Negative Turbulent Magnetic Diffusivity β effect in a Magnetically Forced System
MMNRAS , 1–14 () Preprint 9 February 2021 Compiled using MNRAS L A TEX style file v3.0
Negative Magnetic Diffusivity in a Magnetically Dominant System
Kiwan Park, ★ Myung-Ki Cheon, † Soongsil University, 369, Sangdo-ro, Dongjak-gu, Seoul, Republic of Korea
Accepted XXX. Received YYY; in original form ZZZ
ABSTRACT
We study the large scale dynamo process in a system forced by helical magnetic energy. The dynamo process is basicallynonlinear, but can be linearized with pseudo scholars 𝛼 & 𝛽 and large scale magnetic field B . A coupled semi-analytic equationsbased on statistical mechanics are used to investigate the exact evolution of 𝛼 & 𝛽 . This equation set requires only magnetic helicityand magnetic energy. They are fundamental physics quantities that can be obtained from the dynamo simulation or observationwithout any artificial modification or assumption. 𝛼 effect is thought to be related to magnetic field amplification. However, inreality it converges to 𝑧𝑒𝑟𝑜 very quickly without a significant contribution to B field amplification. Conversely, 𝛽 effect for themagnetic diffusion maintains a negative value, which plays a key role in the amplification with Laplacian ∇ → − 𝑘 . In addition,negative magnetic diffusion accounts for the attenuation of plasma kinetic energy when the system is saturated. The negativemagnetic diffusion is from the interaction of advective term − U · ∇ B and the strongly helical field. When plasma velocity field U is divided into the poloidal component U 𝑝𝑜𝑙 and toroidal one U 𝑡𝑜𝑟 in the absence of reflection symmetry, they interact with B · ∇ U and − U · ∇ B to produce 𝛼 effect and (negative) 𝛽 effect, respectively. We discussed this process using the theoreticalmethod and intuitive field structure model. Key words: magnetohydrodynamics(MHD) – turbulence – plasmas–magnetic forcing–dynamo– 𝛼 & 𝛽 effect Most celestial plasma systems are constrained by magnetic field 𝐵 . However, despite the ubiquitous existence of 𝐵 field, its rolein the astrophysical system is not yet completely understood. Briefly, 𝐵 field takes energy from the turbulent plasma (dynamo),and the amplified field back reacts to the system (magnetic back reaction). Through this mutual interaction, 𝐵 field controls therate of formation of a star and accretion disk (Balbus & Hawley 1991; Machida et al. 2005). Also, the balanced pressure betweenthe magnetic field and plasma can decide the stability of the system (see sausage, kink, or Kruskal-Schwarzschild instability, seeBoyd & Sanderson (2003)).The amplification of 𝐵 field in plasma requires seed magnetic field. However, the origin of seed field (primordial magneticfield, 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 theinteraction of the electromagnetic field and the gravitational field. The breaking of the conformal symmetry is to consider theElectro-Magnetic (EM) coupling to coupling to scalar field (Martin & Yokoyama 2008; Subramanian 2016), coupling to themodified general relativity f(R) theory, coupling to pseudo scalar field and so on. The PMF strength could be generated byquantum perturbations and has been estimated as 10 − 𝑛𝐺 − 𝑛𝐺 (Yamazaki et al. 2012).The second one is based on the cosmological Quantum Chromo Dynamics (QCD) phase transition ( ∼ ∼ − nG by the quark-hadron and 10 − nG -10 − 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 ★ E-mail: [email protected] † E-mail: [email protected] © The Authors a r X i v : . [ phy s i c s . p l a s m - ph ] F e b Kiwan Park et al. vorticity, which arises from the non-zero electron and proton fluid angular velocities by the different masses of proton andelectron in the gravitational field (Harrison’s mechanism, Harrison (1970)). The PMF is thought to be about 10 − 𝑛𝐺 The second and third scenarios are thought to occur on a correlation scale smaller than the Hubble radius, by which we expecta suitable field generated by another dynamical effect, for instance, Biermann battery mechanism (Biermann 1950). When thehot ionized particles (plasma) collide mutually, the perturbed electron density ∇ 𝑛 𝑒 and pressure ∇ 𝑝 𝑒 (or temperature ∇ 𝑇 𝑒 ) can bemisaligned. This instability −∇ 𝑝 𝑒 / 𝑛 𝑒 𝑒 can drive currents to generate magnetic fields. Also, the collisionless neutrino interactionwith charged leptons at the early epoch is thought to have generated primordial magnetic helicity, the measure of twist andlinkage of magnetic fields (cid:104) A · B (cid:105) ( B = ∇ × A ) (Semikoz & Sokoloff 2005). But, since the neutrino interaction exists not onlyin the early universe epoch but are also abundant in the present Universe including the Sun, lepton-neutrino interaction is one ofthe promising candidates of (origin) magnetic field generation.As the number of charged particles increased with time, their collective motion became more important than the quantum fluc-tuation effect. Their aggregative motion formed a flow, which interacted with the seed magnetic fields leading to the amplification(dynamo) or decay of 𝐵 field according to the various conditions. The evolution of magnetic field is now explained with Faraday’slaw 𝜕 B / 𝜕𝑡 = −∇ × E combined with Ohm’s law 𝜂 J = ( E + U × B ) in the level of magnetohydrodynamics (MHD). This com-bined equation, i.e., magnetic induction equation implies that any electromagnetic instability such as Biermann’s battery effector lepton-neutrino interaction can be merged into electromotive force (EMF) in the equation: 𝜕 B / 𝜕𝑡 = ∇ × ( U × B − 𝜂 J + f 𝑚𝑎𝑔 ) .This type of dynamo process is called magnetic forcing dynamo (MFD) in comparison with kinetic forcing dynamo (KFD).The induction, amplification, and propagation of 𝐵 field in plasma are restricted by the massive particle motion throughelectromotive force (EMF, ∼ U × B omitting ∫ 𝑑𝜏 ) and dissipation effect. So, dynamo in plasma is not guaranteed. Moreover,without some specific conditions, 𝐵 field coupled with the massive charged particles usually cascades toward the smaller scaleregime and finally disappears at the dissipation scale (small scale dynamo, SSD). In contrast, with some specific condition suchas helicity, shear, or instability, the field can be transferred to the larger scale (large scale dynamo, LSD, see (Krause & Rädler1980; Moffatt 1978; Brandenburg & Subramanian 2005; Balbus & Hawley 1991; Park & Blackman 2012a)). These featuresdistinguish the magnetic field in plasma from the one as an electromagnetic wave in free space.Basically, dynamo is a nonlinear phenomenon of which exact solution is not yet known. Numerical simulation is required, andtheoretical analysis is limitedly available. But, when the field is helical, the dynamo process, i.e., EMF can be linearized withpseudo scholar 𝛼 & 𝛽 and large scale magnetic field B . 𝛼 & 𝛽 are conceptually and logically inferred quantities and their exactforms are not yet known. Only, their sketchy representations can be derived with closure theory and function reiterative method,e.g., mean field theory (MFT, Moffatt (1978)), eddy damped quasi normal markovianized approximation (EDQNM, Pouquetet al. (1976)), direct interactive approximation (DIA, Yoshizawa (2011)). Physically, 𝛼 effect is thought to arise with Coriolisforce and buoyancy (kinetic helicity ‘ −(cid:104) u · ∇ × u (cid:105) ) and gradually becomes quenched by current helicity ‘ (cid:104) b · ∇ × b (cid:105) ’ generatedby the growing magnetic back reaction . The superposition of these two effects qualitatively explains how the magnetic fieldgrows and finally becomes saturated. However, for some systems like Solar(stellar) corona or a jet structure above the accretiondisk, it is hard to expect that such helical kinetic motion exists and triggers the dynamo process. Rather, the transferred helicalmagnetic field is more likely to play a key role in dynamo.In the analytic derivation of 𝛼 effect, there is no preference or constraint between −(cid:104) u · ∇ × u (cid:105) and (cid:104) b · ∇ × b (cid:105) . Also, to producecurrent helicity in a lab., J is transmitted along magnetic field. These theoretical and experimental examples show that helicalmagnetic forcing dynamo (HMFD) is not forbidden but a possible process. However, there are a couple of things to be solved inHMFD. If current helicity is a unique component in 𝛼 , magnetic field grows without stop. The growth rate ∼ 𝛼 should be largerthan the dissipation rate 𝜂 to arise magnetic field, and their initial large-small relationship does not change no matter how bigthe field grows. To prevent this catastrophic amplification, there should be some constraining effect such as kinetic helicity orsomething else. However, since helical magnetic field ∇ × B = 𝜆 B nullifies Lorentz force J × B that drives plasma motion, thegeneration of helical velocity field U by B looks contradictory. And even if the generation of U is explained, there remain trickyissues in the conservation and chirality of helicity.To explain the inconsistency, we apply the analytic method and field structure model used in HKFD (Park 2020) to themagnetically forced system (HMFD). We show numerical results for the evolving 𝐵 field and its inverse cascade to the large J , 𝜂, E , U are current density, magnetic diffusivity, electric field, and plasma velocity. The over bar in the variable 𝑋 means the large scale quantity, and the small letter 𝑥 means the quantity in the small (turbulent) scale regime. And, the anglebracket indicates its spatial average over the large scale regime: ( / 𝐿 ) ∫ 𝐿 − 𝐿 𝑋 𝑑 r . We assume (cid:104) 𝑋 (cid:105) ∼ 𝑋 MNRAS000
