha =0$ or $\alpha =1/2$ (Graham, Cameron & Schüssler Reference Graham, Cameron and Schüssler2010; Aluie Reference Aluie2013; Grete et al. Reference Grete, O'Shea, Beckwith, Schmidt and Christlieb2017). Given this level of additional complexity regarding energetic transfers, we restrict the scope of this paper to shell-to-shell magnetic helicity transfers, where only the velocity field, without the density, enters into play.

Some insight into the impact of the physics underlying energetic exchanges on the magnetic helicity shell-to-shell transport can be determined by noticing that

(4.10)\begin{align} \partial_t {{\mathcal{H}}^M_K} &= 2\langle {\boldsymbol{b}}_K \boldsymbol{\cdot} \partial_t {\boldsymbol{a}} \rangle, \end{align}

(4.11)\begin{align} &= 2\langle {\boldsymbol{b}}_K \boldsymbol{\cdot} \mathrm{\textbf{rot}^{{-}1}}(\partial_t{\boldsymbol{b}}) \rangle, \end{align}

(4.12)\begin{align} &= 2\langle {\boldsymbol{b}}_K \boldsymbol{\cdot} \mathrm{\textbf{rot}^{{-}1}}(-({\boldsymbol{v}}\boldsymbol{\cdot}\boldsymbol{\nabla}){\boldsymbol{b}}+({\boldsymbol{b}}\boldsymbol{\cdot}\boldsymbol{\nabla}){\boldsymbol{v}}-{\boldsymbol{b}} \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}) \rangle, \end{align}

where $\mathrm{\textbf {rot}^{-1}}(\boldsymbol {f})$ returns the solenoidal field $\boldsymbol {g}$ whose curl is the solenoidal part of field $\boldsymbol {f}$ (defined as $\boldsymbol {\hat {g}}_{\boldsymbol {k}}={\textrm {i} \boldsymbol {k} \times \boldsymbol {\hat {f}}_{\boldsymbol {k}}}/{k^2}$ in Fourier space). The first equality is obtained through the divergence theorem and the fact that $\langle \,\boldsymbol {f}_K \boldsymbol {\cdot } \boldsymbol {g}_Q \rangle =0$ for any two fields when $K\neq Q$ since they have no wavenumber in common. Motivated by the different energetic transfer terms (relations (4.5) and (4.6)), this leads to the decomposition ${\mathcal {T}}^{{{\mathcal {H}}^M}}(Q,K)={\mathcal {U}}^{{{\mathcal {H}}^M}}_{b \to b}(Q,K)+{\mathcal {U}}^{{{\mathcal {H}}^M}}_{v \xrightarrow [T]{} b}(Q,K)+{\mathcal {U}}^{{{\mathcal {H}}^M}}_{v \xrightarrow [P]{} b}(Q,K)$ with

(4.13)$$\begin{gather} {\mathcal{U}}^{{{\mathcal{H}}^M}}_{b \to b}(Q,K) = 2\langle {\boldsymbol{b}}_K \boldsymbol{\cdot} \mathrm{\textbf{rot}^{{-}1}}( (-{\boldsymbol{v}} \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{b}}_Q - \tfrac{1}{2} {\boldsymbol{b}}_Q \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}) \rangle, \end{gather}$$

(4.14)$$\begin{gather}{\mathcal{U}}^{{{\mathcal{H}}^M}}_{v \xrightarrow[T]{} b}(Q,K) = 2\langle {\boldsymbol{b}}_K \boldsymbol{\cdot} \mathrm{\textbf{rot}^{{-}1}}( ({\boldsymbol{b}}_Q \boldsymbol{\cdot} \boldsymbol{\nabla}) {\boldsymbol{v}}) \rangle, \end{gather}$$

(4.15)$$\begin{gather}{\mathcal{U}}^{{{\mathcal{H}}^M}}_{v \xrightarrow[P]{} b}(Q,K) = 2\langle {\boldsymbol{b}}_K \boldsymbol{\cdot} \mathrm{\textbf{rot}^{{-}1}}( - \tfrac{1}{2} {\boldsymbol{b}}_Q \boldsymbol{\nabla} \boldsymbol{\cdot} {\boldsymbol{v}}) \rangle. \end{gather}$$

Contrary to their sum, the ${\mathcal {U}}^{{{\mathcal {H}}^M}}_{b \to b}(Q,K)$, ${\mathcal {U}}^{{{\mathcal {H}}^M}}_{v \xrightarrow [T]{} b}(Q,K)$ and ${\mathcal {U}}^{{{\mathcal {H}}^M}}_{v \xrightarrow [P]{} b}(Q,K)$ functions on their own cannot be interpreted as magnetic helicity shell-to-shell transfers, since they are not antisymmetric. However, they still shed some light on the impact of magnetic $\leftrightarrow$ magnetic energy transfers and kinetic $\leftrightarrow$ magnetic energy transfers due to magnetic tension and pressure on the magnetic helicity dynamics, as described in § 6.4.

