Scaling of Turbulent Viscosity and Resistivity: Extracting a Scale-dependent Turbulent Magnetic Prandtl Number

We analyze the incompressible MHD equations with a constant density $\rho$

where ${\bf u}$ is the velocity, and ${\bf B}$ is the magnetic field normalized by $\sqrt{4\pi\rho}$ to have Alfvén (velocity) units. $p$ is pressure, ${\bf J}={\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\times$}}{\bf B}$ is (normalized) current density, f is external forcing, $\nu$ and $\eta$ are microscopic viscosity and resistivity, respectively.

We use the coarse-graining method to analyze the flow and define the turbulent magnetic Prandtl number. A coarse-grained field in $n$-dimensions $\overline{f}_{\ell}({\bf x})=\int d^{n}{\bf r}\,G_{\ell}({\bf x}-{\bf r})f({\bf r})$ contains modes at length-scales greater than $\ell$, where $G_{\ell}({\bf r})\equiv\ell^{-n}G({\bf r}/\ell)$ is a normalized kernel with its main weight in a ball of diameter $\ell$. The coarse-grained MHD equations for $\overline{\textbf{u}}_{\ell}$, $\overline{\textbf{B}}_{\ell}$, and the quadratic MHD invariants were shown by Aluie (2017). Hereafter, we drop subscript $\ell$ when possible.

The coarse-grained kinetic energy (KE) and magnetic energy (ME) density balance (at scales $>\ell$) are,

where ${\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\cdot$}}[\cdots]$ denotes spatial transport terms, ${\bf S}=({\mbox{\boldmath$\nabla$}}{\bf u}+{\mbox{\boldmath$\nabla$}}{\bf u}^{T})/2$ is the strain-rate tensor, $\overline{\bf f}{\mbox{\boldmath$\cdot$}}\overline{\bf u}$ is the energy injection rate at forcing scale $\ell_{f}=2\pi/k_{f}$ ($k_{f}$ are the modes of the forcing ${\bf f}$). Microscopic dissipation terms $\nu|\overline{{\bf S}}|^{2}$ and $\eta|\overline{{\bf J}}|^{2}$ are mathematically guaranteed (Aluie, 2017; Eyink, 2018) and numerically demonstrated (Zhao & Aluie, 2018; Bian & Aluie, 2019) to be negligible at scales $\ell\gg{\left(\ell_{\nu},\ell_{\eta}\right)}$, where $\ell_{\nu}$ and $\ell_{\eta}$ are the viscous and resistive length scales, respectively.

The KE cascade term $\overline{\Pi}^{u}_{\ell}\equiv-\overline{\bf S}_{\ell}\mathbin{:}\overline{\boldsymbol{\tau}}_{\ell}$ in eq. (4) quantifies the KE transfer across scale $\ell$, where $\overline{\tau}_{ij}\equiv\tau_{\ell}(u_{i},u_{j})-\tau_{\ell}(B_{i},B_{j})$ is the sum of subscale Reynolds and Maxwell stresses generated by scales $<\ell$ acting on the large-scale strain $\overline{S}_{ij}$, resulting in “turbulent viscous dissipation” to scales $<\ell$. Subscale stress is defined as $\tau_{\ell}(f,g)=\overline{\left(fg\right)}_{\ell}-\overline{f}_{\ell}\overline{g}_{\ell}$ for any two fields $f$ and $g$. Similarly, ME casade term $\overline{\Pi}^{b}_{\ell}\equiv-\overline{{\bf J}}_{\ell}{\mbox{\boldmath$\cdot$}}\overline{\boldsymbol{\varepsilon}}_{\ell}$ in eq. (5) quantifies the ME transfer across scale $\ell$, where the subscale electromotive force (EMF) $\overline{\boldsymbol{\varepsilon}}_{\ell}\equiv\overline{{\bf u}{\mbox{\boldmath$\times$}}{\bf B}}-\overline{{\bf u}}{\mbox{\boldmath$\times$}}\overline{{\bf B}}$ is (minus) the electric field generated by scales $<\ell$ acting on the large-scale current $\overline{\bf J}={\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\times$}}\overline{\bf B}$, resulting in “turbulent Ohmic dissipation” to scales $<\ell$. Both $\overline{\Pi}^{u}_{\ell}$ and $\overline{\Pi}^{b}_{\ell}$ appear as sinks in the energy budgets of large scales $>\ell$ and as sources in the energy budgets of small scales $<\ell$ (Aluie, 2017).

