Negative Turbulent Magnetic Diffusivity $\beta$ effect in a Magnetically Forced System

The amplification of $B$ field in plasma usually requires seed magnetic field. However, the origin of seed field (primordial magnetic field, PMF) is still under debate. At present, its cosmological origins are divided into the era of inflationary genesis and post-inflationary magneto-genesis.

The first inflationary scenario generates the very large scale PMF, but it needs the breaking of conformal symmetry by the interaction of the electromagnetic field and the gravitational field. The breaking of the conformal symmetry is to consider the Electro-Magnetic (EM) coupling to scalar field [4, 5], coupling to the modified general relativity f(R) theory, coupling to pseudo scalar field and so on. The PMF strength could be generated by quantum fluctuations and has been estimated as $10^{-5}nG-1nG$[6].

The second scenario is based on the cosmological Quantum Chromo Dynamics (QCD) phase transition ($\sim$250MeV) [7, 8] and the electroweak phase transition ($\sim$100MeV). The PMF could be generated by collision and percolation of some bubbles from the first order transition and estimated as $10^{-7}$nG by the quark-hadron and $10^{-14}$nG - $10^{-8}$nG order by the electroweak transition.

The third scenario can occur during or after the epoch of photon last scattering. The PMF can be produced by non-vanishing vorticity, which arises from the non-zero electron and proton fluid angular velocities by the different masses of proton and electron in the gravitational field (Harrison’s mechanism, [9]). The PMF is thought to be about $10^{-9}nG$

The second and third scenarios are thought to occur on a correlation scale smaller than the Hubble radius, by which we expect a suitable field generated by another dynamical effect, for instance, Biermann battery mechanism [10]. When the hot ionized particles (plasma) collide mutually, the fluctuating electron density $\nabla n_{e}$ and pressure $\nabla p_{e}$ (or temperature $\nabla T_{e}$) can be misaligned. This instability $-\nabla p_{e}/n_{e}e$ can drive currents to generate magnetic fields. Also, the neutrino interaction with charged leptons at the early epoch is thought to have generated primordial magnetic helicity, the measure of twist and linkage of magnetic fields ($H_{M}=\langle{\bf A}\cdot{\bf B}\rangle$, ${\bf B}=\nabla\times{\bf A}$) [11]. But, since the neutrino interaction exists not only in the early universe epoch but also are abundant in the present Universe including the Sun, lepton-neutrino interaction in primordial plasma can be one of the promising candidates of (origin) magnetic field generation.

The PMF may have scalar, vector and tensor fluctuations, which may affect CMB observables such as the CMA anisotropies and the matter power spectrum. Also the helical PMF may produce parity-odd cross correlations which results in the non-Gaussian CMB. The PMF effect on the BBN has been shown to be not large enough to explain the Li7 problem [12]. But a recent paper [13] assumed inhomogeneous PMF, which causes inhomogeneous temperature and non-Maxwell Boltzmann (MB) velocity distributions of baryons, and affect the relevant nuclear reaction on the BBN epoch.

These quantum fluctuating effects on magnetic field generation were followed by that of the collective plasma particles, i.e., fluid. The statistical feature of charged particles constraining one another became more important. Their aggregative motion formed a flow, which interacted with the seed magnetic fields, led to the amplification (dynamo) of $B$ field or its decay according to the various conditions. The evolution of magnetic field is now explained with Faraday equation ${\partial\bf B}/\partial t=-\nabla\times{\bf E}$ combined with Ohm’s law $\eta{\bf J}=({\bf E}+{\bf U}\times{\bf B})$ in the level of magnetohydrodynamics (MHD).²²2${\bf J},\,\eta,\,{\bf E},\,{\bf U}$ are current density, magnetic diffusivity, electric field, and plasma velocity. This combined equation, i.e., magnetic induction equation implies that any electromagnetic instability such as Biermann’s battery effect or lepton-neutrino interaction can be merged into electromotive force (EMF, $\sim{\bf U}\times{\bf B}$ omitting $\int d\tau$) in the equation: ${\partial\bf B}/\partial t=\nabla\times({\bf U}\times{\bf B}-\eta{\bf J}+{\bf f}_{mag})$. Then, the provided magnetic energy grows and propagates in the plasma system. This type of process is called magnetic forcing dynamo (MFD).

