Effect of Turbulent Kinetic Helicity on Diffusive $\beta$ effect for Large Scale Dynamo

Most celestial plasma systems are significantly influenced by magnetic fields, which play a crucial role in their dynamics. Magnetic fields extract energy from turbulent plasma, and the amplified fields react back on the system, constraining its evolution. On a macroscopic scale, magnetic fields are known to regulate the rate of star formation and the development of accretion disks [1, 2]. Additionally, the balance between magnetic pressure and plasma pressure can determine the stability of the system, as seen in phenomena such as the sausage, kink, or Kruskal-Schwarzschild instabilities [3]. Furthermore, the ubiquitous magnetic fields, having existed since just after the Big Bang, may have also played a critical role in nucleosynthesis, the production of matter in the Universe. Perturbed electrons, influenced by the Lorentz force, increase electron density through superposition, reducing the potential barrier between interacting nuclei and facilitating fusion [4, 5]. Magnetic fields are a nearly unique physical entity capable of influencing a plasma system while maintaining electrical neutrality.

According to Maxwell’s theory, magnetic fields in free space propagate infinitely while gradually decreasing in amplitude or field density. However, in plasma, where massive charged particles interact with the magnetic field, this propagation requires significantly more energy to overcome the interference from these heavy particles. Often, magnetic eddies, combined with particle motion, expand by reducing their eddy scales alongside fluid eddies. From the perspective of electromagnetism, this behavior contrasts with the intrinsic nature of magnetic fields. For magnetic eddies to grow in scale, specific conditions must be met to overcome the eddy turnover time or the inertia of massive particle eddies.

The amplification of magnetic fields migrating toward either larger or smaller scales is referred to as the dynamo. This process involves the conversion of kinetic energy into magnetic energy and its subsequent transport within the plasma system. The converted $B$-field cascades toward either the large-scale or small-scale regime, both processes involving the induction of the $B$-field through electromotive force (EMF, $(\mathbf{U}\times\mathbf{B})$, where $\mathbf{U}$ represents fluid velocity). The migration of magnetic energy toward the large-scale regime is known as an ’inverse cascade,’ resulting in a large-scale dynamo (LSD). In contrast, the ’cascade of energy’ refers to a small-scale dynamo (SSD). The cascade of energy to smaller scales is commonly observed in hydrodynamics (HD) and magnetohydrodynamics (MHD). However, the inverse cascade of energy requires more stringent conditions, driven by factors such as helicity, differential rotation [6, 7, 8, 9], or magnetorotational instability (MRI), also known as the Balbus-Hawley instability [1]. In SSD, non-helical magnetic energy ($E_{M}$) cascades toward smaller scales, resulting in the peak of $E_{M}$ forming between the injection and dissipation scales [10]. If the growth rate of the magnetic field depends on magnetic resistivity, the process is called a slow dynamo; otherwise, it is known as a fast dynamo. The migration and amplification of the magnetic field are influenced by various factors, and the critical conditions for these processes remain an ongoing topic of debate.

In the presence of a helical magnetic field ($\nabla\times\mathbf{B}=\lambda\mathbf{B}$), the turbulent electromotive force $\langle\mathbf{u}\times\mathbf{b}\rangle$ is known to be expressed as $\alpha\overline{\mathbf{B}}-\beta\nabla\times\overline{\mathbf{B}}$, regardless of whether the turbulence is driven by kinetic or magnetic forces. Determining the $\alpha$ and $\beta$ coefficients is crucial for explaining the evolution of the solar magnetic fields and other astrophysical ones. For instance, the magnetic induction equation reveals that the evolutions of poloidal and toroidal magnetic fields are constrained by the $\alpha$ coefficient [11, 12]. A precise understanding of these coefficients is valuable for forecasting space weather, which has significant implications for Earth’s climate.

