2. Hosking integral and scaling for the Hall cascade

The Hosking integral $I_{H}$ is defined as the asymptotic limit of the magnetic helicity density correlation integral ${\mathcal {I}}_{H}(R)$ for scales $R$ large compared with the correlation length of the turbulence, $\xi$, but small compared with the system size $L$. The original work on this integral is that by Hosking & Schekochihin (Reference Hosking and Schekochihin2021), who subsequently applied it to the magnetic field decay in the early universe (Hosking & Schekochihin Reference Hosking and Schekochihin2022); see also Brandenburg, Kahniashvili & Tevzadze (Reference Brandenburg, Kahniashvili and Tevzadze2015) and Brandenburg & Kahniashvili (Reference Brandenburg and Kahniashvili2017) for earlier work where inverse cascading of magnetically dominated non-helical turbulence was first reported. The function ${\mathcal {I}}_{H}(R)$ is given by

(2.1)\begin{equation} {\mathcal{I}}_{H}(R)=\int_{V_R}\,{\rm d}^3r \langle h(\boldsymbol{x})h(\boldsymbol{x}+\boldsymbol{r})\rangle, \end{equation}

where $V_R$ is the volume of a ball of radius $R$ and $h=\boldsymbol {A}\boldsymbol {\cdot }\boldsymbol {B}$ is the magnetic helicity density with $\boldsymbol {A}$ being the magnetic vector potential, so $\boldsymbol {B}={\boldsymbol {\nabla }}\times \boldsymbol {A}$. We recall that $\langle \cdots\kern0.7pt \rangle$ denotes averaging over $\boldsymbol {x}$. The function ${\mathcal {I}}_{H}(R)$ is thus the integral over the volume $V_R=4{\rm \pi} R^3/3$. For small $R$, it increases proportional to $R^3$, but for large $R$, it levels off. This is the value of $R$ which determines the Hosking integral $I_{H}\equiv {\mathcal {I}}_{H}(R)$. In practice, it is chosen empirically and must still be small compared with the size of the domain; see Hosking & Schekochihin (Reference Hosking and Schekochihin2021) for various examples and Zhou et al. (Reference Zhou, Sharma and Brandenburg2022) for a comparison of different computational techniques for obtaining ${\mathcal {I}}_{H}(R)$.

What matters for the Hall cascade is the fact that the dimensions of $h$ are $[h]=[B]^2[x]$, and therefore the dimensions of ${\mathcal {I}}_{H}$ and $I_{H}$ are

(2.2)\begin{equation} [I_{H}]=[B]^4[x]^5. \end{equation}

However, as already noted in B20, using $e=1.6\times 10^{-19}\ {\rm A} {\rm s}$, $\mu _0=4{\rm \pi} \times 10^{-7}\ {\rm T}\ {\rm m}\ {\rm A}^{-1}$ and $n_e\approx 2.5\times 10^{40}\ {\rm m}^{-3}$ for neutron star crusts, we have $en_e\mu _0\approx 5\times 10^{15}\ {\rm T}\ {\rm s}\ {\rm m}^{-2}$, and therefore

(2.3)\begin{equation} \frac{B}{en_e\mu_0}=\frac{B}{5\times10^{15}\ {\rm T}} \frac{{\rm m}^2}{{\rm s}}, \end{equation}

which is why we say $B$ has dimensions of ${\rm m}^2\ {\rm s}^{-1}$.Footnote ¹ Therefore, the dimensions of $I_{H}$ are

(2.4)\begin{equation} [I_{H}]=[x]^{13}[{\rm s}]^{{-}4}. \end{equation}

Thus, given that $I_{H}={\rm const.}$ in the limit of small magnetic diffusivity, a self-similar evolution must imply that all relevant length scales, and in particular the magnetic correlation length $\xi (t)$, must increase with time like $\xi \sim t^{4/13}$. Since $4/13\approx 0.31$, this is indeed close to the behaviour $\xi \sim t^q$ with $q\approx 0.3$ found in § 3.2 of B20 (their Run B), as already highlighted in the introduction of the present paper.