Term $\overline{S}_{ij}\overline{B}_{i}\overline{B}_{j}$ quantifies KE-to-ME conversion at all scales $>\ell$ and appears as a sink in eq. (4) and a source in eq. (5). Bian & Aluie (2019) showed that $\langle\overline{S}_{ij}\overline{B}_{i}\overline{B}_{j}\rangle$ ($\langle...\rangle$ denotes a spatial average) is a large-scale process, which only operates at the largest scales in the inertial-inductive range (which was called the “conversion range”) and vanishes at intermediate and small scales in the inertial-inductive range (which was called the “decoupled range”). In the decoupled range, $\langle\overline{\Pi}^{u}_{\ell}\rangle$ and $\langle\overline{\Pi}^{b}_{\ell}\rangle$ become constant as a function of scale (i.e., scale-independent). The observation of constant KE and ME fluxes $\langle\overline{\Pi}^{u}_{\ell}\rangle$ and $\langle\overline{\Pi}^{b}_{\ell}\rangle$ is important since it indicates separate conservative cascades of each of KE and ME, which arises asymptotically at high Reynolds number regardless of forcing, external magnetic field, and microscopic magnetic Prandtl number.

Oftentimes, turbulence is modeled as a diffusive process via effective (or turbulent) transport coefficients. For example, mixing length or eddy viscosity models represent the subscale stress $\overline{\boldsymbol{\tau}}_{\ell}$, due to scales $<\ell$, as $\overline{\tau}_{ij}=-2\nu_{t}^{\bf x}\overline{S}_{ij}$, where $\nu_{t}^{\bf x}$ is a turbulent viscosity¹¹1Strictly speaking, the eddy viscosity definition is $\mathring{\overline{\tau}}_{ij}=-2\nu_{t}\overline{S}_{ij}$, where $\mathring{\overline{\tau}}_{ij}=\overline{\tau}_{ij}-\delta_{ij}\overline{\tau}_{kk}/3$ is the traceless part of the stress. For our incompressible flow analysis here, which is based on the energy flux, this distinction does not matter. (e.g., Pope (2001) and references therein). Similarly, the subscale EMF can be modeled as $\overline{\boldsymbol{\varepsilon}}=-\eta_{t}^{\bf x}\overline{\bf J}+\alpha\overline{{\bf B}}$, where the $\eta_{t}^{\bf x}\overline{\bf J}$ term models the subscales as turbulent resistive diffusion (Miesch et al., 2015) and the $\alpha\overline{{\bf B}}$ term is the $\alpha$-effect of dynamo theory (Moffatt, 1978). The $\alpha$-effect is expected to play a role in flows where the driving mechanism is helical. To simplify our analysis and the presentation of our approach, we shall neglect the $\alpha\overline{{\bf B}}$ term and assume that the subscale EMF can be modeled solely as Ohmic diffusion, $\overline{\boldsymbol{\varepsilon}}=-\eta_{t}^{\bf x}\overline{\bf J}$. Note that $\nu_{t}^{\bf x}({\bf x},t,\ell)$ and $\eta_{t}^{\bf x}({\bf x},t,\ell)$ are generally functions of space ${\bf x}$, length scale $\ell$, and time.

A main goal of this paper is extracting the turbulence transport coefficients, $\nu_{t}^{\bf x}$ and $\eta_{t}^{\bf x}$, as a function of length-scale. However, we do not pursue a phenomenological analysis similar to that of Smagorinsky (1963) or of a mixing length framework (Tennekes & Lumley, 1972) in part because we lack a consensus MHD turbulence theory analogous to that of Kolmogorov (1941). To achieve our goal, we shall instead analyze the energy budgets resultant from the eddy viscosity model. Within our coarse-graining framework, this is equivalent to having the rate of energy cascading to scales smaller than $\ell$ equal a turbulent dissipation acting on scales $>\ell$:

These two relations are definitions for $\nu_{t}$ and $\eta_{t}$. Note that unlike in relation $\overline{\tau}_{ij}=-2\nu_{t}^{\bf x}\overline{S}_{ij}$, the turbulence transport coefficients in eqs. (6)-(7) are defined using scalar quantities $\langle\overline{\Pi}^{u}_{\ell}\rangle$, $\langle\overline{\Pi}^{b}_{\ell}\rangle$, $\langle|\overline{\bf S}_{\ell}|^{2}\rangle$, and $\langle|\overline{{\bf J}}_{\ell}|^{2}\rangle$. For homogeneous turbulence considered in this study, we rely on spatial averages, $\langle\dots\rangle$, rendering $\nu_{t}$ and $\eta_{t}$ independent of location ${\bf x}$ but still a function of scale $\ell$.

Consistent with the eddy viscosity hypothesis, eq. (6) (eq. (7)) models the kinetic (magnetic) energy cascading from scales $>\ell$ to smaller scales as effectively being dissipated by a turbulent viscosity (resistivity). From these, we can also extract a scale-dependent turbulent magnetic Prandtl number,

What power-law scaling can we expect these turbulent transport coefficients to have? It is possible to relate $\nu_{t}$ and $\eta_{t}$ to energy spectra. Indeed, the space-averaged turbulent dissipation can be expressed in terms of energy spectra:

where $E^{u}(k)$ ($E^{b}(k)$) is the kinetic (magnetic) energy spectrum with (dimensionless) wavenumber $k=L/\ell$ for a periodic domain of size $L$.

The scaling of spectra in turn are related to the scaling of velocity and magnetic field increments (Aluie, 2017)

where increment $\delta f(x;\ell)=f(x+\ell)-f(x)$ (see details in Eyink (2005); Aluie & Eyink (2010); Aluie (2017)). From eq. (11), the kinetic and magnetic energy spectra scale as

The relation between increments and spectra does not make any assumptions about the specific exponent values, only that they are $\sigma_{u,b}<1$ (Sadek & Aluie, 2018). Scaling exponents $\sigma_{u}$ and $\sigma_{b}$ are a measure of smoothness of the velocity and magnetic fields, respectively (see Fig. 1 and related discussion in Aluie (2017)). A value of $\sigma=1$ indicates that the field is very smooth (e.g., of a laminar flow) with a spectrum decaying as $k^{-3}$ or steeper. Canonical hydrodynamic turbulence has intermediate smoothness, with $\sigma_{u}=1/3$ according to Kolmogorov (1941) (K41). The larger is the value of $\sigma$, the smoother is the field.

For sufficiently high Reynolds number flows, Bian & Aluie (2019) showed that each of $\langle\overline{\Pi}^{u}_{\ell}\rangle$ and $\langle\overline{\Pi}^{b}_{\ell}\rangle$ become constant, independent of scale in the decoupled range. From definitions (6)-(8), and considering the scaling relations discussed above, we can infer that the turbulent transport coefficients vary with scale as follows:

for scales $k$ in the decoupled range. It is possible to obtain scaling relations (13) from either the scaling of spectra in eqs. (9)-(10), or the scaling of coarse-grained strain and current, $|\overline{{\bf S}}_{\ell}|\sim\delta u(\ell)/\ell$ and $|\overline{{\bf J}}_{\ell}|\sim\delta B(\ell)/\ell$ (Eyink et al., 2013; Aluie, 2017). Eq. (13) highlights that $Pr_{t}$ is independent of scale only if $\sigma_{u}=\sigma_{b}$.

Regardless of the specific value, and consistent with existing MHD turbulence phenomenologies, we expect that $\sigma_{u,b}<1$. Indeed, a $\sigma_{u,b}\geq 1$ would correspond to a smooth flow that is inconsistent with the qualitative expectation of a ‘rough’ or ‘fractal’ turbulent flow. Therefore, relations (13) indicate that $\nu_{t}$ and $\eta_{t}$ decay as $\ell\to 0$ (or $k\to\infty$). This is consistent with physical expectations since the ‘eddies’ effecting the turbulent transport become weaker at smaller scales, yielding smaller transport coefficients.