We study the large scale dynamo process in a system forced by helical magnetic energy. The dynamo process is basicallynonlinear, but can be linearized with pseudo scholars 𝛼 & 𝛽 and large scale magnetic field B . A coupled semi-analytic equationsbased on statistical mechanics are used to investigate the exact evolution of 𝛼 & 𝛽 . This equation set requires only magnetic helicityand magnetic energy. They are fundamental physics quantities that can be obtained from the dynamo simulation or observationwithout any artificial modification or assumption. 𝛼 effect is thought to be related to magnetic field amplification. However, inreality it converges to 𝑧𝑒𝑟𝑜 very quickly without a significant contribution to B field amplification. Conversely, 𝛽 effect for themagnetic diffusion maintains a negative value, which plays a key role in the amplification with Laplacian ∇ → − 𝑘 . In addition,negative magnetic diffusion accounts for the attenuation of plasma kinetic energy when the system is saturated. The negativemagnetic diffusion is from the interaction of advective term − U · ∇ B and the strongly helical field. When plasma velocity field U is divided into the poloidal component U 𝑝𝑜𝑙 and toroidal one U 𝑡𝑜𝑟 in the absence of reflection symmetry, they interact with B · ∇ U and − U · ∇ B to produce 𝛼 effect and (negative) 𝛽 effect, respectively. We discussed this process using the theoreticalmethod and intuitive field structure model. Key words: magnetohydrodynamics(MHD) – turbulence – plasmas–magnetic forcing–dynamo– 𝛼 & 𝛽 effect Most celestial plasma systems are constrained by magnetic field 𝐵 . However, despite the ubiquitous existence of 𝐵 field, its rolein the astrophysical system is not yet completely understood. Briefly, 𝐵 field takes energy from the turbulent plasma (dynamo),and the amplified field back reacts to the system (magnetic back reaction). Through this mutual interaction, 𝐵 field controls therate of formation of a star and accretion disk (Balbus & Hawley 1991; Machida et al. 2005). Also, the balanced pressure betweenthe magnetic field and plasma can decide the stability of the system (see sausage, kink, or Kruskal-Schwarzschild instability, seeBoyd & Sanderson (2003)).The amplification of 𝐵 field in plasma requires seed magnetic field. However, the origin of seed field (primordial magneticfield, 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 theinteraction of the electromagnetic field and the gravitational field. The breaking of the conformal symmetry is to consider theElectro-Magnetic (EM) coupling to coupling to scalar field (Martin & Yokoyama 2008; Subramanian 2016), coupling to themodified general relativity f(R) theory, coupling to pseudo scalar field and so on. The PMF strength could be generated byquantum perturbations and has been estimated as 10 − 𝑛𝐺 − 𝑛𝐺 (Yamazaki et al. 2012).The second one is based on the cosmological Quantum Chromo Dynamics (QCD) phase transition ( ∼ ∼ − nG by the quark-hadron and 10 − nG -10 − 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 ★ E-mail: [email protected] † E-mail: [email protected] © The Authors a r X i v : . [ phy s i c s . p l a s m - ph ] F e b Kiwan Park et al. vorticity, which arises from the non-zero electron and proton fluid angular velocities by the different masses of proton andelectron in the gravitational field (Harrison’s mechanism, Harrison (1970)). The PMF is thought to be about 10 − 𝑛𝐺 The second and third scenarios are thought to occur on a correlation scale smaller than the Hubble radius, by which we expecta suitable field generated by another dynamical effect, for instance, Biermann battery mechanism (Biermann 1950). When thehot ionized particles (plasma) collide mutually, the perturbed electron density ∇ 𝑛 𝑒 and pressure ∇ 𝑝 𝑒 (or temperature ∇ 𝑇 𝑒 ) can bemisaligned. This instability −∇ 𝑝 𝑒 / 𝑛 𝑒 𝑒 can drive currents to generate magnetic fields. Also, the collisionless neutrino interactionwith charged leptons at the early epoch is thought to have generated primordial magnetic helicity, the measure of twist andlinkage of magnetic fields (cid:104) A · B (cid:105) ( B = ∇ × A ) (Semikoz & Sokoloff 2005). But, since the neutrino interaction exists not onlyin the early universe epoch but are also abundant in the present Universe including the Sun, lepton-neutrino interaction is one ofthe promising candidates of (origin) magnetic field generation.As the number of charged particles increased with time, their collective motion became more important than the quantum fluc-tuation effect. Their aggregative motion formed a flow, which interacted with the seed magnetic fields leading to the amplification(dynamo) or decay of 𝐵 field according to the various conditions. The evolution of magnetic field is now explained with Faraday’slaw 𝜕 B / 𝜕𝑡 = −∇ × E combined with Ohm’s law 𝜂 J = ( E + U × B ) in the level of magnetohydrodynamics (MHD). This com-bined equation, i.e., magnetic induction equation implies that any electromagnetic instability such as Biermann’s battery effector lepton-neutrino interaction can be merged into electromotive force (EMF) in the equation: 𝜕 B / 𝜕𝑡 = ∇ × ( U × B − 𝜂 J + f 𝑚𝑎𝑔 ) .This type of dynamo process is called magnetic forcing dynamo (MFD) in comparison with kinetic forcing dynamo (KFD).The induction, amplification, and propagation of 𝐵 field in plasma are restricted by the massive particle motion throughelectromotive force (EMF, ∼ U × B omitting ∫ 𝑑𝜏 ) and dissipation effect. So, dynamo in plasma is not guaranteed. Moreover,without some specific conditions, 𝐵 field coupled with the massive charged particles usually cascades toward the smaller scaleregime and finally disappears at the dissipation scale (small scale dynamo, SSD). In contrast, with some specific condition suchas helicity, shear, or instability, the field can be transferred to the larger scale (large scale dynamo, LSD, see (Krause & Rädler1980; Moffatt 1978; Brandenburg & Subramanian 2005; Balbus & Hawley 1991; Park & Blackman 2012a)). These featuresdistinguish the magnetic field in plasma from the one as an electromagnetic wave in free space.Basically, dynamo is a nonlinear phenomenon of which exact solution is not yet known. Numerical simulation is required, andtheoretical analysis is limitedly available. But, when the field is helical, the dynamo process, i.e., EMF can be linearized withpseudo scholar 𝛼 & 𝛽 and large scale magnetic field B . 𝛼 & 𝛽 are conceptually and logically inferred quantities and their exactforms are not yet known. Only, their sketchy representations can be derived with closure theory and function reiterative method,e.g., mean field theory (MFT, Moffatt (1978)), eddy damped quasi normal markovianized approximation (EDQNM, Pouquetet al. (1976)), direct interactive approximation (DIA, Yoshizawa (2011)). Physically, 𝛼 effect is thought to arise with Coriolisforce and buoyancy (kinetic helicity ‘ −(cid:104) u · ∇ × u (cid:105) ) and gradually becomes quenched by current helicity ‘ (cid:104) b · ∇ × b (cid:105) ’ generatedby the growing magnetic back reaction . The superposition of these two effects qualitatively explains how the magnetic fieldgrows and finally becomes saturated. However, for some systems like Solar(stellar) corona or a jet structure above the accretiondisk, it is hard to expect that such helical kinetic motion exists and triggers the dynamo process. Rather, the transferred helicalmagnetic field is more likely to play a key role in dynamo.In the analytic derivation of 𝛼 effect, there is no preference or constraint between −(cid:104) u · ∇ × u (cid:105) and (cid:104) b · ∇ × b (cid:105) . Also, to producecurrent helicity in a lab., J is transmitted along magnetic field. These theoretical and experimental examples show that helicalmagnetic forcing dynamo (HMFD) is not forbidden but a possible process. However, there are a couple of things to be solved inHMFD. If current helicity is a unique component in 𝛼 , magnetic field grows without stop. The growth rate ∼ 𝛼 should be largerthan the dissipation rate 𝜂 to arise magnetic field, and their initial large-small relationship does not change no matter how bigthe field grows. To prevent this catastrophic amplification, there should be some constraining effect such as kinetic helicity orsomething else. However, since helical magnetic field ∇ × B = 𝜆 B nullifies Lorentz force J × B that drives plasma motion, thegeneration of helical velocity field U by B looks contradictory. And even if the generation of U is explained, there remain trickyissues in the conservation and chirality of helicity.To explain the inconsistency, we apply the analytic method and field structure model used in HKFD (Park 2020) to themagnetically forced system (HMFD). We show numerical results for the evolving 𝐵 field and its inverse cascade to the large J , 𝜂, E , U are current density, magnetic diffusivity, electric field, and plasma velocity. The over bar in the variable 𝑋 means the large scale quantity, and the small letter 𝑥 means the quantity in the small (turbulent) scale regime. And, the anglebracket indicates its spatial average over the large scale regime: ( / 𝐿 ) ∫ 𝐿 − 𝐿 𝑋 𝑑 r . We assume (cid:104) 𝑋 (cid:105) ∼ 𝑋 MNRAS000 , 1–14 () egative Magnetic Diffusivity 𝛽 effect in a Magnetically Dominant System (a) (b)(c) (d) Figure 1.