There have been efforts to calculate these coefficients [13, 14, 15]. Currently, only approximate forms of the $\alpha$ and $\beta$ coefficients can be obtained through dynamo theories such as mean field theory (MFT), the eddy-damped quasi-normal Markovian (EDQNM) approximation, or the direct interaction approximation (DIA). These theories generally suggest that $\alpha$ is related to residual helicity $\langle\mathbf{b}\cdot(\nabla\times\mathbf{b})\rangle-\langle\mathbf{u}\cdot(\nabla\times\mathbf{u})\rangle$. The $\beta$ coefficient, on the other hand, is related to turbulent energy $\langle u^{2}\rangle+\langle b^{2}\rangle$.

Additional theoretical and experimental work has been conducted on negative magnetic diffusivity [16], and references therein. These studies are based on $\alpha-\alpha$ correlations in strong helical systems. Additionally, [17] experimentally found that turbulent magnetic diffusivity $\eta_{\text{turb}}$ was negative. They argued that the net diffusivity $\eta_{\text{turb}}+\eta$ became positive again. However, we believe that the overall dynamo effect weakened in their work. Negative magnetic diffusivity was also observed in another liquid sodium experiment [18], where it was found that small-scale turbulent fluctuations ($\sim u$) contribute to the negative magnetic diffusivity in the interior region. Recently, [15] suggested an iterative removal of sources (IROS) method that uses the time series of the mean magnetic field and current as inputs. This approach is quite mathematical, producing detailed components of $\alpha_{ij}$ and $\beta_{ij}$. We also investigate the effect of magnetic diffusion $\beta$ on the helical large-scale dynamo. We also aim to make this theoretically refined approach applicable to experiments or observations.

In Section 2, we describe the numerical model and code. Chapter 3 presents the results of numerical calculations, while Chapter 4 explains the related theory. Some of these theories have been introduced in our previous work ([19]¡ references therein), but for consistency and readability, they are explained again in more detail with additional content. We also include some IDL scripts used to create plots from the numerical data. The final chapter provides a summary. The most important feature of this paper is its use of large-scale magnetic energy and magnetic helicity data, which are relatively easy to measure, to apply theoretical methods for obtaining $\alpha$ and $\beta$ coefficients. This approach is then verified using turbulent kinetic data and is employed to directly reproduce the evolution of the large-scale magnetic field obtained numerically for comparison.

We employed the $\mathrm{PENCIL\,\,CODE}$ [9] for our numerical simulations. The computational domain is a periodic cube with dimensions of $(2\pi)^{3}$, discretized into a mesh of $400^{3}$ grid points. The code is used to solve the set of magnetohydrodynamic (MHD) equations, which are given as follows:

The symbols used in the equations are defined as follows: ‘$\rho$’ denotes the ‘density’, ‘$\bf U$’ represents the ‘velocity’, ‘$\bf B$’ corresponds to the ‘magnetic field’, ‘$\bf A$’ is the ‘vector potential’, and ‘${\bf J}$’ stands for the ‘current density’. The symbol ‘$D/Dt$’ refers to the ‘advective derivative’, defined as $\partial/\partial t+{\bf U}\cdot{\bf\nabla}$. The term ‘$\eta$’ (=$c^{2}/4\pi\sigma$) represents the molecular magnetic diffusivity, where $c$ is the speed of light and $\sigma$ is the conductivity.