5. Helical transfers

The helical decomposition is based on the diagonalisation of the curl operator in Fourier space $\boldsymbol {\hat {f}} \to \textrm {i} \boldsymbol {k} \times \boldsymbol {\hat {f}}$, which possesses the eigenvalues $(0,+k,-k)$ (with $k=|\boldsymbol {k}|$) associated, respectively, with the unitary eigenvectors $\boldsymbol {{{\hat {h}}}}^0_{\boldsymbol {k}} = \boldsymbol {k}/k$ and (Waleffe Reference Waleffe1992; Brandenburg & Subramanian Reference Brandenburg and Subramanian2005)

(5.1)\begin{equation} \boldsymbol{{{\hat{h}}}}^\pm_{\boldsymbol{k}} = \frac{1}{\sqrt{2}}{\frac{\boldsymbol{k} \times (\boldsymbol{k} \times {\boldsymbol{{\hat{e}}}}) \mp \textrm{i} k (\boldsymbol{k} \times {\boldsymbol{{\hat{e}}}})}{k^2\sqrt{1-(\boldsymbol{k} \boldsymbol{\cdot} {\boldsymbol{{\hat{e}}}}/k)^2}}}, \end{equation}

with ${\boldsymbol {{\hat {e}}}}$ an arbitrary vector of unitary length non-parallel to $\boldsymbol {k}$. This decomposition can be viewed as an extension of the Helmholtz decomposition in a solenoidal (orthogonal to $\boldsymbol {k}$) and a compressive (parallel to $\boldsymbol {k}$) part. The plane orthogonal to $\boldsymbol {k}$ is spanned by $(\boldsymbol {{{\hat {h}}}}^+_{\boldsymbol {k}},\boldsymbol {{{\hat {h}}}}^-_{\boldsymbol {k}})$ so that the solenoidal modes are further decomposed into circularly polarised waves with opposite polarity, corresponding in configuration space to flow lines forming either a right-handed or a left-handed helix. Thus, the $(\boldsymbol {{{\hat {h}}}}^0_{\boldsymbol {k}},\boldsymbol {{{\hat {h}}}}^+_{\boldsymbol {k}},\boldsymbol {{{\hat {h}}}}^-_{\boldsymbol {k}})$ eigenvectors form an orthonormal basis in Fourier space. The study of helical mode interactions allows one to find possibilities for inverse transfer dynamics in three-dimensional hydrodynamic turbulence, in the absence of rotation or confinement (quasi-two-dimensional flow) (Waleffe Reference Waleffe1992; Biferale, Musacchio & Toschi Reference Biferale, Musacchio and Toschi2012), which may be relevant for turbulence models in large-eddy simulations (Alexakis Reference Alexakis2017). Extensions to the MHD case seem promising to clarify the intertwined dynamics of magnetic and kinetic helicities, for example analytically (Linkmann et al. Reference Linkmann, Berera, McKay and Jäger2016, Reference Linkmann, Sahoo, McKay, Berera and Biferale2017), through direct numerical simulations (Linkmann et al. Reference Linkmann, Sahoo, McKay, Berera and Biferale2017; Linkmann & Dallas Reference Linkmann and Dallas2017) and shell models (Lessinnes, Plunian & Carati Reference Lessinnes, Plunian and Carati2009; Rathmann & Ditlevsen Reference Rathmann and Ditlevsen2019) – see Alexakis & Biferale (Reference Alexakis and Biferale2018) for a review.