Plot (a) & (c) show the logarithmic evolution of energy and helicity of the system forced with right handed helical magnetic energy ( 𝑓 ℎ = + 𝑓 ℎ = − 𝑘 = 𝑘 = (cid:104) 𝑏 (cid:105) , and the red dotted one means its helical contribution 𝑘 (cid:104) a · b (cid:105) . In forcing scale (k=5), magnetic energy and its helical part are practically the same so that the corresponding lines are overlapped. On the other hand,the black solid line indicates kinetic energy (cid:104) 𝑢 (cid:105) , and the black dotted line indicates its helical part (cid:104) u · ∇ × u (cid:105)/ 𝑘 . In (a) & (c) kinetic and magnetic helicityclearly show up, but those in (b) & (d) are not shown except some part of large scale kinetic helicity. This indicates that the polarization of helicity in HMFD,except the large scale velocity, is consistently decided by forcing magnetic field. scale regime. And then, we show the evolving profile of 𝛼 & 𝛽 along with the growth of 𝐵 field and investigate their physicalfeatures and mutual relations. We discuss the parameterization of EMF: (cid:104) u × b (cid:105) ∼ ∫ 𝑑𝜏 ( 𝛼 B − 𝛽 ∇ × B ) and compare themusing numerical data and analytic approach. Using field structure model, we explain the intuitive meaning of 𝛼 effect and how 𝛽 becomes negative. Then, to supplement them, we derive 𝛽 coefficient again when the field is helical. 𝛽 effect also explains howthe plasma kinetic energy is suppressed when the system is forced by helical magnetic energy. This work focuses on the physicalmechanism of helical forcing dynamo which occurs in the fundamental level of astro-plasma system. MNRAS , 1–14 ()
Kiwan Park et al. (a) (b)
Figure 2.
The evolution of helicity ratio: 𝑘 (cid:104) A · B (cid:105)/(cid:104) 𝐵 (cid:105) for magnetic energy and helicity, (cid:104) U · ∇ × U (cid:105)/ 𝑘 (cid:104) 𝑈 (cid:105) for kinetic energy and helicity (k=1, 5, 8). (a) 𝑓 ℎ of large scale magnetic field (red thick line) converges to 1. (b) 𝑓 ℎ of large scale magnetic field converges to -1. The basic MHD equations are composed of continuity, momentum, and magnetic induction equation as follows: 𝜕 𝜌𝜕𝑡 = − U · ∇ 𝜌 − 𝜌 ∇ · U , (1) 𝜕 U 𝜕𝑡 = − U · ∇ U − ∇ ln 𝜌 + 𝜌 J × B + 𝜈 (cid:0) ∇ U + ∇∇ · U (cid:1) + f 𝑘𝑖𝑛 , (2) 𝜕 A 𝜕𝑡 = (cid:104) U × B (cid:105) − 𝜂 ∇ × B + f 𝑚𝑎𝑔 . (3) (cid:18) ⇒ 𝜕 B 𝜕𝑡 = ∇ × (cid:104) U × B (cid:105) + 𝜂 ∇ B + ∇ × f 𝑚𝑎𝑔 (cid:19) , (4)Here, the symbols 𝜌 , 𝜈 , and 𝜂 indicate density, kinematic viscosity, and magnetic diffusivity. ‘ U ’ is in the units of sound speed 𝑐 𝑠 , and ‘ B ’ is normalized by ( 𝜌 𝜇 ) / 𝑐 𝑠 ( 𝜇 : magnetic permeability in vacuum.) The fields U , B can be separated into thelarge scale fields U , B and turbulent small scale ones u , b . Analytically, the evolution of B can be represented as follows: 𝜕 B 𝜕𝑡 ∼ ∇ × (cid:104) u × b (cid:105) + 𝜂 ∇ B , (5) ∼ ∇ × ( 𝛼 B ) + ( 𝛽 + 𝜂 )∇ B (6)When these equations are numerically solved with Pencil-code. (Brandenburg 2001)(see the manual http://pencil-code.nordita.org),the system is driven by a forcing source like: f ( 𝑘, 𝑡 ) = 𝑖 k ( 𝑡 ) × ( k ( 𝑡 ) × ˆe ) − 𝜆 | k ( 𝑡 )|( k ( 𝑡 ) × ˆe ) 𝑘 ( 𝑡 ) √ + 𝜆 √︁ − ( k ( 𝑡 ) · e ) / 𝑘 ( 𝑡 ) . (7)‘ 𝑘 ’ is a wavenumber, ‘ ˆe ’ is an arbitrary unit vector, ‘ 𝜙 ( 𝑡 ) ’ is a random phase ( | 𝜙 ( 𝑡 )| ≤ 𝜋 ). This forcing function can be locatedat Eq.(2) (KFD), or Eq.(3) (MFD). The helical forcing or nonhelical one is decided by ‘ 𝜆 ’. If 𝜆 is +(−)
1, the forcing energy isfully right (left) handed helical 𝑖 k × f = ± 𝑘 f . If 𝜆 is 0, f is fully nonhelical.Usually, the forcing function in nature is different from Gaussian random type (white noise) like Eq.(7). For example,Biermann’s effect is represented as f 𝑚𝑎𝑔 = ∇ 𝑝 𝑒 /( 𝑛 𝑒 𝑒 )(→ ∇ 𝑛 𝑒 × ∇ 𝑝 𝑒 /( 𝑛 𝑒 𝑒 )) , which is a typical example of nonhelical magneticforcing dynamo (NHMFD). Also, lepton-neutrino interaction produces the electromagnetic instability like below: f 𝑚𝑎𝑔 = − 𝐺 𝐹 √ | 𝑒 | 𝑛 𝑒 ∑︁ 𝜈 𝑎 𝑐 𝑎𝐴 (cid:20) ( 𝑛 − + 𝑛 + ) ˆ b 𝜕𝛿𝑛 𝜈 𝑎 𝜕𝑡 + ( 𝑁 − + 𝑁 + )∇( ˆ b · 𝛿 j 𝜈 𝑎 ) (cid:21) . (8) MNRAS000