We highlight a technical, albeit important aspect of scaling relations (13). Our coefficients seem to scale with the inverse of coarse-grained strain and current magnitudes, $\nu_{t}\sim|\overline{{\bf S}}_{\ell}|^{-2}\sim\ell^{2-2\sigma_{u}}$ and $\eta_{t}\sim|\overline{{\bf J}}_{\ell}|^{-2}\sim\ell^{2-2\sigma_{b}}$, but do not appear to depend on the subscale stress and EMF, $\overline{\boldsymbol{\tau}}_{\ell}$ and $\overline{\boldsymbol{\varepsilon}}_{\ell}$, respectively. At face value, this result seems counter-intuitive wherein $\sigma\to 1$ associated with smoother fields and weaker ‘eddies’ leads to an increase rather than a drop in the turbulent coefficient values in eq. (13). However, a key assumption in arriving at relations (13) is that fluxes $\langle\overline{\Pi}^{u}_{\ell}\rangle$ and $\langle\overline{\Pi}^{b}_{\ell}\rangle$ are constant, independent of scale. For scale-independent fluxes to be established, consistent with a persistent cascade to arbitrarily small scales (in the $Re\to\infty$ limit), $\sigma_{u}$ and $\sigma_{b}$ have to take on fixed values that are yet to be determined and agreed upon by the community. If $\sigma_{u,b}$ were to be somehow increased above those values, the cascade would shut down (fluxes would decay with $k$) before carrying the energy all the way to dissipation scales (Aluie, 2017). For scale-dependent fluxes, relations (13) have to be modified to also include the scaling of $\langle\overline{\Pi}^{u}_{\ell}\rangle$ and $\langle\overline{\Pi}^{b}_{\ell}\rangle$ (see Aluie (2017) for details).

To infer the scaling of turbulence transport coefficients, the approach we adopt in this paper circumvents using values of $\sigma_{u}$ and $\sigma_{b}$ (in the asymptotic $Re\to\infty$ limit) from a specific MHD phenomenology–whether it exists or not– by relying on results from Bian & Aluie (2019) of scale-independent fluxes $\langle\overline{\Pi}^{u}_{\ell}\rangle$ and $\langle\overline{\Pi}^{b}_{\ell}\rangle$.

Under K41 scaling $\sigma_{u}=1/3$ (Kolmogorov, 1941), our scaling of $\nu_{t}\propto\ell^{2-2\sigma_{u}}\propto\ell^{4/3}$ is equivalent to that from mixing length theory $\nu_{t}=\ell^{2}|\overline{{\bf S}}|\propto\ell^{\sigma_{u}+1}\propto\ell^{4/3}$ (Smagorinsky, 1963). Our analysis is also compatible with different scaling theories and observations in MHD turbulence (Iroshnikov, 1963; Kraichnan, 1965; Goldreich & Sridhar, 1995; Boldyrev, 2005, 2006; Boldyrev & Perez, 2009; Boldyrev et al., 2011). For example, solar wind observations (Podesta et al., 2007; Borovsky, 2012) suggest that $E^{u}(k)\sim k^{-3/2}$ for the kinetic energy spectrum, corresponding to $\delta u(\ell)\sim\ell^{1/4}$, and $E^{b}(k)\sim k^{-5/3}$ for the magnetic energy spectrum, corresponding to $\delta B(\ell)\sim\ell^{1/3}$, yield

for $k$ in the decoupled scale-range.

Instead of analyzing the energy budgets to determine $\nu_{t}$, $\eta_{t}$, $Pr_{t}$ and their scaling, we can alternatively focus on the budgets for vorticity and current. Similar to eqs. (4)-(5), we can derive the budgets

Here, $\overline{Z}_{\ell}=\overline{\boldsymbol{\omega}}{\mbox{\boldmath$\cdot$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\times$}}({\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\cdot$}}\overline{\boldsymbol{\tau}})$ and $\overline{Y}_{\ell}=-\overline{\bf J}{\mbox{\boldmath$\cdot$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\times$}}{\mbox{\boldmath$\nabla$}}{\mbox{\boldmath$\times$}}\overline{\boldsymbol{\varepsilon}}$ are the only “scale-transfer” terms in the coarse-grained eqs. (15)-(16) involving the interaction of subscale terms $\overline{\boldsymbol{\tau}}_{\ell}$ and $\overline{\boldsymbol{\varepsilon}}_{\ell}$ with large-scale quantities (here, $\overline{\boldsymbol{\omega}}$ and $\overline{\bf J}$) to cause transfer across scale $\ell$. From the models $\overline{\boldsymbol{\tau}}_{\ell}=-2\nu_{t}\overline{\bf S}_{\ell}$ and $\overline{\boldsymbol{\varepsilon}}_{\ell}=-\eta_{t}\overline{\bf J}_{\ell}$, we have alternate definitions for the turbulent transport coefficients:

Note that unlike energy, vorticity and current density are not ideal invariants and, therefore, do not undergo a cascade in the manner energy does. Yet, to the extent $\nu_{t}$ and $\eta_{t}$ are able to capture the subscale physics embedded in $\overline{\boldsymbol{\tau}}_{\ell}$ and $\overline{\boldsymbol{\varepsilon}}_{\ell}$, it is reasonable to expect that the turbulent transport coefficients are consistent with the budget of any quantity derived from the underlying dynamics.

In Fig. 1, we compare $\nu_{t}$ and $\eta_{t}$ when calculated from eqs. (17)-(18) to those obtained from the energy budgets in eqs. (6)-(7). We find that the two definitions yield fairly similar results with slight quantitative differences. This consistency lends support to our approach of using the energy budgets to calculate $\nu_{t}$ and $\eta_{t}$ (eqs. (6)-(7)) and make inferences about turbulent diffusion or dissipation of quantities other than energy.

It is almost always the case that astrophysical systems of interest are at sufficiently high Reynolds numbers (both magnetic and hydrodynamic) that it is impossible to simulate the entire dynamic range of scales that exist (Miesch et al., 2015). In practice, most simulations are either explicit or implicit Large eddy simulations (LES), resolving only the large-scale dynamics (Meneveau & Katz, 2000). The former include explicit terms in the equations being solved that model the unresolved subgrid physics, whereas the latter rely on the numerical scheme to act as a de facto model for such missing physics. Our analysis here can offer guidance for tuning the turbulent coefficients when conducting explicit Large Eddy Simulations using eddy diffusivity models. It can also offer us insight into whether relying on a similar scheme and grid for simulating both the momentum and magnetic fields is justified.

In the inertial-inductive range, using eq. (6), $|\overline{{\bf S}}_{\ell}|\sim\delta u(\ell)/\ell$, and the Ansatz (Aluie, 2017)

where $u_{rms}=\langle|{{\bf u}}|^{2}\rangle^{1/2}$, $L$ is a characteristic large scale such as the integral scale or that of the domain, and we ignore intermittency corrections, we then have

In an LES with grid spacing $\Delta x$, the turbulent viscosity accounting for subgrid scales should be evaluated at a coarse-graining scale $\ell_{c}=L/k_{c}$ proportional to $\Delta x$ (Pope, 2001), where $k_{c}=L/\ell_{c}$ is a dimensionless cutoff wavenumber:

for $\left(\ell_{\nu},\ell_{\eta}\right)\ll\ell_{c}\ll L$, where dimensionless constant $C_{u}$ defined as the proportionality factor of the relation

Figure 6 in the Appendix shows that $C_{u}$ is indeed a proportionality constant that is scale-independent within the decoupled range, taking on values from 2 to 5 in various simulated flows.

Similarly, the turbulent resistivity at the cutoff wavenumber is

where $B_{rms}=\langle|{\bf B}-{\bf B}_{0}|^{2}\rangle^{1/2}$ (${\bf B}_{0}$ is the uniform external magnetic field), and dimensionless constant $C_{b}$ defined as

Fig. 6 in the Appendix shows that $C_{b}$ is indeed a proportionality constant that is scale-independent within the decoupled range, taking on values from 10 to 15 in various simulated flows.

If the grid is sufficiently fine to resolve some of the scales in the decoupled range, then eqs. (21),(23) simplify to

with scale-independent fluxes $\langle\overline{\Pi}^{u}_{\ell}\rangle=\varepsilon_{u}$ and $\langle\overline{\Pi}^{b}_{\ell}\rangle=\varepsilon_{b}$. These are the KE and ME cascade rates, which were found by Bian & Aluie (2019) to reach equipartition in the decoupled range, $\varepsilon_{u}=\varepsilon_{b}=\varepsilon/2$, half the total energy cascade rate, $\varepsilon$.