In comparison with kinetic forcing dynamo (KFD), MFD has its own characteristic features. For example, seed magnetic field is not necessary, and magnetic helicity is not conserved. The most unusual thing is that plasma kinetic energy ($E_{V}=U^{2}/2$) is not converted into magnetic energy ($E_{M}=B^{2}/2$). Rather, partial magnetic energy is converted into kinetic energy through Lorentz force. The converted velocity field forms EMF, which transfers magnetic energy. This energy convert process minimizes the thermal dissipation in plasma so that the amplification of magnetic field in MFD is more efficient than that of KFD. In our case, we used only 20% of forcing strength compared to KFD, but the saturated magnetic energy is larger. MFD itself is one of the prominent processes that connect the cosmological magnetic field and the seed magnetic field for KFD.

Basically, dynamo is a nonlinear phenomenon. However, when the field is helical ($\nabla\times{\bf F}=\lambda{\bf F}$), the dynamo process, i.e., EMF can be linearized with $\alpha$ & $\beta$ and large scale magnetic field $\overline{\bf B}$. At present, the exact analytic forms of $\alpha$ & $\beta$ are unknown. Only, their sketchy representations can be derived with closure theory and function reiterative method, e.g., mean field theory (MFT, [14]), eddy damped quasi normal markovianized approximation (EDQNM, [15]), direct interactive approximation (DIA, [16]). Physically, $\alpha$ effect is thought to arise with Coriolis force and buoyancy (kinetic helicity ‘$-\langle{\bf u}\cdot\nabla\times{\bf u}\rangle$) and gradually becomes quenched by current helicity ‘$\langle{\bf b}\cdot\nabla\times{\bf b}\rangle\,(=k^{2}\langle{\bf a}\cdot\nabla\times{\bf b}\rangle$, k: wavenumber in Fourier space)’ generated by the growing magnetic back reaction³³3The over bar in the variable $\overline{X}$ means the large scale quantity, and the small letter $x$ means the quantity in the small (turbulent) scale regime. And, the angle bracket indicates its spatial average over the large scale regime: $(1/2L)^{3}\int^{L}_{-L}{X}d{\bf r}$. We assume $\langle\overline{X}\rangle\sim\overline{X}$. Adding these two effects qualitatively explains how the large scale magnetic field grows and finally becomes saturated. However, for some systems like Solar(stellar) corona or a jet structure above the accretion disk, it is hard to expect that such helical kinetic motion exists and triggers the dynamo process. Rather, the transferred helical magnetic field $(\nabla\times{\bf B}=\lambda{\bf B}$) from the structures is more likely to play a key role in dynamo.

Theoretically, $\alpha$ effect is derived from the differentiation of EMF over time: $\partial\langle{\bf u}\times{\bf b}\rangle/\partial t$. $\partial{\bf u}/\partial t$ and $\partial{\bf b}/\partial t$ are replaced by MHD equations leading to $\alpha\sim\int dt(-\langle{\bf u}\cdot\nabla\times{\bf u}\rangle+\langle{\bf b}\cdot\nabla\times{\bf b}\rangle)$. During this analytic calculation, there is no constraint on the derivation of $-\langle{\bf u}\cdot\nabla\times{\bf u}\rangle$ and $\langle{\bf b}\cdot\nabla\times{\bf b}\rangle$. The calculation is done only with tensor identity and the assumption of isotropy. This implies that we can drive the system either kinetic energy or magnetic energy. Current density $\bf J$ is the source of magnetic field as Biot-Savart law and Maxwell theory indicate. Moreover, to produce magnetic helicity in a lab., $\bf J$ is transmitted along magnetic field. These theoretical and experimental examples show that helical magnetic forcing dynamo (HMFD) is not forbidden but a feasible process in nature.