1, the forcing energy isfully right (left) handed helical 𝑖 k × f = ± 𝑘 f . If 𝜆 is 0, f is fully nonhelical.Usually, the forcing function in nature is different from Gaussian random type (white noise) like Eq.(7). For example,Biermann’s effect is represented as f 𝑚𝑎𝑔 = ∇ 𝑝 𝑒 /( 𝑛 𝑒 𝑒 )(→ ∇ 𝑛 𝑒 × ∇ 𝑝 𝑒 /( 𝑛 𝑒 𝑒 )) , which is a typical example of nonhelical magneticforcing dynamo (NHMFD). Also, lepton-neutrino interaction produces the electromagnetic instability like below: f 𝑚𝑎𝑔 = − 𝐺 𝐹 √ | 𝑒 | 𝑛 𝑒 ∑︁ 𝜈 𝑎 𝑐 𝑎𝐴 (cid:20) ( 𝑛 − + 𝑛 + ) ˆ b 𝜕𝛿𝑛 𝜈 𝑎 𝜕𝑡 + ( 𝑁 − + 𝑁 + )∇( ˆ b · 𝛿 j 𝜈 𝑎 ) (cid:21) . (8) MNRAS000 , 1–14 () egative Magnetic Diffusivity 𝛽 effect in a Magnetically Dominant System Its axial vector term is represented as f 𝑚𝑎𝑔 = 𝛼 (cid:48) B , where 𝛼 (cid:48) isv(Semikoz 2004; Semikoz & Sokoloff 2005) : 𝛼 (cid:48) ∼ 𝑙𝑛 √ 𝜋 (cid:18) − 𝑇𝑚 𝑝 𝜆 𝜈𝑓 𝑙𝑢𝑖𝑑 (cid:19) 𝛿𝑛 𝜈 𝑛 𝜈 . (9)Although this function is not the same as Gaussian type, 𝛼 (cid:48) is the result of ˆ b · 𝛿 j 𝜈 𝑎 that produces HMFD.We prepared for the two systems with a unit magnetic Prandtl number 𝑃𝑟 𝑀 = 𝜂 / 𝜈 = 𝜂 = 𝜈 = × − , and numericalresolution is 400 . They were forced by Eq.(7) with fully helical magnetic energy ( 𝜆 = + − 𝑘 = (cid:104) 𝑈 (cid:105) , magnetic energy (cid:104) 𝐵 (cid:105) , kinetic helicity (cid:104) U · ∇ × U (cid:105) , magnetic helicity (cid:104) A · B (cid:105) . The stability ofcode and data have been verified. The system in Fig.1(a), 1(c) is forced by the fully positive (right-handed) helical magnetic energy (red dashed line, helicity ratioof forcing energy: 𝑓 ℎ ≡ 𝑘 𝑓 (cid:104) a · b (cid:105)/(cid:104) 𝑏 (cid:105) = , 𝑘 𝑓 = 𝑓 ℎ = − 𝑈 is ∼ × − , and magneticReynolds number is defined as 𝑅𝑒 𝑀 ≡ 𝑈 𝐿 / 𝜂 ∼ 𝜋 /
3, where 𝐿 , 𝜂 are 2 𝜋 and 6 × − , respectively. In HMFD, the least amountof magnetic energy is transferred to plasma.In Fig.1(a), large scale magnetic energy (cid:104) 𝐵 (cid:105) ( 𝑘 =
1, solid line) grows to be saturated at 𝑡 ∼ (cid:104) 𝐵 (cid:105) , the largescale magnetic helicity (cid:104) A · B (cid:105) (dashed line) evolves keeping the relation of (cid:104) 𝐵 (cid:105) ≥ 𝑘 (cid:104) A · B (cid:105) ( 𝑘 = (cid:104) 𝑈 (cid:105) inthe large scale grows keeping (cid:104) 𝑈 (cid:105) ≥ (cid:104) U · ∇ × U (cid:105)/ 𝑘 . But the direction of kinetic helicity fluctuates from positive to negative asthe discontinuous cusp line implies in this log-scaled plot. Similarly, Fig.1(c) shows that evolving small scale magnetic energy (cid:104) 𝑏 (cid:105) and kinetic energy (cid:104) 𝑢 (cid:105) with their helical part (cid:104) u · ∇ × u (cid:105)/ 𝑘 and 𝑘 (cid:104) a · b (cid:105) ( 𝑘 ≥ 𝑓 ℎ ≡ 𝑘 (cid:104) a · b (cid:105)/(cid:104) 𝑏 (cid:105) and kinetic helicity ratio (cid:104) u · ∇ × u (cid:105)/ 𝑘 (cid:104) 𝑢 (cid:105) for 𝑘 =1, 5, 8.Left and right panel are for the right handed case ( 𝑓 ℎ =
1) and left handed one ( 𝑓 ℎ = − 𝐵 is eventually saturated at 𝑓 ℎ = + − − 𝑈 is as low as ∼ .
25 ( − . J × B ∼ 𝛼 & 𝛽 effect and the large scale magnetic energy 2 𝐸 𝑀 . Left (right) panel shows theevolution of 𝐸 𝑀 , 𝛼 and 𝛽 effect for 𝑓 ℎ = 𝑓 ℎ = − 𝛼 effect for 𝑓 ℎ = 𝑧𝑒𝑟𝑜 as 𝐸 𝑀 gets saturated. In contrast, 𝛼 effect for 𝑓 ℎ = − 𝛼 effect is quenched much earlierthan the slowly evolving 𝐸 𝑀 . The decreasing oscillation in both cases implies that 𝛼 effect does not play a decisive role in thegrowth of the large scale magnetic field. Conversely, 𝛽 retains the negative value in both cases and has a much larger size thanthat of 𝛼 . This negative 𝛽 , combined with the negative Laplacian ∇ → − 𝑘 in Fourier space, can be considered as the actualsource of the large scale magnetic field. This is contradictory to the current dynamo theory concluding that 𝛽 is always positiveto diffuse magnetic energy. We will show that this conventional inference is valid only for the ideally isotropic system with re-flection symmetry. When the symmetry is broken, ‘ u ’ in the small scale regime can yield the anti-diffusing effect of magnetic field.In Fig.4, we compared ∇ × 𝐸 𝑀 𝐹 (black solid line) with ∇ × ( 𝛼 B − ∇ × B ) (red dashed line) using Eq.(6) and Eq.(25), (26).They are quite close to each other except the saturation regime. As the field becomes saturated, 𝐸 𝑀 and 𝐻 𝑀 are so close that Fermi constant 𝐺 𝐹 = − / 𝑚 𝑝 ( 𝑚 𝑝 : proton mass); 𝑐 𝑎𝐴 = ∓ . 𝑎 : electron, muon, tau; (−) : electron, (+) ’: muon or tau); 𝛿𝑛 𝜈 𝑎 :neutrino density asymmetry; 𝛿 j 𝜈 𝑎 (neutrino current asymmetry); 𝑛 ± ∼ ( | 𝑒 | 𝐵 / 𝜋 ) 𝑇 𝑙𝑛 𝜆 𝜈𝑓 𝑙𝑢𝑖𝑑 ∼ 𝑡 is a scaleof neutrino fluid inhomogeneity. MNRAS , 1–14 () Kiwan Park et al. (a) (b)
Figure 3. 𝛼 ( 𝑡 ) , 𝛽 ( 𝑡 ) , and (cid:104) 𝐵 (cid:105) for 𝑓 ℎ = −
1. The small and early quenching 𝛼 effect shows its limited effect on the growth of the large scale magneticenergy 𝐸 𝑀 . the logarithmic function diverges. However, the coincidence in the transient regime shows how reliable the equations are. 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 discussthe mechanism in detail soon.Fig.7 is for the typical kinetic small scale dynamo. Nonhelical random kinetic energy was given to 𝑘 =
5. The plot includeslarge scale kinetic energy 𝐸 𝑉 and magnetic energy 𝐸 𝑀 . 𝑅𝑒 𝑀 is approximately 80. In comparison with LSD, 𝐸 𝑀 grows a littlebit, and 𝐸 𝑉 is not quenched. Most magnetic energy is transferred to the small scale regime, and its peak is located at 𝑘 ∼ . 𝑠 → .