These equations are presented in dimensionless form. The velocity field and magnetic field are expressed in units of the sound speed ‘$c_{s}$’ and $(\rho_{0}\,\mu_{0})^{1/2}c_{s}$, respectively. This scaling is derived from the relations $E_{M}\sim B^{2}/\mu_{0}$ and $E_{V}\sim\rho_{0}U^{2}$, where ‘$\mu_{0}$’ and ‘$\rho_{0}$’ represent the magnetic permeability in free space and the initial density, respectively. It is important to note that the plasma system is weakly compressible, implying ‘$\rho\sim\rho_{0}$’. The term ‘${\bf f}_{kin}(x,t)$’ denotes a forcing function, defined as $N\,{\bf f}(t)\,\exp\,[i\,{\bf k}_{f}(t)\cdot{\bf x}+i\phi(t)]$, where $N$ is a normalization factor, ${\bf f}$ is the forcing magnitude, and ${\bf k}_{f}(t)$ is the forcing wave number. The code randomly selects one of 20 vectors from the ${\bf k}_{f}$ vector set at each time step. For simplicity, ‘$c_{s}$’, ‘$\mu_{0}$’, and ‘$\rho_{0}$’ are set to ‘1’, rendering the equations dimensionless.

The forcing function ${\bf f}(t)$ is defined as $f_{0}\mathbf{f}_{k}(t)$:

‘$k$’ is a wavenumber defined as $2\pi/l$ ($l$: scale size). $k=1$ indicates the large scale regime, and $k>2$ means the wavenumber in the small (turbulent) scale regime. $\lambda=\pm 1$ generates fully right (left) handed helical field $\nabla\times{\bf f}_{k}\rightarrow i\mathbf{k}\times\mathbf{f}_{k}\rightarrow\pm k\mathbf{f}_{k}$. And, ‘$\mathbf{\hat{e}}$’ is an arbitrary unit vector. We gave fully helical kinetic energy ($\lambda=1$) at $\langle k\rangle_{ave}\equiv k_{f}\sim 5$. This forcing function was located at Eq. (2) with $f_{0}=0.07$ for the helical kinetic forcing dynamo (HKFD). But, $\lambda=0$ yields a nonhelical forcing source. Note that Reynolds rule is not applied to this energy source: $\langle f\rangle\neq 0$.

The MHD equation set primarily consists of differential equations, each requiring initial values for their solution. Notably, an initial seed magnetic field of $B_{0}\sim 10^{-4}$ was introduced into the system. However, the influence of this seed field diminishes rapidly due to the presence of the forcing function and the lack of memory in the turbulent flow. As we will observe, the small-scale magnetic energy is initially stronger during the very early stages, before decreasing in the subsequent simulation steps.

We have driven the plasma system with parameters $\nu=\eta=0.006$ using helical kinetic energy input. The helicity ratio is $f_{h}=1$ (fully helical), and the energy is injected at a forcing scale eddy with $k=5$. Since our goal was to determine $\alpha$ and $\beta$, we selected the most common conditions to minimize unnecessary complexities, such as those caused by an unbalanced dissipation scale with large or small magnetic Prandtl number $Pr_{M}=\nu/\eta$. In this section, we briefly introduce the numerical results shown in Figs. 1–3, and we will discuss their physical implications using theoretical methods in the next section.

Fig. 1(a) shows the energy evolutions in MHD turbulence dynamo. The externally given kinetic energy drives the plasma at approximately $U_{\text{rms}}$, and this kinetic motion is converted into magnetic energy $B_{\text{rms}}$. The initial magnetic field is weak but eventually surpasses the kinetic energy and becomes saturated. The conversion and amplification processes are fundamentally nonlinear and occur through the electromotive force $\langle\mathbf{U}\times\mathbf{B}\rangle$. Howeve, as Fig. 1(b) indicates, most energy is located at the forcing scale.

Fig. 2(a) shows the large-scale kinetic energy ($\overline{E}_{V}$, black dot-dashed line) and magnetic energy ($\overline{E}_{M}$, red solid line), both of which are multiplied by 10 for clarity. As expected, $\overline{E}_{M}$ rises at $t\sim 100{-}150$ before reaching saturation, while $\overline{E}_{V}$ remains negligibly small. We also include the evolving profiles of $\alpha$ (dotted line) and $\beta$ (dashed line) coefficients, obtained from the large-scale magnetic energy and magnetic helicity ($\overline{H}_{M}$). The $\alpha$ coefficient oscillates, while $\beta$ remains slightly negative. Both coefficients converge to zero as $\overline{E}_{M}$ saturates. Near saturation, $\alpha$ and $\beta$ oscillate rapidly in this nonlinear stage, so we applied a smoothing technique of IDL, averaging over approximately 10 points.