However, there are a couple of things to be made clear in HMFD. The growth rate ($\alpha$) should be larger than the dissipation rate to arise magnetic field. If current helicity is a unique component in $\alpha$ effect, magnetic field grows without stop. To prevent this catastrophic amplification, there should be sort of a constraining effect such as kinetic helicity. However, since helical magnetic field $\nabla\times{\bf B}\sim{\bf B}$ nullifies Lorentz force ${\bf J}\times{\bf B}$, the generation of helical velocity field by ${\bf B}$ looks contradictory. Moreover, even if the generation of $\bf U$ is explained, there remain tricky issues in the conservation and chirality of helicity. For example, if the system is forced by right handed($+$) helical kinetic energy, left handed($-$) magnetic helicity is generated and inversely cascaded. And, this ($-$) magnetic helicity in the large scale produces ($+$) magnetic helicity in the small-scale regime to conserve magnetic helicity in the system. However, in HMFD, if ($+$) helical magnetic energy drives the system, ($+$) magnetic helicity is generated in the whole scale.

Besides, the scale invariant helical magnetic field, i.e., magnetic helicity provides us with information on the horizon during inflation. The correlation length of PMF is constrained by hubble horizon, beyond which the correlation vanishes with sudden cut-off. Therefore, within the subhorizon with the condition of ${\bf B}\cdot{\bf\hat{n}}=0$, magnetic helicity becomes gauge free. However, the correlation length scale will be smaller than the typical galaxy scale. Even if the scale had been expanded through MHD process, the strength of magnetic field would have become too weak. An efficient dynamo process by helicity leading to both inverse cascade of magnetic energy and scale expansion is required.

In the present paper, to explain the inconsistent features depending on the forcing method, we apply the analytic method and field structure model used in helical kinetic forcing dynamo [17] to helical magnetic forcing dynamo. Followed by this introductory section, numerical method and related MHD equations for simulation are introduced in chapter 2. In chapter 3, we show numerical results for the evolving ${B}$ field and its inverse cascade to the large scale regime. And then, we show the evolving profile of $\alpha$ & $\beta$ along with the growth of $B$ field and investigate their physical features and mutual relations. In chapter 4, we discuss the parameterisations of EMF: $\langle{\bf u}\times{\bf b}\rangle\sim\int d\tau(\alpha\overline{\bf B}-\beta\nabla\times\overline{\bf B}$) and compare them using numerical data and analytic approach. Using field structure model, we explain the intuitive meaning of $\alpha$ effect and how $\beta$ becomes negative. Then, to supplement them, we derive $\beta$ coefficient again when the field is helical. $\beta$ effect also explains how the plasma velocity field is suppressed when the system is forced by helical magnetic field. This work focuses on the physical mechanism of helical forcing dynamo which occurs in the fundamental level of astro-plasma system. In chapter 5, we summarize our work.

The fields $\bf U$, $\bf B$ can be separated into the large scale fields $\overline{\bf U}$, $\overline{\bf B}$ and turbulent small scale ones $\bf u$, $\bf b$. Analytically, the evolution of $\overline{\mathbf{B}}$ can be represented as follows:

Since $\alpha$&$\beta$ are turbulent quantities, their curl effect over the large scale is negligible. Then, with some rearrangement, we can derive Eq. (2.6). And, this diffusion equation can be analytically solved with some appropriate closure theory. However, since Eq. (2.1)-(2.3) should be numerically solved, we used Pencil-Code [18](see the manual http://pencil-code.nordita.org). Pencil-code is a 6^th order finite-difference code for compressible fluid dynamics (CFD) with magnetic field. The code solves vector potential ‘$\bf A$’ in Eq. (2.3) instead of magnetic field ‘${\bf B}$’ Eq. (2.4). Solving ‘$\bf A$’, the condition of divergence free magnetic field ($\nabla\cdot{\bf B}=0$) is met without any numerical manipulation. Moreover, magnetic helicity $H_{M}(=\langle{\bf A}\cdot{\bf B}\rangle)$ as well as magnetic energy $E_{M}(=\langle B^{2}/2\rangle)$ can be calculated free from the gauge issue with the assumption of the simply connected volume bounded by a magnetic surface (${\bf B}\cdot\hat{n}=0$.)

We initially gave a random seed magnetic field $B_{0}\sim 10^{-4}$ (no unit) to a cube simulation box $(8\pi^{3}$) with a periodic boundary condition. But, what determines the evolution of the system is the forcing method. We used a forcing function as follows:

This is the Fourier transformed function represented by a wavenumber ‘$k$’, helicity ratio controller $\lambda$, and arbitrary unit vector ‘$\mathbf{\hat{e}}$’. ‘$k$’ is inversely proportional to the eddy scale $l\sim 1/k$. For instance, $k=1$ indicates the large scale regime, and $k>2$ indicates the small (turbulent) scale regime. We gave helical magnetic field (energy) at the randomly chosen wavenumber $k$, which is constrained by $\langle k\rangle_{ave}\equiv k_{f}\sim 5$. This forcing function can be located at Eq. (2.2) (KFD), or Eq. (2.3) (MFD). And, if $\lambda$ is $+(-)1$, the forcing energy is fully right (left) handed helical field $i\mathbf{k}\times\mathbf{f}=\pm k\mathbf{f}$. But, $\lambda=0$ yields a nonhelical forcing source.

Of course, the forcing source in nature is different from Eq.(2.7). In Appendix, we introduced two examples of magnetic forcing dynamo $f_{mag}$: Biermann battery effect and neutrino-lepton interaction. One of the most essential differences between them is magnetic helicity that decides the direction of magnetic energy transfer. Biermann effect does not have helicity so that the magnetic energy cascades toward the smaller scale regimes (small scale dynamo). In contrast, neutrino-lepton interaction yields magnetic helicity, i.e., $\alpha$ effect, which can play an important role in large scale dynamo.

We prepared for the two systems with a unit magnetic Prandtl number $Pr_{M}=\eta/\nu=1$. $\eta=\nu=6\times 10^{-3}$, and numerical resolution is $400^{3}$. They were forced by Eq. (2.7) with fully helical magnetic field ($\lambda=+1$ or $-1$ at $k=5$). We used basic data set such as kinetic energy $E_{V}(=\langle U^{2}\rangle/2)$, magnetic energy $E_{M}(=\langle B^{2}\rangle/2)$, kinetic helicity $\langle{\bf U}\cdot\nabla\times{\bf U}\rangle$, and magnetic helicity $\langle{\bf A}\cdot{\bf B}\rangle$. The stability of code and data have been verified.

In Fig. 1(a), large scale magnetic energy $\langle{\overline{B}}^{2}\rangle$ (=$2\overline{E}_{M}$, $k=1$, solid line) grows to be saturated at $t\sim 100$. Along with $\langle{\overline{B}}^{2}\rangle$, the large scale magnetic helicity $\langle{\overline{\bf A}}\cdot{\overline{\bf B}}\rangle$(dashed line) evolves keeping the relation of $\langle{\overline{B}}^{2}\rangle\geq k\langle{\overline{\bf A}}\cdot{\overline{\bf B}}\rangle$($k=1$). Also, kinetic energy $\langle{\overline{U}}^{2}\rangle$ in the large scale grows keeping $\langle{\overline{U}}^{2}\rangle\geq\langle{\overline{\bf U}}\cdot\nabla\times{\overline{\bf U}}\rangle/k$. But the direction of kinetic helicity fluctuates from positive to negative as the discontinuous cusp line implies in this log-scaled plot. Similarly, Fig. 1(c) shows that evolving small scale magnetic energy $\langle b^{2}\rangle$ and kinetic energy $\langle u^{2}\rangle$ with their helical part $\langle{{\bf u}}\cdot\nabla\times{{\bf u}}\rangle/k$ and $k\langle{{\bf a}}\cdot{{\bf b}}\rangle$ ($k\geq 2$). The fields grow and get saturated like the large scale field, but the saturation occurs earlier than that of the large scale field because of their smaller eddy turnover time.