53 years), and the vertical line indicates the ‘latitude’ (0 − 𝜋 /
2: northern hemisphere, 𝜋 / − 𝜋 :southern hemisphere). The color indicates the phase of a net magnetic field (toroidal B 𝑡𝑜𝑟 + poloidal B 𝑝𝑜𝑙 ). The simulationin the left panel is the reproduction of Jouve et al. (2008) without Babcock effect. It shows the period of 16.31 years for theone complete cycle of solar magnetic field: amplification-annihilation-reverse. On the contrary, in right panel, the tidal effect ofplanets on the Solar tachocline is added to 𝛼 . The period elevates up to 21.74 years without manipulating the numerical variables.These plots show the practical applicability of Eq.(4). We have shown that magnetic energy can be inversely cascaded in the system forced by the helical magnetic energy ( ∇ × B = 𝜆 B ).HMFD has apparently contradictory features. Helical magnetic energy, which makes Lorentz force 𝑧𝑒𝑟𝑜 , exerts a force on thesystem leading to the generation of helical kinetic energy with the same chirality. Moreover, the conservation of magnetichelicity is not valid anymore. We study the internal interaction of U & B using field structure model and analytic methodbeyond conventional theory and phenomenological rope dynamo model. Although MHD has some hydrodynamic features, thegeneration and transport of 𝐵 field are innately electromagnetic phenomena constrained by the plasma motion. Note that plasmakinetic energy is converted into magnetic energy only through EMF U × B ∼ 𝜂 J ∼ E , which is different from mechanical force. 𝛼 effect The right handed magnetic structure in Fig.5(a) is composed of the toroidal magnetic component b 𝑡𝑜𝑟 and poloidal part b 𝑝𝑜𝑙 .Statistically, b 𝑡𝑜𝑟 and b 𝑝𝑜𝑙 are not distinguished in the homogeneous and isotropic system. But, if we remove reflection symmetry MNRAS000
2: northern hemisphere, 𝜋 / − 𝜋 :southern hemisphere). The color indicates the phase of a net magnetic field (toroidal B 𝑡𝑜𝑟 + poloidal B 𝑝𝑜𝑙 ). The simulationin the left panel is the reproduction of Jouve et al. (2008) without Babcock effect. It shows the period of 16.31 years for theone complete cycle of solar magnetic field: amplification-annihilation-reverse. On the contrary, in right panel, the tidal effect ofplanets on the Solar tachocline is added to 𝛼 . The period elevates up to 21.74 years without manipulating the numerical variables.These plots show the practical applicability of Eq.(4). We have shown that magnetic energy can be inversely cascaded in the system forced by the helical magnetic energy ( ∇ × B = 𝜆 B ).HMFD has apparently contradictory features. Helical magnetic energy, which makes Lorentz force 𝑧𝑒𝑟𝑜 , exerts a force on thesystem leading to the generation of helical kinetic energy with the same chirality. Moreover, the conservation of magnetichelicity is not valid anymore. We study the internal interaction of U & B using field structure model and analytic methodbeyond conventional theory and phenomenological rope dynamo model. Although MHD has some hydrodynamic features, thegeneration and transport of 𝐵 field are innately electromagnetic phenomena constrained by the plasma motion. Note that plasmakinetic energy is converted into magnetic energy only through EMF U × B ∼ 𝜂 J ∼ E , which is different from mechanical force. 𝛼 effect The right handed magnetic structure in Fig.5(a) is composed of the toroidal magnetic component b 𝑡𝑜𝑟 and poloidal part b 𝑝𝑜𝑙 .Statistically, b 𝑡𝑜𝑟 and b 𝑝𝑜𝑙 are not distinguished in the homogeneous and isotropic system. But, if we remove reflection symmetry MNRAS000 , 1–14 () egative Magnetic Diffusivity 𝛽 effect in a Magnetically Dominant System (a) (b) Figure 4.
Comparison of EMF and 𝛼 & 𝛽 approximation. Instead of conventional definitions, Eq. (25), (26) were used. Note that the average is taken for thelarge scale regime 𝑘 =
1. (a) Helical Magnetic forcing (b) Induced helical kinetic forcing
Figure 5.
The externally provided helical magnetic energy (a) induce kinetic helicity (b) with the same handedness. from this system, b 𝑡𝑜𝑟 and b 𝑝𝑜𝑙 become independent components playing different roles with u .The interaction between u and b 𝑡𝑜𝑟 yields current density, i.e., u × b 𝑡𝑜𝑟 ∼ j , 𝑑𝑜𝑤𝑛 and j , 𝑢 𝑝 in the front and back. Thesetwo components induce a new magnetic field b 𝑖𝑛𝑑 . At the same time, u × b 𝑝𝑜𝑙 generates j . This current density forms the righthanded magnetic helicity with b 𝑖𝑛𝑑 : (cid:104) j · b 𝑖𝑛𝑑 (cid:105) → 𝑘 (cid:104) a · b 𝑖𝑛𝑑 (cid:105) , which is a (pseudo) scalar to be added to the system. There isalso a possibility that u and b 𝑖𝑛𝑑 induce j yielding the left handed magnetic helicity (cid:104) j · b 𝑝𝑜𝑙 (cid:105) . However, the induced magneticfield from j is weakened by the externally provided b 𝑡𝑜𝑟 .On the other hand, j × b 𝑡𝑜𝑟 and j × b 𝑝𝑜𝑙 generate Lorentz force toward − u , which may look just suppressing plasma motion.However, j × b 𝑡𝑜𝑟 at the right and left end yields an rotation effect, which is toward ‘ − u ’. This rotation with those two interactionsgenerates the right handed kinetic helicity (cid:104) u · ∇ × u (cid:105) . The interaction between the current density and magnetic field makestwo effects. Magnetic pressure effect −∇ 𝐵 / j × b 𝑡𝑜𝑟 and j × b 𝑝𝑜𝑙 suppresses the plasma motion with thermal pressure −∇ 𝑃 . And, j × b 𝑡𝑜𝑟 creates a rotational force to form kinetic helicity with the two suppressing effects. It should be noted that j is not directly induced from b . As Fourier transformed Lorentz force j ( p ) × b ( q ) ∼ 𝜕 u ( k )/ 𝜕𝑡 shows, the wavenumbers ‘ p ’ and‘ q ’ are constrained by the relation of p + q = k . MNRAS , 1–14 ()
Kiwan Park et al.
Figure 6.
More detailed field structure based on
𝐸 𝑀 𝐹 : ∇ × ( U × B ) = − U · ∇ B + B · ∇ U . These structures correspond to Fig.5(b). The left field structurein Fig.6 is for the early time regime while 𝐵 < 𝑏 . Right structure is for the magnetic back reaction with 𝐵 (cid:38) 𝑏 at later time regime, which is negligible in themagnetically forced system. The symbol ‘ (cid:203) ’ means the direction ( − ˆ 𝑧 ) of induced current density J ∼ U × B , and its size indicates the relative strength. Figure 7.