Using the helical decomposition, the velocity and magnetic fields read (Lessinnes et al. Reference Lessinnes, Plunian and Carati2009)

(5.2)$$\begin{gather} {\boldsymbol{b}} = \sum_{\boldsymbol{k}} \sum_{s_k} b^{s_k}_{\boldsymbol{k}} \boldsymbol{{{{\hat{h}}}^{s_k}_{\boldsymbol{k}}}} \, \textrm{e}^{\textrm{i}\boldsymbol{k} \boldsymbol{\cdot} \boldsymbol{x}}, \end{gather}$$

(5.3)$$\begin{gather}{\boldsymbol{v}} = \sum_{\boldsymbol{p}} \sum_{s_p} v^{s_p}_{\boldsymbol{p}} \boldsymbol{{{{\hat{h}}}^{s_p}_{\boldsymbol{p}}}} \, \textrm{e}^{\textrm{i}\boldsymbol{p} \boldsymbol{\cdot} \boldsymbol{x}}, \end{gather}$$

with $s_k \in \{+,-\}$ corresponding to the positively and negatively helical modes and $s_p \in \{+,-,0\}$ corresponding to both helical modes and the compressive mode along $\boldsymbol {k}$. Since the magnetic field is solenoidal, it cannot have components along $\boldsymbol {{{\hat {h}}}}^0_{\boldsymbol {k}}$.

Plugging the helical decomposition (5.2) and (5.3) into (2.3) in the absence of forcing leads to (Lessinnes et al. Reference Lessinnes, Plunian and Carati2009; Linkmann et al. Reference Linkmann, Berera, McKay and Jäger2016, Reference Linkmann, Sahoo, McKay, Berera and Biferale2017)

(5.4)\begin{equation} \partial_t b^{s_k}_{\boldsymbol{k}}=s_k k \sum_{\boldsymbol{k}+\boldsymbol{p}+\boldsymbol{q}=0} \sum_{s_p,s_q} v^{s_p*}_{\boldsymbol{p}} b^{s_q*}_{\boldsymbol{q}} g^{s_k,s_p,s_q}_{k,p,q}, \end{equation}

with $g^{s_k,s_p,s_q}_{k,p,q}$ a geometric factor depending on the triad shape and the helical components considered. Its modulus $G^{s_k,s_p,s_q}_{k,p,q}$ can be viewed as a weight of the respective helical triad interaction. With regard to a given wavenumber $k=|\boldsymbol {k}|$, the geometric factor maps a weight to every possible shape of a nonlinear interaction triangle formed by the corresponding wavevectors $\boldsymbol {k}+\boldsymbol {p}+\boldsymbol {q}=0$. This shape is uniquely determined by $k$ and the leg ratios $p/k$ and $q/k$. The weight signature on the strip defined by the triad condition in the $(p/k,q/k)$ plane (see § 6.3) gives an indication of the level of spectral locality of the most dominant interactions distinguishing the different helical contributions. As shown in § 6, it is interesting to observe that the nonlinear transfer of magnetic helicity comprises three clearly separable interaction families. Those are (i) a local direct transfer linked with a direct magnetic energy cascade due to the advective flux of magnetic helicity (transfer contribution ${\mathcal {U}}^{{{\mathcal {H}}^M}}_{b \to b}$, see relation (4.13)), and (ii) a local inverse transfer due to advection and predominantly shear Alfvénic magnetic tension effects. (iii) Pronounced nonlocal inverse transfer, the hallmark of inverse magnetic helicity dynamics, is generated through the small-scale velocity field, coupling the small- and large-scale magnetic fields through advective flux, magnetic stretching and magnetic pressure. More details are given in § 6.4.

The geometric factor has been derived in Waleffe (Reference Waleffe1992) for triads in the incompressible hydrodynamic case. Since the magnetic field is solenoidal, these derivations remain formally valid in incompressible MHD as noted in Lessinnes et al. (Reference Lessinnes, Plunian and Carati2009). In the same way, they remain valid in compressible MHD when one considers only solenoidal velocity modes $s_p \in \{+,-\}$, which leads to (Waleffe Reference Waleffe1992)

(5.5)\begin{equation} G^{s_k,s_p \in \{+,-\},s_q}_{k,p,q}=\frac{|s_k k+s_p p+s_q q|\sqrt{2k^2p^2+2p^2q^2+2q^2k^2-k^4-p^4-q^4}}{2kpq}. \end{equation}

These derivations are repeated in Appendix A and remain valid in compressible MHD when considering only solenoidal modes $s_k,s_p,s_q\in \{+,-\}$. The only difference that appears in this case is that the velocity field can have compressive components ($s_p=0$). An extension of these derivations for compressible flows is also presented in Appendix A, leading for $s_p=0$, $s_k=S$ and $s_q=\pm S$ to

(5.6)\begin{equation} G^{s_k=S,s_p=0,s_q={\pm} S}_{k,p,q}=\frac{|(q\mp k)(p^2-k^2-q^2 \mp 2qk)|}{2kpq}. \end{equation}