Eqs. (21),(23) (and eqs. (25),(26)) connect the scaling of turbulent transport coefficients with the scaling of velocity and magnetic spectra, and are compatible with different MHD scaling theories. For example, if $E^{u}(k)\sim k^{-5/3}$ (corresponding to $\delta u(\ell)\sim\ell^{1/3}$) as in the theory by Goldreich & Sridhar (1995), eq. (21) reduces to the turbulent viscosity model of Verma (2001a) derived from an RG analysis (see also Verma & Kumar (2004)).

We have tested the scaling of $Pr_{t}$ under different microscopic $Pr_{m}$ flow conditions. We remind the reader that our results here pertain to the decoupled range, which is within the inertial-inductive range. These scales are immune from the direct influence of both resistivity and viscosity. We do not expect our results here (and those of Bian & Aluie (2019) upon which this analysis is based) to hold in the viscous-inductive (Batchelor) range at high $Pr_{m}$, or in the inertial-resistive range at low $Pr_{m}$. From practical modeling considerations, such as when simulating a galactic accretion disc at global scales, grid-resolution constraints render it virtually impossible to have $\Delta x\sim\ell$ within the viscous-inductive range. Therefore, the restriction of our analysis to inertial-inductive scales is still pertinent to modeling as well as being of theoretically import.

Fig. 1(a)(b) shows the case (Run IV) of $Pr_{m}=2$, where we find a scaling of $Pr_{t}$ similar to the case of unity microscopic $Pr_{m}$. We also conduct simulations (Table 1) at $Pr_{m}$ = 0.1, 5, and 10, albeit at a lower resolution of $512^{3}$ due to the computational overhead required by non-unity $Pr_{m}$. Fig. 3 shows that scale-dependence of $Pr_{t}$ is consistent with that of unity-$Pr_{m}$ runs, although the scaling is not as clear. Due to the lower resolution, the decoupled range is barely established in the non-unity $Pr_{m}$ cases. Non-unity $Pr_{m}$ simulations require even larger Reynolds numbers to achieve a significant decoupled range and still make an allowance for a viscous-inductive or an inertial-resistive range of scales. This is beyond our computing capability for this work.

Our results also suggest that within the limited dynamic range of our simulations, increasing the external B-field strength from 0 (Run I) to 2 (Run V) to 10 (Run II) seems to change the $Pr_{t}$ scaling slightly from $k^{-1/3}$ to $k^{-1/6}$ due to $\sigma_{u}$ increasing from 1/6 to 1/4 (see Fig. 7 in Appendix). However, we do not believe this trend will persist at asymptotically high-$Re$ as we discuss in the following subsection. We also note that our analysis here does not distinguish the anisotropy in turbulent transport. Our effective transport coefficients in this paper are isotropic even though the underlying turbulent flow may be anisotropic such as in Runs II and V (see Fig. 5 in Appendix). We hope this work is extended to anisotropic turbulent transport in future studies.

Can we expect the scaling of $Pr_{t}$ in Fig. 1(a),(b), which is in support of our relations in eq. (13), to extrapolate to the wide dynamical ranges (high-$Re$) that exist in many astrophysical systems of interest?

Figure 4 (and Fig. 9 in Appendix) examines the scaling of $Pr_{t}(k)$ at different Reynolds numbers. Each panel shows results from a suite of simulations under the same parameters except for $Re$ (or grid-resolution). The plots show that $Pr_{t}(k)$ takes on a value between 1 and 2 at the smallest scales within the inertial-inductive range, regardless of $Re$ (also Fig. 16 and Table 3 in Appendix). These scales near $\ell_{d}$ are bordering the dissipation range. The reason $Pr_{t}(k\approx L/\ell_{d})\approx 1\text{~{}to~{}}2$ can be understood from definition (8) of $Pr_{t}=\left(\langle\overline{\Pi}^{u}_{\ell}\rangle/\langle\overline{\Pi}^{b}_{\ell}\rangle\right)\left(\langle|\overline{\bf J}_{\ell}|^{2}\rangle/\langle 2|\overline{\bf S}_{\ell}|^{2}\rangle\right)$. Due to equipartition of the cascades in the decoupled range, we have $\langle\overline{\Pi}^{u}_{\ell}\rangle/\langle\overline{\Pi}^{b}_{\ell}\rangle\approx 1$, whereas $\langle|\overline{\bf J}_{\ell}|^{2}\rangle/\langle 2|\overline{\bf S}_{\ell}|^{2}\rangle\approx 1\text{~{}to~{}}2$, as is clear from Fig. 2 (and Figs. 3,8). The latter can also be inferred from comparing the spectra $E^{u}(k^{\prime})$ and $E^{b}(k^{\prime})$ in Figs. 10-11 via eqs. (9)-(10).