Nonhelical kinetic forcing small scale dynamo. Nonhelical kinetic energy was given to k=5. 𝜂 = 𝜈 = . 𝑅𝑒 𝑀 is ∼
80. Left panel shows the temporal evolution of 𝐸 𝑉 and 𝐸 𝑀 . Rght panel shows their spectra at 𝑡 = 𝐸 𝑀 is located between theforcing scale and dissipation scale. The induced right handed kinetic helicity in Fig.5(b) again generates j (cid:48) in the front and back. j (cid:48) , 𝑓 𝑟𝑜𝑛𝑡 and j (cid:48) , 𝑏𝑎𝑐𝑘 induce b (cid:48) 𝑖𝑛𝑑 , which generates j (cid:48) with u 𝑝𝑜𝑙 . Then, j (cid:48) forms the left handed magnetic helicity with b (or B ). Also, u 𝑝𝑜𝑙 × b yields j (cid:48) leading to right handed magnetic helicity. If all interactions are summed up, magnetic helicity in the system is +|(cid:104) j · b 𝑖𝑛𝑑 (cid:105)|−|(cid:104) j · b 𝑝𝑜𝑙 (cid:105)|−|(cid:104) j (cid:48) · b (cid:105)|+|(cid:104) j (cid:48) · b (cid:48) 𝑖𝑛𝑑 (cid:105)| qualitatively. Comparing this result with 𝛼 quenching in Fig.3(a), we mayquestion what indeed amplifies magnetic field and determines the net magnetic helicity. There is one more term to be considered. 𝛽 effect Fig.6 is the more detailed right handed helical kinetic structure of Fig.5(b). It is based on the geometrical meaning of‘ ∇ × ( u × B ) ∼ B · ∇ u − u · ∇ B ’. Here, we name ‘ − u · ∇ B ’ as ‘local transfer (advective) term’, and we call ‘ B · ∇ u ’ ‘nonlocal transferterm’. This nonlocal transfer term actually corresponds to b (cid:48) 𝑖𝑛𝑑 in Fig.5(b), but local transfer term − u · ∇ B is new. The symbol‘ ⊗ ’ means the direction of current density ‘ J ’ heading for − ˆ 𝑧 . Its distribution is spatially inhomogeneous so that the nontrivialcurl effect generates the magnetic fields toward ˆ 𝑥 (locally transferred) and ˆ 𝑦 (nonlocally transferred). Their net magnetic field b 𝑛𝑒𝑡 becomes a new seed field for the next dynamo process. As the net magnetic field grows, it approaches to the velocity field‘ u ’ so that ‘ u × b 𝑛𝑒𝑡 ’ itself decreases. The field gets saturated eventually if there is not any other reason e.g., frozen field or helicity. u 𝑝𝑜𝑙 interacts with ∫ 𝑑𝜏 B · ∇ u to induce j (cid:48) , which yields the left handed helicity with B 𝑝𝑜𝑙 . u 𝑝𝑜𝑙 also interacts with B 𝑝𝑜𝑙 MNRAS000
80. Left panel shows the temporal evolution of 𝐸 𝑉 and 𝐸 𝑀 . Rght panel shows their spectra at 𝑡 = 𝐸 𝑀 is located between theforcing scale and dissipation scale. The induced right handed kinetic helicity in Fig.5(b) again generates j (cid:48) in the front and back. j (cid:48) , 𝑓 𝑟𝑜𝑛𝑡 and j (cid:48) , 𝑏𝑎𝑐𝑘 induce b (cid:48) 𝑖𝑛𝑑 , which generates j (cid:48) with u 𝑝𝑜𝑙 . Then, j (cid:48) forms the left handed magnetic helicity with b (or B ). Also, u 𝑝𝑜𝑙 × b yields j (cid:48) leading to right handed magnetic helicity. If all interactions are summed up, magnetic helicity in the system is +|(cid:104) j · b 𝑖𝑛𝑑 (cid:105)|−|(cid:104) j · b 𝑝𝑜𝑙 (cid:105)|−|(cid:104) j (cid:48) · b (cid:105)|+|(cid:104) j (cid:48) · b (cid:48) 𝑖𝑛𝑑 (cid:105)| qualitatively. Comparing this result with 𝛼 quenching in Fig.3(a), we mayquestion what indeed amplifies magnetic field and determines the net magnetic helicity. There is one more term to be considered. 𝛽 effect Fig.6 is the more detailed right handed helical kinetic structure of Fig.5(b). It is based on the geometrical meaning of‘ ∇ × ( u × B ) ∼ B · ∇ u − u · ∇ B ’. Here, we name ‘ − u · ∇ B ’ as ‘local transfer (advective) term’, and we call ‘ B · ∇ u ’ ‘nonlocal transferterm’. This nonlocal transfer term actually corresponds to b (cid:48) 𝑖𝑛𝑑 in Fig.5(b), but local transfer term − u · ∇ B is new. The symbol‘ ⊗ ’ means the direction of current density ‘ J ’ heading for − ˆ 𝑧 . Its distribution is spatially inhomogeneous so that the nontrivialcurl effect generates the magnetic fields toward ˆ 𝑥 (locally transferred) and ˆ 𝑦 (nonlocally transferred). Their net magnetic field b 𝑛𝑒𝑡 becomes a new seed field for the next dynamo process. As the net magnetic field grows, it approaches to the velocity field‘ u ’ so that ‘ u × b 𝑛𝑒𝑡 ’ itself decreases. The field gets saturated eventually if there is not any other reason e.g., frozen field or helicity. u 𝑝𝑜𝑙 interacts with ∫ 𝑑𝜏 B · ∇ u to induce j (cid:48) , which yields the left handed helicity with B 𝑝𝑜𝑙 . u 𝑝𝑜𝑙 also interacts with B 𝑝𝑜𝑙 MNRAS000 , 1–14 () egative Magnetic Diffusivity 𝛽 effect in a Magnetically Dominant System (or b ) and ∫ 𝑑𝜏 (− u · ∇ B ) yielding j (cid:48) and j (cid:48) , respectively. The polarization of (cid:104) j (cid:48) · ∫ 𝑑𝜏 B · ∇ u (cid:105) is always opposite (+) to thatof (cid:104) j (cid:48) · B 𝑝𝑜𝑙 (cid:105) (−) , but that of (cid:104) j (cid:48) · ∫ 𝑑𝜏 B · ∇ u (cid:105) depends on the relative value of − u · ∇ B . When this locally transferred field isweak, j (cid:48) is parallel to ∫ 𝑑𝜏 ( B · ∇ u ) producing the oppositely polarized ( + ) magnetic helicity with reference to (cid:104) j (cid:48) · B 𝑝𝑜𝑙 (cid:105) (−) .However, as the strength of B grows to surpass ‘ | b | ’, ‘ − u · ∇ B ’ turns over so that the direction of j (cid:48) is opposite to ∫ B · ∇ u 𝑑𝜏 yielding the left handed (−) magnetic helicity. This is the result of magnetic back reaction. However, this effect is negligible inHMFD where the helical magnetic energy is continuously provided by the external source. Therefore, net magnetic helicity is‘ +|(cid:104) j · b 𝑖𝑛𝑑 (cid:105)| − |(cid:104) j · b 𝑝𝑜𝑙 (cid:105)| − |(cid:104) j (cid:48) · B 𝑝𝑜𝑙 ( b )(cid:105)| + |(cid:104) j (cid:48) · ∫ 𝑑𝜏 B · ∇ u (cid:105)| ± |(cid:104) j (cid:48) · ∫ 𝑑𝜏 B · ∇ u (cid:105)| ’. The last term representing 𝛽 effectsubstantially amplifies the large scale magnetic field when 𝛼 effect becomes negligible (Fig.3(a), 3(b)). 𝛼 & 𝛽 In the helical dynamo, the basis of Eq.(6) from Eq.