The shell-to-shell transfer approach described in § 4 can be extended to helically decomposed shell-to-shell transfers. The magnetic field is projected in Fourier space on the helical eigenvectors $\boldsymbol {{{\hat {h}}}}^+_{\boldsymbol {k}}$ and $\boldsymbol {{{\hat {h}}}}^-_{\boldsymbol {k}}$ and then transformed into configuration space, giving ${\boldsymbol {b}}^+$ and ${\boldsymbol {b}}^-$, respectively. In the same way, the velocity field is decomposed in ${\boldsymbol {v}}^+$, ${\boldsymbol {v}}^-$ and ${\boldsymbol {v}}^0$, with ${\boldsymbol {v}}^0$ its compressive part (obtained through a projection along $\boldsymbol {{{\hat {h}}}}^0_{\boldsymbol {k}}$ in Fourier space). The magnetic helicity transfer function (relation (4.2)) is then extended to

(5.7)\begin{equation} {\mathcal{T}}^{{{\mathcal{H}}^M}}(Q,P,K)=\sum_{s_K \in \{+,-\}}\sum_{s_P \in \{+,0,-\}}\sum_{s_Q \in \{+,-\}} {\mathcal{T}}^{{{\mathcal{H}}^M}}_{s_Ks_Ps_Q}, \end{equation}

with 12 different helical contributions

(5.8)\begin{equation} {\mathcal{T}}^{{{\mathcal{H}}^M}}_{s_Ks_Ps_Q}(Q,P,K)=2\langle {\boldsymbol{b}}^{s_K}_{K} \boldsymbol{\cdot} ({\boldsymbol{v}}^{s_P}_{P} \times {\boldsymbol{b}}^{s_Q}_{Q}) \rangle. \end{equation}

The six helical transfer functions for which $s_K=s_Q$ can be interpreted as antisymmetric magnetic helicity transfers between shells ${\mathcal {T}}^{{{\mathcal {H}}^M}}_{s_Ks_Ps_K}(Q,P,K)=-{\mathcal {T}}^{{{\mathcal {H}}^M}}_{s_Ks_Ps_K}(K,P,Q)$. These terms are labelled $+{+}+,\ +{-}+,\ +{0}+,\ -{+}-,\ -{-}-$ and $-{0}-$ in the following, where the first letter corresponds to $s_K$, the second to $s_P$ and the third to $s_Q$. However, the terms where $s_K \neq s_Q$, labelled henceforth as ‘heterochiral’ are not antisymmetric on their own but need to be considered in pairs, as in Alexakis (Reference Alexakis2017): the three ‘symmetrised’ terms ${\mathcal {T}}^{{{\mathcal {H}}^M}}_{Ss_PS}={\mathcal {T}}^{{{\mathcal {H}}^M}}_{+s_P-}+{\mathcal {T}}^{{{\mathcal {H}}^M}}_{-s_P+}$ verify the antisymmetric property and can thus be interpreted as magnetic helicity transfers between shells. They are noted similarly $S{+}S,\ S{-}S$ and $S{0}S$.

Even though the triad helical geometric factor $G^{s_k,s_p,s_q}_{k,p,q}$ is strictly speaking only valid for single triad interactions, one can expect the helically decomposed shell-to-shell transfers to be weighted essentially by the same geometric factor, especially for $K, P, Q$ large enough.

In order to relate more easily to physical quantities, the Fourier spectra shown in the present work are normalised by the isothermal sound speed squared, i.e. the specific kinetic energy spectrum is $E^V_K=\langle|\boldsymbol{v}_K|^2\rangle/(2c^2_s)$, the magnetic energy $E^M_K=\langle|\boldsymbol{b}_K|^2\rangle/(2\rho_0 c^2_s)$ with $\rho _0=1$ the mean density in the system, since ${\boldsymbol {b}}$ has the same dimension as $\sqrt {\rho }{\boldsymbol {v}}$ and similarly $H^M_K=\langle \boldsymbol{a}_K \boldsymbol{\cdot} \boldsymbol{b}_K\rangle/(\rho_0 c^2_s)$. Projecting the Fourier coefficients on the helical eigenvectors, one can obtain the spectral repartition of energy among the helical modes. For example, the positive helical specific kinetic energy is $E^{V,+}_K=\langle|\boldsymbol{v}^+_K|^2\rangle/(2c^2_s)$ and similarly for the negative helical specific kinetic energy $E^{V,-}$ and its compressive part $E^{V,0}$.