Based on these observations, it is physically reasonable to assume that for a sufficiently wide dynamical range (or large $Re$), $\langle|\overline{\bf J}_{\ell_{d}}|^{2}\rangle/\langle 2|\overline{\bf S}_{\ell_{d}}|^{2}\rangle$ converges to a constant when $\ell\approx\ell_{d}$ (or $k\approx k_{d}$), independent of the dynamical range extent (i.e., independent of $Re$). That is, the ratio $\langle|\overline{\bf J}_{\ell_{d}}|^{2}\rangle/\langle 2|\overline{\bf S}_{\ell_{d}}|^{2}\rangle$ has the same value under successive refinement of the grid. This effectively provides us with a conceptual boundary condition on $Pr_{t}(k)$ at those smallest scales $k\approx k_{d}$.

These logical considerations, when combined with the scaling $Pr_{t}(k)\sim k^{-2(\sigma_{b}-\sigma_{u})}$ in eq. (13) (with empirical support in Figs. 1(a)(b),7), may lead us at face value to the uncomfortable conclusion that at any fixed $k$ in the inertial-inductive range, $Pr_{t}(k)$ will keep increasing with increasing $Re$ (or higher resolution) as Fig. 4 (and Fig. 9 in Appendix) seems to indicate. That is unless the MHD dynamics eventually yields $\sigma_{u}=\sigma_{b}$ in this asymptotic limit, i.e., at successively higher resolution simulations. Indeed, there are indications from Figs. 4,9 that the $Pr_{t}(k)$ sensitivity to $Re$ decreases with increasing $Re$ as we now discuss.

At first glance, Fig. 4 seems to indicate that $Pr_{t}(k)$ at any fixed wavenumber, e.g., $k=50$ within the decoupled range, $Pr_{t}(k=50)$ increases with increasing resolution. Yet, as we shall now argue, Fig. 4 highlights how certain metrics such as $Pr_{t}$ in our simulations, which are very high-resolution by today’s standards, are still not fully converged to the high-$Re$ limit. From the definition of $Pr_{t}$ in eq. (8), this increase can only be due to an increase of the cascade ratios $\langle\overline{\Pi}^{u}_{\ell}\rangle/\langle\overline{\Pi}^{b}_{\ell}\rangle$, or the current-to-strain ratio, $\langle|\overline{\bf J}_{\ell}|^{2}\rangle/\langle 2|\overline{\bf S}_{\ell}|^{2}\rangle$, or both. We find in Fig. 15 that the latter accounts for most of this increase. Fig. 15 suggests that the cascade ratio $\langle\overline{\Pi}^{u}_{\ell}\rangle/\langle\overline{\Pi}^{b}_{\ell}\rangle$ is fairly converged with resolution in our largest simulations at $\ell=L/50$. Physically, we expect $\langle|\overline{\bf J}_{\ell}|^{2}\rangle/\langle 2|\overline{\bf S}_{\ell}|^{2}\rangle$ to also converge since the ratio depends on the strain and current (or equivalently, the spectra) at scales larger than $L/50$. These should not remain sensitive to the smallest scales in a simulation once a sufficiently high resolution has been achieved. Fig. 15 in Appendix indicates that the high resolution of our simulations is still not sufficient for the convergence of these quantities ($\langle 2|\overline{\bf S}_{\ell}|^{2}\rangle$ and $\langle|\overline{\bf J}_{\ell}|^{2}\rangle$). Ignoring convergence trends under the guise of “having conducted the highest resolution simulation to date” can be rife with pitfalls. In general, when analyzing simulations of turbulent flows, it is vitally important to study trends as a function of Reynolds number and check if the phenomenon under study persists and can be extrapolated to the large Reynolds numbers present in nature.