To demonstrate that the energy spectra $E(k,t)$ at different times are indeed self-similar, we collapse them on top of each other by plotting them versus $k\xi (t)$. Here, $\xi (t)=\mathcal {E}^{-1}\int k^{-1} E(k,t)\,{\rm d} k$ is the weighted integral of $k^{-1}$, where the spectra are normalized such that $\int E(k,t)\,{\rm d} k=\mathcal {E}(t)$ is the magnetic energy density. This ensures that the maxima of $E(k\xi )$ are always approximately near $k\xi (t)=1$. In addition, we must also compensate for the decay in amplitude by multiplying the spectra by a time-dependent function, e.g. $\xi (t)^\beta$, where $\beta$ is a suitable exponent, so that the compensated spectra all have the same height. In this way, we find a universal spectral function by plotting

(2.5)\begin{equation} [\xi(t)]^\beta E(k\xi(t),t)\equiv\phi(k\xi(t)).\end{equation}

As an example, we show in figure 1 the compensated spectra for Run B of B20, which we discuss in more detail below. At this point, we just note that these were solutions to (1.1), where the initial condition consists of a non-helical magnetic field with a spectrum $E(k)\propto k^4$ for $k\ll k_0$, with $k_0$ being the initial peak wavenumber. For $k\gg k_0$, we assume a decaying spectrum, here with $E(k)\propto k^{-5/3}$, although this particular choice of the exponent was not important. After some time, the spectral slopes of both subranges change: at small $k$, the spectrum steepens from $k^4$ to $k^5$. Beyond the peak, it falls off with a $k^{-7/3}$ inertial range, as was already found by Biskamp, Schwarz & Drake (Reference Biskamp, Schwarz and Drake1996).

Figure 1. Compensated spectra for Run B of B20, which corresponds to Run B1 in the present paper. Here, $\beta =1.7$ has been used as the best empirical fit parameter.

To determine the theoretically expected value of $\beta$, we invoke the condition that the compensated spectra be invariant under rescaling, $t\to \tau t'$, $\boldsymbol {x}\to \tau ^q\boldsymbol {x}'$, where $\tau$ is an arbitrary scaling factor. We recall that the dimensions of $E(k,t)$ are $[x]^5[t]^{-2}$, so rescaling yields a factor $\tau ^{5q-2}$. In addition, expressing $E'$ in terms of its universal spectral function $\phi (k\xi )$, the factor $\xi ^\beta$ on the left-hand side of (2.5) produces a factor $\xi ^{-\beta }$ on the right-hand side of (2.5), and therefore, after rescaling, a $\tau ^{-\beta q}$ factor, i.e.

(2.6)\begin{equation} E(k)\to\tau^{5q-2} E'(k)\propto \xi(t)^{-\beta}\tau^{-\beta q}\phi(k\xi). \end{equation}

Therefore, $5q-2=-\beta q$ must be satisfied in order that the compensated spectra remain invariant under rescaling. Thus, $\beta =2/q-5$, as already found in B20. Inserting now $q=4/13$ yields $\beta =3/2$. The total magnetic energy density is therefore

(2.7)\begin{equation} \mathcal{E}=\int \xi^{-\beta} \phi(k\xi)\,{\rm d} k= \xi^{-(\beta+1)} \int \phi(k\xi)\,{\rm d}(k\xi) \propto t^{-(\beta+1) q}, \end{equation}

and since $\mathcal {E}\propto t^{-p}$, we haveFootnote ² $p=(\beta +1) q$. Using $\beta =2/q-5$, we have $p=2 (1-2q)$; see (28) of B20. For $q=4/13\approx 0.31$, we have $p=2 (1-8/13)=10/13\approx 0.78$, which is not quite as close to the value reported in B20 as that of $q$, but this could be ascribed to the lack of scale separation and also the magnetic field no longer being strong enough so that the Lundquist number,Footnote ³

(2.8)\begin{equation}{\rm Lu}=B_{{\rm rms}}/en_e\mu_0\eta, \end{equation}

is no longer in the asymptotic regime. This also resulted in the empirical value of $\beta$ being slightly larger than the theoretical one, as we will see next.

3. Comparison with simulations for different diffusivities

In B20, various simulations of the Hall cascade have been presented, including forced and decaying simulations, helical and non-helical ones, with constant and time-varying magnetic diffusivities, with and without stratification, etc. The main purpose of that work was to understand the dissipative losses that would lead to resistive heating in the crust of a neutron star. One of those simulations is particularly relevant for the present paper: his Run B, which had a relatively strong initial magnetic field, no helicity, large scale separation and a magnetic diffusivity that decreased with time in a power law fashion, allowing the simulation to retain a higher Lundquist number as the magnetic field decreases.

In the present paper, we analyse his Run B, which is here called Run B1. It is actually a new run, because we now have calculated the Hosking integral during run time. We also compare with another run (Run B2), where we decreased the magnetic diffusivity by a factor of two. As in B20, $\eta$ is assumed to decrease with time proportional to $t^{-3/7}$. We kept, however, the same resolution of $1024^3$ mesh points for both runs, but we must keep in mind that this can lead to artefacts resulting from a poorly resolved diffusive subrange for Run B2.