(5) is the replacement of small scale EMF (cid:104) u × b (cid:105) with 𝛼 B − 𝛽 ∇ × B . Thisrelation can be approximately derived using sort of a function iterative method with some appropriate closure theories such asMFT(Blackman & Field 2002), EDQNMPouquet et al. (1976), DIA(Yoshizawa 2011). All theories show qualitatively the sameresults; but, they have their own limitations, too. For example, for MFT, the variable 𝑋 is divided into the mean (large) scalequantity X and small (turbulent) one x . Then, they are taken average over the large scale (cid:104)·(cid:105) , and calculated with Reynolds ruleand tensor identity. 𝛼 = ∫ 𝑡 (cid:0) (cid:104) j · b (cid:105) − (cid:104) u · ∇ × u (cid:105) (cid:1) 𝑑𝜏, (10) 𝛽 = ∫ 𝑡 (cid:104) 𝑢 (cid:105) 𝑑𝜏. (11)During the analytic calculation, some turbulent variables with the triple correlation or higher order terms are derived. They aredropped with Reynolds rule or simply ignored with the assumption of being small. This may cause the increasing discrepancybetween the real system and MFT as 𝑅𝑒 𝑀 grows. Also, the eddy turnover time ‘ 𝑡 ’ appears with integration, but at present, thereis no good method to calculate it except some simple dimensional analysis or experimental approach.Another issue is the existence of large scale plasma motion U . If U × B is averaged over large scale and applied with Reynoldsrule, two terms remain: 𝜉 ∼ U × B + (cid:104) u × b (cid:105) . In principle, they should be replaced by 𝛼 B − 𝛽 ∇ × B . But, U × B is usuallyexcluded with Galilean transformation. However, U in simulation and observation does not disappear, rather its effect can growwith the increasing B . Eq.(10), (11) are actually over simplified results.In DIA, those issues are included in formal Green function 𝐺 with statistical second order relation. (cid:104) 𝑋 𝑖 ( 𝑘 ) 𝑋 𝑗 (− 𝑘 )(cid:105) = ( 𝛿 𝑖 𝑗 − 𝑘 𝑖 𝑘 𝑗 𝑘 ) 𝐸 𝑋 ( 𝑘 ) + 𝑖 𝑘 𝑙 𝑘 𝜖 𝑖 𝑗𝑙 𝐻 𝑋 ( 𝑘 ) (12) ((cid:104) 𝑋 (cid:105) = ∫ 𝐸 𝑋 ( 𝑘 ) 𝑑𝑘, (cid:104) X · ∇ × X (cid:105) = ∫ 𝐻 𝑋 ( 𝑘 ) 𝑑𝑘 ) And, 𝛼 & 𝛽 in DIA are 𝛼 = ∫ 𝑑 k ∫ 𝑡 ( 𝐺 𝑀 (cid:104) j · b (cid:105) − 𝐺 𝐾 (cid:104) u · ∇ × u (cid:105)) 𝑑𝜏, (13) 𝛽 = ∫ 𝑡 ( 𝐺 𝐾 (cid:104) 𝑢 (cid:105) + 𝐺 𝑀 (cid:104) 𝑏 (cid:105)) 𝑑𝜏. (14)They are quite similar to those of MFT except 𝐺 function and turbulent magnetic energy (cid:104) 𝑏 (cid:105) in 𝛽 . 𝛼 coefficient implies itsquenching as 𝐺 𝑀 (cid:104) j · b (cid:105) → 𝐺 𝐾 (cid:104) u · ∇ × u (cid:105) . Also, the 𝛽 effect depends on the turbulent energy including 𝑏 . Since DIA calculateskinetic approach and counter kinetic (magnetic) one separately, Eq.(12) yields (cid:104) 𝑢 (cid:105) and (cid:104) 𝑏 (cid:105) in 𝛽 ( x = u , b ). 𝛼 & 𝛽 calculated with EDQNM approximation show more or less similar physical properties such as quenching 𝛼 and energydependent positive 𝛽 (Pouquet et al. 1976). 𝛼 = ∫ ∞ Θ 𝑘 𝑝𝑞 ( 𝑡 ) (cid:0) (cid:104) j · b (cid:105) − (cid:104) u · ∇ × u (cid:105) (cid:1) 𝑑𝑞, (15) 𝛽 = ∫ ∞ Θ 𝑘 𝑝𝑞 ( 𝑡 )(cid:104) 𝑢 (cid:105) 𝑑𝑞, (16)where triad relaxation time Θ 𝑘 𝑝𝑞 = ( − 𝑒𝑥 𝑝 (− 𝜇 𝑘 𝑝𝑞 𝑡 ))/ 𝜇 𝑘 𝑝𝑞 and eddy damping operator 𝜇 𝑘 𝑝𝑞 are used. Formally, DIA orEDQNM is free from the nonlinear effects neglected in MFT. However, still there are unknown Green function, triad relaxation MNRAS , 1–14 () Kiwan Park et al. time, and eddy damping rate including eddy turnover time.Furthermore, the small scale EMF (cid:104) u × b (cid:105) used in two scale MFT and DIA is not well defined quantity. It is inferred from X − X which is supposed to be in the range of 𝑘 ≥ 𝛼 & 𝛽 with the conventional MFT shows that u & b (or 𝛼 & 𝛽 ) existonly around the forcing scale. The whole turbulent range yields much larger growth of B than actual value (Fig.1, Park &Blackman (2012b), Fig.1b, Park (2017)). Kolmogorov’s inertia range seems to separate the range of u & b for 𝛼 & 𝛽 from otherdissipation scale. However, it is not clear whether the latter just dissipates or plays some other roles in dynamo. At least, theydo not amplify B directly. But, since the exact range cannot be found with theory, we may question if 𝛼 & 𝛽 (or u & b ) are justconceptual quantities. Statistically, however, it makes sense to substitute 𝛼 & 𝛽 and B for EMF. And, the result is associated withthe statistical correlation Eq.(12). If we apply Moffatt (1978)’s assumption 𝐸 𝑀 𝐹 𝑖 ∼ 𝛼 𝑖 𝑗 B 𝑗 + 𝛽 𝑖 𝑗𝑘 ∇ 𝑘 B 𝑗 to magnetic inductionequation, we get 𝜕 B 𝜕𝑡 ∼ ∇ × ( 𝛼 B − ( 𝛽 + 𝜂 )∇ × B ) + ∇ · B . (17)This shows that the growth rate of vector B is represented by its curl and divergence with appropriate coefficients, i.e., Helmholtztheory.This formal equation can be applied to the practical dynamo phenomenon such as Solar dynamo. If the equation is divided intothe poloidal component and toroidal one, two coupled equations from magnetic induction equation are derived (Charbonneau2014): 𝜕 𝐴𝜕𝑡 = ( 𝜂 + 𝛽 ) (cid:18) ∇ − 𝜛 (cid:19) 𝐴 − u 𝑝 𝜛 · ∇( 𝜛 𝐴 ) + 𝛼𝐵 𝑡𝑜𝑟 , (18) 𝜕𝐵 𝑡𝑜𝑟 𝜕𝑡 = ( 𝜂 + 𝛽 ) (cid:18) ∇ − 𝜛 (cid:19) 𝐵 𝑡𝑜𝑟 + 𝜛 𝜕 ( 𝜛𝐵 𝑡𝑜𝑟 ) 𝜕𝑟 𝜕 ( 𝜂 + 𝛽 ) 𝜕𝑟 (19) − 𝜛 u 𝑝 · ∇ (cid:18) 𝐵 𝑡𝑜𝑟 𝜛 (cid:19) − 𝐵 𝑡𝑜𝑟 ∇ · u 𝑝 + 𝜛 (∇ × ( 𝐴 ˆ 𝑒 𝜙 )) · ∇ 𝛀 +∇ × ( 𝛼 ∇ × ( 𝐴 ˆ 𝑒 𝜙 )) , where B 𝑝𝑜𝑙 = ∇ × A , 𝜛 = 𝑟 𝑠𝑖𝑛𝜃 , and Ω is the angular velocity from convetive motion U = r × 𝛀 . This equation set reproducesthe periodic solar magnetic field: amplification-annihilation-reverse. In Appendix, we show Solar dynamo simulation. Fig.A1(a)includes the reproduction of Jouve et al. (2008)’s work. 𝛼 & 𝛽 were chosen for the critical dynamo yielding 16.3 year period.Stefani et al. (2016) added the effect of synchronized helicity oscillation from planets to 𝛼 effect and solved the 1D equation toget ∼