Fig. 1(b), 1(d) show the growth of kinetic and magnetic energy in the system forced by the fully left-handed magnetic energy. Basically, they evolve consistently in comparison with Fig. 1(a), 1(c). But, the kinetic helicity and magnetic helicity are invisible in this logarithmic plot indicating their left-handed (negative) chirality. The direction of helicity in the magnetically forced system tends to be consistent with that of forcing energy. This is the opposite tendency of HKFD (helical kinetic forcing dynamo). Nonetheless, the practically same growth of energy shows that the chirality of the forcing energy is not a determinant to the evolution of the plasma system.

Fig. 2 shows the evolving magnetic helicity ratio $f_{h}\equiv k\langle{\bf a}\cdot{\bf b}\rangle/\langle b^{2}\rangle$ and kinetic helicity ratio $\langle{\bf u}\cdot\nabla\times{\bf u}\rangle/k\langle u^{2}\rangle$ for $k$=1, 5, 8. Left and right panel are for the right handed case ($f_{h}=1$) and left handed one ($f_{h}=-1$), respectively. The helicity ratio of large scale ${\overline{B}}$ is eventually saturated at $f_{h}=+1$ ($-1$), and that of the small scale u & b reaches to the value less (larger) than ‘$1$ ($-1$)’. However, the helicity ratio of ${\overline{U}}$ is as low as $\sim 0.25$ ($-0.25$). The magnetic helicity ratio less than ‘1’ in the small scale regime shows that the small scale magnetic field substantially accelerates the large scale plasma motion. Plasma in the large scale regime is driven by Lorentz force ${\bf J}(p)\times{\bf B}(q)$, where the wave numbers are constrained by $p+q=1$. This implies that the eddies associated with $p$, $q$ are very close to each other in the small scale regime and nearly out of phase. Their helicity ratios smaller than $1$ implies Lorentz force forcing the large scale eddy meaningfully grows. But, the effect of the large-scale magnetic field on the plasma is limited: $\overline{\bf J}\times\overline{\bf B}\sim 0$. The saturated helicity ratio ‘$f_{h}=$1’ for the large scale field indicates that $k$ for $\overline{\bf B}$ is definitely ‘1’. Also, the initial largest energy level with ‘$k=5$’ implies that the scale with this wavenumber is forced by an external energy source, which is consistent with our code setting.

Fig. 3 includes the temporally evolving $\alpha$ & $\beta$ effect and the large scale magnetic energy $2\overline{E}_{M}$. Left (right) panel shows the evolution of $E_{M}$, $\alpha$ and $\beta$ effect for $f_{h}=1$ ($f_{h}=-1$). $\alpha$ effect for $f_{h}=1$ positively oscillates and decreases significantly as $E_{M}$ gets saturated. In contrast, $\alpha$ effect for $f_{h}=-1$ negatively oscillates before it disappears. $\alpha$ effect is quenched much earlier than the slowly evolving $E_{M}$. The decreasing oscillation in both cases implies that $\alpha$ effect does not play a decisive role in the growth of the large scale magnetic field. Conversely, $\beta$ retains the negative value in both cases and has a much larger size than that of $\alpha$. This negative $\beta$, combined with the negative Laplacian $\nabla^{2}\rightarrow-k^{2}$ in Fourier space, can be considered as the actual source of the large scale magnetic field. This is contradictory to the current dynamo theory concluding that $\beta$ is always positive to diffuse magnetic energy. We will show that this conventional inference is valid only for the ideally isotropic system with reflection symmetry. When the symmetry is broken, ‘$\bf u$’ in the small scale regime can yield the anti-diffusing effect of magnetic field.