Fig. 2(b) presents the spectra of kinetic energy $E_{V}$ (black dashed line) and kinetic helicity $H_{V}$ (red solid line) in Fourier space at $t=0.2$ (lowest), 3.2, 100, and 1440 (highest). The forcing scale is $k=5$, and the smallest dissipation scale is $k_{\text{max}}=200$. Since the profiles for $k>100$ are just extensions of flat noise lines, we cut the spectrum at $k\sim 100$ for clarity. The plot indicates that kinetic energy injected at the forcing scale cascades toward smaller scales (larger $k$). Unlike kinetic energy, kinetic helicity has polarity. In some scale regimes, $H_{V}$ is negative, so we plotted its absolute value.

Fig. 2(c) shows the spectra of magnetic energy $2(E_{M}$ (black dashed line) and magnetic helicity $H_{M}$ (red solid line). Similar to Fig. 1(b), we used the absolute value for magnetic helicity, as it also has polarity. Notably, $\overline{H}_{M}$ at large scales ($k=1$) is negative, while $H_{M}$ at other scales is positive, indicating the conservation of magnetic helicity. The strongest $\overline{E}_{M}$, which indicates an inverse cascade, is accompanied by this oppositely polarized $\overline{H}_{M}$, characterizing the helical kinetic forcing dynamo. The temporally evolving energy profile at $k=1$ is also shown in Fig. 1(a).

Fig. 2(d) includes $k\langle B^{2}\rangle$ (red solid) and $\langle\mathbf{J}\cdot\mathbf{B}\rangle$ (black dashed). Since magnetic fields in the small-scale regime are much weaker compared to those in the large scale, we have used the amplified magnetic helicity and magnetic energy with the wavenumber $k$. Current helicity $\langle\mathbf{J}\cdot\mathbf{B}\rangle=k^{2}\langle\mathbf{A}\cdot\mathbf{B}\rangle$ exhibits similar characteristics to magnetic helicity $\langle\mathbf{A}\cdot\mathbf{B}\rangle$, apart from the magnitude difference due to the $k^{2}$ factor. This plot illustrates that while the magnetic helicity at the forcing scale is positive, the large-scale magnetic helicity grows to be negative. This demonstrates the conservation of magnetic helicity and the relationship between helicity and energy.

Fig. 3(a) displays $E_{V}$ (black dashed line) and $E_{M}$ (red solid line) together. It is evident that the kinetic energy injected at $k=5$ is converted into magnetic energy and undergoes an inverse cascade to larger scales. Due to the complex energy transfer, Kolmogorov’s $k^{-5/3}$ scaling is not observed; instead, the spectrum is much steeper, approximately $k^{-4}$. These processes are inherently nonlinear, making them challenging to understand. However, they may be linearized using the $\alpha$ and $\beta$ parameters to make the dynamics more intuitively understandable.

Fig. 3(b) shows the helicity ratios. The kinetic helicity ratio, $f_{hk}$, was calculated using $\langle\mathbf{U}\cdot\bm{\omega}\rangle/k\langle U^{2}\rangle$, where $\bm{\omega}$ is the vorticity $\nabla\times\mathbf{U}$. The magnetic helicity ratio, $f_{hm}$, was calculated using $k\langle\mathbf{A}\cdot\mathbf{B}\rangle/\langle B^{2}\rangle$. The helicity ratios were computed for $k=1$(large scale), $k=5$(forcing small scale), and $k=8$(small scale). Since we applied fully helical kinetic energy, $f_{hk}$ at $k=5$ remains 1. The other kinetic helicity ratios behave similarly to that at the forcing scale, except at the large scale. For magnetic helicity, the polarity depends on the wavenumber. $f_{hm}$ at $k=1$ (large scale) saturates at ‘-1’, while $f_{hm}$ at smaller scales, $k=5$ and $k=8$, becomes positive. These opposite signs clearly demonstrate the conservation of magnetic helicity. Of course, since we externally impose helical energy, the total magnetic helicity is not exactly zero. However, the tendency toward conservation is maintained. And, it should be noted that a helicity ratio less than 1 indicates the production of nonhelical components, which is a natural occurrence in turbulence.