In table 1, we compare several characteristic parameters: the start and end times, $t_1$ and $t_2$, respectively, of the interval for which averaged data have been accumulated, non-dimensional measures of the magnetic diffusivity, the magnetic field strength and the dissipation, $\tilde {\eta }$, $\tilde {B}_{\rm rms}$ and $\tilde {\epsilon }$, respectively, and the instantaneous scaling exponents $p$, $q$ and $\beta$. For $\tilde {\eta }$, $\tilde {B}_{\rm rms}$ and $\tilde {\epsilon }$, we compute the following time-averaged ratios:

(3.1a–c)\begin{equation} \tilde{\eta}\equiv\langle t\eta/\xi^2\rangle_t,\quad \tilde{B}_{\rm rms}\equiv\langle B_{{\rm rms}}/(en_e\mu_0\eta)\rangle_t,\quad{\rm and}\quad \tilde{\epsilon}\equiv\langle \epsilon/(e^2 n_e^2 \mu_0 \eta^3/\xi^2)\rangle_t, \end{equation}

where $\xi (t)={\mathcal {E}}^{-1}\int k^{-1} E(k,t)\,{\rm d} k$ is the correlation length and $\epsilon =\eta \mu _0\langle \boldsymbol {J}^2\rangle$ is the magnetic dissipation with $\eta =\eta (t)$, as noted above. These were also computed in B20. Time is given in diffusive units, $[t]=(\eta k_0^2)^{-1}$. In the runs of series B of B20, the value of $k_0$ is 180 times larger than the lowest wavenumber $k_1\equiv 2{\rm \pi} /L$ of our cubic domain of size $L^3$.

Table 1. Summary of runs discussed in this paper.

It turns out that a lower resistivity is important for obtaining the expected scaling. We therefore now consider Run B2, where ${\rm Lu}\approx 1300$. The result is shown in figure 2, where we used $\beta =1.6$ as the best fit, which is still slightly larger than the expected value of 3/2, but it goes in the right direction. Therefore, we show in figure 2 the resulting compensated spectra for Run B2, where the magnetic diffusivity is half that of Run B1 and ${\rm Lu}$ is now twice as large a before; see table 1. We use the Pencil Code (Pencil Code Collaboration et al. 2021) and, in both cases, we use a resolution of $1024^3$ mesh points.

Figure 2. Compensated spectra for Run B2.

Another comparison between Runs B1 and B2 is shown in figure 3, where we compare their evolution in the $pq$ diagram. While Run B1 clearly evolves along the $\beta \approx 1.7$ line, Run B2 tends to be closer to the $\beta \approx 3/2$ line. Note also that both runs settle near the $p=2(1-2q)$ self-similarity line (B20), although we begin to see departures near the end of the run, which is due to the finite size of the domain.

Figure 3. The $pq$ diagrams for Runs B1 (open red symbols) and B2 (closed blue symbols). Larger symbols denote later times.

Finally, we show in figure 4 the scalings of $I_{H}(t)$, where we see that the decay exponent $p_{H}\equiv -{\rm d}\ln I_{H}/{\rm d}\ln t$ is approximately $p_{H}\approx 0.16$ for Run B1 and approximately $0.11$ for Run B2. Earlier work by Zhou et al. (Reference Zhou, Sharma and Brandenburg2022) showed that $p_{H}$ decreases as the Lundquist number increases, and is, in MHD, approximately 0.2 for ${\rm Lu}\approx 10^3$, and decreases to $p_{H}\approx 0.01$ for ${\rm Lu}\approx 4\times 10^7$. Such large values can currently only be obtained with magnetic hyper-diffusivity (Hosking & Schekochihin Reference Hosking and Schekochihin2021; Zhou et al. Reference Zhou, Sharma and Brandenburg2022), but this has not been attempted in the present work.

Figure 4. Evolution of $I_{H}(t)$, showing first a slight increase and then a decline proportional to $t^{-0.16}$ for Run B1 and proportional to $t^{-0.11}$ for Run B2. The inset shows the evolution of ${\mathcal {I}}_{H}(R;t)$ as a function of $R$ for increasing values of $t$ (indicated by the arrow) for Run B2. The abscissae of the main plot and the inset are normalized by $\eta k_0^2$ and $k_0$, respectively.

As already noted by Zhou et al. (Reference Zhou, Sharma and Brandenburg2022), there is an initial increase in $I_{H}(t)$. This is explained by the fact that the magnetic field obeys Gaussian statistics initially, but not during the later evolution. The inset shows the $R$ dependence of ${\mathcal {I}}_{H}(R;t)$ for different $t$. The relevant value of $R$ is deemed to be at the location where the local slope of ${\mathcal {I}}_{H}(R)$ is minimum at late times.