In Fig. 4, we compared $\nabla\times\langle{\bf u}\times{\bf b}\rangle$ (black solid line) with $\nabla\times(\alpha\overline{\bf B}-\beta\nabla\times\overline{\bf B})$ (red dashed line) to verify Eq. (2.6) and Eq. (4.15), (LABEL:betaSolution3). The former uses only the simulation data for the large scale magnetic energy $\overline{E}_{M}$, but the latter requires the data of large scale magnetic helicity $\overline{H}_{M}$ in addition to $\overline{E}_{M}$. Although they use different types of data and formulas, they are coincident in the transient mode ($t<\sim 100$ and in the range of $10^{-8}-10^{-2}$). Note that Eq. (4.15), (LABEL:betaSolution3) are valid until the system gets saturated where $\overline{H}_{M}\sim 2\overline{E}_{M}$. As Fig. 1, 2 show, $\overline{H}_{M}$ is different from $2\overline{E}_{M}$ in the transient state. As the field becomes saturated, $\overline{E}_{M}$ and $\overline{H}_{M}$ are so close that the logarithmic function diverges. For $f_{h}=-1$, we used absolute values for a clear comparison.

Fig. 5, 6 show field structure models. They are introduced to explain the dynamo process in an intuitive way. We will discuss the mechanism in detail soon.

Fig. 7 is for the typical kinetic small scale dynamo. Nonhelical random velocity field was driven at $k=5$. The plot includes large scale kinetic energy $E_{V}$ and magnetic energy $E_{M}$. $Re_{M}$ is approximately $80$. In comparison with LSD, $E_{M}$ grows a little bit, and $E_{V}$ is not quenched. Most magnetic energy is transferred to the small scale regime, and its peak is located at $k\sim 10$. These plots are to compare Fig. 1(a), 1(b).

Fig. 8, 8 in appendix include the simulations of solar magnetic field using Eq. (4.9) and (4.10). The horizontal axis means ‘scaled time’ ($0.01s\rightarrow 15.53$ years), and the vertical line indicates the ‘latitude’ ($0-\pi/2$: northern hemisphere, $\pi/2-\pi$: southern hemisphere). The color indicates the phase of a net magnetic field (toroidal ${\bf B}_{tor}$ + poloidal ${\bf B}_{pol}$). The simulation in the left panel is the reproduction of [19] without Babcock effect. It shows the period of 16.31 years for the one complete cycle of solar magnetic field: amplification-annihilation-reverse. On the contrary, in right panel, the tidal effect of planets on the Solar tachocline is added to $\alpha$. The period elevates up to 21.74 years without manipulating the numerical variables. These plots show the practical applicability of Eq. (2.4).

Fig. 9 in appendix shows the time evolving $1/3\langle u^{2}\rangle$ and $1/6\langle{\bf u}\cdot\nabla\times{\bf u}\rangle$. This plot is to make sure the overall $\beta$ effect in HMFD is negative. We will also briefly introduce Kraichnan’s result [20]. He expanded $\alpha$ with the unknown time constant $\tau_{2}$ and correlation factor $A(x-x^{\prime})$, and they work as a negative magnetic diffusivity.

Fig. 9 in appendix shows the effect of helical velocity field in magnetic diffusivity. Fully helical kinetic forcing is turned off at $t\sim 200$ followed by nonhelical kinetic forcing where $\langle{\bf u}\cdot{\nabla\times\bf u}\rangle=0$. After $t\sim 200$, turbulent magnetic diffusivity becomes positive, which diffuses magnetic field in the system. The growth of large scale magnetic field is suppressed.