Fig. 3(c) shows and compares the profiles of $\alpha$ and $\frac{1}{3}\left(\langle\mathbf{j}\cdot\mathbf{b}\rangle-\langle\mathbf{u}\cdot\bm{\omega}\rangle\right)$. The $\alpha$ profile was calculated using $\overline{E}_{M}$ and $\overline{H}_{M}$. The $\alpha$ profile from the large-scale magnetic data oscillates and converges to zero early. In contrast, the conventional integrand constituting $\alpha$¹¹1For simplicity, we assumed a correction time of 1, such that $\int^{\tau}f\,dt\rightarrow f\tau\rightarrow f$. remains negative for a longer period and saturates at a decreased negative value as $\overline{B}$ saturates. Since the system was driven with helical kinetic energy, kinetic helicity is naturally larger than current helicity. We tested the profiles for $k=2-4$, $k=2-6$, and $k=2-k_{\text{max}}$. The forcing scale regime, including $k=5$, primarily determines the profile.

Fig. 3(d) shows the profiles of $\beta$ derived from large-scale magnetic data ($\overline{E}_{M}$ and $\overline{H}_{M}$) and from small-scale kinetic data $\frac{1}{3}\langle u^{2}\rangle-\frac{l}{6}\langle\mathbf{u}\cdot\bm{\omega}\rangle$. For simplicity, we also set the correlation time to 1. Compared to the conventional $\beta$, the latter includes the effect of kinetic helicity modified by the correlation length $l/6$. Since the exact method for determining $l$ is not yet known, we used the inverse of the wavenumber $2\pi/k$. At present, the smallest wavenumber in the small-scale regime, $k=2$, provides the best fit to the $\beta$ profiles. When larger wavenumbers are used, the profiles tend to increase and approach zero. Since a given eddy cannot simultaneously possess both toroidal and poloidal components, it can be inferred that there exists a correlation length between the toroidal and poloidal components that constitute the kinetic helicity. Although we have used a trial-and-error method to find $l$, a more precise method and physical analysis are necessary.

Fig. 4(a) shows a comparison between the electromotive force (EMF) $\langle\mathbf{u}\times\mathbf{b}\rangle$ and the linearized EMF, given by $\alpha\mathbf{B}-\beta\nabla\times\mathbf{B}$. The exact profiles of $\langle\mathbf{u}\times\mathbf{b}\rangle$ are not available at present due to the uncertainty in the exact range of the small-scale regime. Instead, we used $\partial\overline{\mathbf{B}}/\partial t-\eta\nabla^{2}\overline{\mathbf{B}}$ (red solid line) as a substitute for $\langle\mathbf{u}\times\mathbf{b}\rangle$ in the small scale regime. This serves as a reasonable approximation for the EMF at small scales. For $\alpha$, we used the large-scale magnetic data $\overline{H}_{M}$ and $\overline{E}_{M}$ for this semi $\alpha$ instead of the conventional approach. And, for semi $\beta$(dotted line), one version is derived from the large-scale magnetic data like $\alpha$, while theo $\beta$ (dot-dashed line) comes from a theoretical approach that includes kinetic helicity and kinetic energy. The EMF grows and saturates around $0.002$, which is compensated by $\eta\nabla^{2}\overline{\mathbf{B}}$ to ensure that $\partial\overline{\mathbf{B}}/\partial t$ approaches zero. Initially, the linearized EMF with large-scale magnetic data (semi $\alpha$ and semi $\beta$ as well as theo $\beta$) closely matches $\langle\mathbf{u}\times\mathbf{b}\rangle$. However, as the nonlinear stage progresses or magnetic effects grow, the EMF with $(\alpha_{\text{semi-theo}},\,\beta_{\text{theo}})$ becomes larger than that of the other set. This indicates that $\beta_{\text{theo}}$ is not sufficiently quenched. In contrast, the EMF with $(\alpha_{\text{semi-theo}},\,\beta_{\text{semi-theo}})$ aligns precisely with $\nabla\times\langle\mathbf{u}\times\mathbf{b}\rangle$. Due to the significant numerical noise, a smoothing function in IDL (averaging over 10 neighboring points) was applied. In contrast, theo-$\beta$ derived from the small-scale kinetic data performs better during this nonlinear stage.

Fig. 4(b) includes the reproduced the large-scale magnetic field $\overline{\mathbf{B}}$ using two approaches: (semi-$\alpha$, semi-$\beta$, black dot-dashed line) and (semi-$\alpha$, theo-$\beta$, red dashed line). These are compared with the numerically calculated $\overline{\mathbf{B}}$ from the code (black solid line). The reproduced fields match quite well with the DNS data, but in the nonlinear stage, the theoretical $\beta$ yields better results than the semi-analytic $\beta$. As noted, during this stage, numerical oscillations caused by the close values of $\langle\overline{\mathbf{B}}^{2}\rangle$ and $\langle\overline{\mathbf{A}}\cdot\overline{\mathbf{B}}\rangle$ increase significantly.

Figs. 5, 6 are designed to examine the dependency of $\alpha$ and $\beta$ on (kinetic) helicity from different perspectives. Fig.  5(a) illustrates the evolution of the profiles of $2\overline{E}_{M}$ and $\overline{H}_{M}$ in a system driven by helical kinetic energy (at $k=5$) for $t<210$, and by nonhelical kinetic energy (also at $k=5$) for $t>210$. Here, helicity is controlled by adjusting $\lambda$ in Eq. 6 from $\lambda=1$ to $0$, while keeping the energy constant. When kinetic helicity vanishes at $t=210$, $\overline{B}$ undergoes a sharp decline as the figure shows. Fig. 5(b) shows the changes in the helicity ratio of kinetic eddies and magnetic eddies across different scales. In the $t<210$ interval, the helical forcing case is similar to Fig. 3(b); however, after helicity is removed, the helicity ratio converges to zero in many cases. Nonetheless, at small scales, a nonzero region persists for some time. In Fig. 5(c), the evolution of $\alpha$ and $\alpha_{MFT}$ is presented, where it is notable that $\alpha_{MFT}$ converges to zero almost simultaneously as kinetic helicity disappears. $\alpha$ has already approached zero, so its change is minimal. However, in the case of $\alpha_{MFT}$, as shown in Fig. 5(b), $\langle{\bf j}\cdot{\bf b}\rangle$ remains nonzero for a time, but $\alpha_{MFT}$ converges to zero more rapidly. This suggests that the defining equation for MFT, Eq. (12), may be insufficient. Fig. 5(d) compares $\beta$, $\beta_{theo}$, and $\beta_{MFT}$. It is noteworthy that all three measures converge when helicity is absent.

Fig. 6(a) compares the source of $\overline{B}$, $\nabla\langle{\bf u}\times{\bf b}\rangle$, with the EMF composed of $\alpha$ and $\beta$. Since helicity was removed before the EMF sufficiently increased, the changes are minor, but it is presented here in the same format as Fig. 4(a) for consistency. Interestingly, as helicity supply ceases, the curl of EMF converges to zero, yet slight oscillations remain. Fig. 6(b) reconstructs $\overline{B}$ using $\alpha$ and $\beta$, confirming a consistent result.