22 year oscillation period. We solved it in 2D ( 𝑟, 𝜃 ) simulation in spherical coordinates. Fig.A1(b) shows that the modified 𝛼 reproduces the period of 21.7 without tuning any code variable. If additional physical effect exists, it can be applied to 𝛼 & 𝛽 rather than EMF. It is also possible to infer 𝛼 & 𝛽 from 𝐸 𝑀 ( 𝑡 ) and 𝐻 𝑀 ( 𝑡 ) . Test Field Method (TFM) was suggested to extract 𝛼 & 𝛽 from the simulation data (Schrinner et al. 2005). If the simulation with theartificial test field 𝐵 𝑇 is repeated, the data set for u & b can be obtained. Then, from 𝜉 𝑖 = (cid:104) u × b (cid:105) 𝑖 = 𝛼 𝑖 𝑗 𝐵 𝑇𝑗 + 𝛽 𝑖 𝑗𝑘 𝜕𝐵 𝑇𝑗 / 𝜕𝑥 𝑘 + 𝛾 𝑖 𝑗𝑙𝑘 ... ,the coefficients can be calculated. TFM provides detailed information on 𝛼 𝑖 𝑗 & 𝛽 𝑖 𝑗𝑘 depending on the component and posi-tion. Especially, TFM also shows that time averaged magnetic diffusion effect, i.e., 𝛽 𝑟 𝜃 , 𝛽 𝑟 𝜙 is practically negative like ourresult(Käpylä et al. 2009; ? ; ? ).However, there are a couple of things to be checked for the validity of 𝐵 𝑇 . If the test magnetic field is applied, the chargedparticle motion parallel to 𝐵 𝑇 is not influenced and can move in a free way. But, the motion perpendicular to 𝐵 𝑇 is constrainedby the field. Larmor radii of the particles decrease as 𝐵 𝑇 increases, and the electric Coulomb effect with binding energy amongcharged particles changes, too. Consequently, the geometry of collected particles will be like a needle. Then, distribution function 𝑓 becomes anisotropic so that 𝑓 should be divided into 𝑓 | | parallel to 𝐵 𝑇 and 𝑓 ⊥ perpendicular to 𝐵 𝑇 . And, if ‘ 𝑓 ’ is anisotropicwith 𝐵 𝑇 , Eq.(1), (2) should be divided into two parts, parallel and perpendicular direction. The simple replacement of EMF with 𝛾 is neglected for simplicityMNRAS000
22 year oscillation period. We solved it in 2D ( 𝑟, 𝜃 ) simulation in spherical coordinates. Fig.A1(b) shows that the modified 𝛼 reproduces the period of 21.7 without tuning any code variable. If additional physical effect exists, it can be applied to 𝛼 & 𝛽 rather than EMF. It is also possible to infer 𝛼 & 𝛽 from 𝐸 𝑀 ( 𝑡 ) and 𝐻 𝑀 ( 𝑡 ) . Test Field Method (TFM) was suggested to extract 𝛼 & 𝛽 from the simulation data (Schrinner et al. 2005). If the simulation with theartificial test field 𝐵 𝑇 is repeated, the data set for u & b can be obtained. Then, from 𝜉 𝑖 = (cid:104) u × b (cid:105) 𝑖 = 𝛼 𝑖 𝑗 𝐵 𝑇𝑗 + 𝛽 𝑖 𝑗𝑘 𝜕𝐵 𝑇𝑗 / 𝜕𝑥 𝑘 + 𝛾 𝑖 𝑗𝑙𝑘 ... ,the coefficients can be calculated. TFM provides detailed information on 𝛼 𝑖 𝑗 & 𝛽 𝑖 𝑗𝑘 depending on the component and posi-tion. Especially, TFM also shows that time averaged magnetic diffusion effect, i.e., 𝛽 𝑟 𝜃 , 𝛽 𝑟 𝜙 is practically negative like ourresult(Käpylä et al. 2009; ? ; ? ).However, there are a couple of things to be checked for the validity of 𝐵 𝑇 . If the test magnetic field is applied, the chargedparticle motion parallel to 𝐵 𝑇 is not influenced and can move in a free way. But, the motion perpendicular to 𝐵 𝑇 is constrainedby the field. Larmor radii of the particles decrease as 𝐵 𝑇 increases, and the electric Coulomb effect with binding energy amongcharged particles changes, too. Consequently, the geometry of collected particles will be like a needle. Then, distribution function 𝑓 becomes anisotropic so that 𝑓 should be divided into 𝑓 | | parallel to 𝐵 𝑇 and 𝑓 ⊥ perpendicular to 𝐵 𝑇 . And, if ‘ 𝑓 ’ is anisotropicwith 𝐵 𝑇 , Eq.(1), (2) should be divided into two parts, parallel and perpendicular direction. The simple replacement of EMF with 𝛾 is neglected for simplicityMNRAS000 , 1–14 () egative Magnetic Diffusivity 𝛽 effect in a Magnetically Dominant System 𝛼 , 𝛽 , and B is not valid anymore. Of course, compared to hydrodynamics, MHD is not free from the anisotropic issue due to theinternally produced magnetic field. But, the generated magnetic fields are coupled with the plasma particles that transfer theirovergrown momentum through frequent collisions. But, if external 𝐵 𝑇 field is applied to the system, the anisotropy issue cannotbe ignored anymore.Moreover, it is not clear if 𝛼 & 𝛽 with 𝐵 𝑇 really describe the system. Even if the system shows a similar result to that of TFMsimulation, there is no guarantee that they are the same.Instead of modifying the system with the artificial application, we can find 𝛼 & 𝛽 using the data of magnetic helicity andmagnetic energy. We pointed out that Eq. (17) is formally consistent with the statistical relation. Then, from Eq.(6) we get B · 𝜕 B 𝜕𝑡 = 𝛼 B · ∇ × B + ( 𝛽 + 𝜂 ) B · ∇ B = 𝛼 J · B − ( 𝛽 + 𝜂 ) B · B ( 𝑘 = )→ 𝜕𝜕𝑡 𝐸 𝑀 = − ( 𝛽 + 𝜂 ) 𝐸 𝑀 + 𝛼𝐻 𝑀 . (20)We can also derive the evolving magnetic helicity as follows. → 𝑑𝑑𝑡 𝐻 𝑀 = 𝛼𝐸 𝑀 − ( 𝛽 + 𝜂 ) 𝐻 𝑀 . (21)These two equations are functions of actual data 𝐸 𝑀 and 𝐻 𝑀 resulting from all internal and external effects. One of a simplemethod to solve this coupled equation set is to diagonalize the matrix using an invertible matrix 𝑃 , [ 𝐻 𝑀 , 𝐸 𝑀 ] ≡ 𝑃 [ 𝐻, 𝐸 ] .Then, (cid:20) 𝜕𝐻𝜕𝑡𝜕𝐸𝜕𝑡 (cid:21) = 𝑃 − (cid:20) − ( 𝛽 + 𝜂 ) 𝛼𝛼 − ( 𝛽 + 𝜂 ) (cid:21) 𝑃 (cid:20) 𝐻𝐸 (cid:21) = (cid:20) 𝜆 𝜆 (cid:21) (cid:20) 𝐻𝐸 (cid:21) . (22) [ 𝐻 𝑀 , 𝐸 𝑀 ] can be found from 𝑃 − [ 𝐻, 𝑀 ] , where the column vector of 𝑃 forms the basis of eigenvectors. The result is,2 𝐻 𝑀 ( 𝑡 𝑛 ) = ( 𝐸 𝑀 ( 𝑡 𝑛 − ) + 𝐻 𝑀 ( 𝑡 𝑛 − )) 𝑒 ∫ 𝑡𝑛 ( 𝛼 − 𝛽 − 𝜂 ) 𝑑𝜏 −( 𝐸 𝑀 ( 𝑡 𝑛 − ) − 𝐻 𝑀 ( 𝑡 𝑛 − )) 𝑒 ∫ 𝑡𝑛 (− 𝛼 − 𝛽 − 𝜂 ) 𝑑𝜏 , (23)4 𝐸 𝑀 ( 𝑡 𝑛 ) = ( 𝐸 𝑀 ( 𝑡 𝑛 − ) + 𝐻 𝑀 ( 𝑡 𝑛 − )) 𝑒 ∫ 𝑡𝑛 ( 𝛼 − 𝛽 − 𝜂 ) 𝑑𝜏 +( 𝐸 𝑀 ( 𝑡 𝑛 − ) − 𝐻 𝑀 ( 𝑡 𝑛 − )) 𝑒 ∫ 𝑡𝑛 (− 𝛼 − 𝛽 − 𝜂 ) 𝑑𝜏 . (24) 𝐻 𝑀 is always smaller than 2 𝐸 𝑀 , which satisfies realizability condition. But, 𝐻 𝑀 → 𝐸 𝑀 as the system is getting saturated. Incase of right handed HMFD, clearly 𝛼 > 𝐻 𝑀 ( 𝑡 𝑛 ) as well as 𝐸 𝑀 ( 𝑡 𝑛 ) is positive. But in case of left handed HMFD, the second term is dominant. This indicates that 𝐻 𝑀 ( 𝑡 𝑛 ) is negative, but 𝐸 𝑀 ( 𝑡 𝑛 ) is positive. On the contrary, in case of positively forced HKFD, 𝛼 is negative so that the second term in each equationis dominant leading to negative 𝐻 𝑀 . Still, 𝐸 𝑀 is not influenced by the chirality of forcing. These inferences are well consistentwith the simulation result of HKFD or HMFD. 𝛼 & 𝛽 from above results are (Park 2020) 𝛼 ( 𝑡 ) = 𝑑𝑑𝑡 𝑙𝑜𝑔 𝑒 (cid:12)(cid:12)(cid:12)(cid:12) 𝐸 𝑀 ( 𝑡 ) + 𝐻 𝑀 ( 𝑡 ) 𝐸 𝑀 ( 𝑡 ) − 𝐻 𝑀 ( 𝑡 ) (cid:12)(cid:12)(cid:12)(cid:12) , (25) 𝛽 ( 𝑡 ) = − 𝑑𝑑𝑡 𝑙𝑜𝑔 𝑒 (cid:12)(cid:12)(cid:0) 𝐸 𝑀 ( 𝑡 ) − 𝐻 𝑀 ( 𝑡 ) (cid:1) (cid:0) 𝐸 𝑀 ( 𝑡 ) + 𝐻 𝑀 ( 𝑡 ) (cid:1)(cid:12)(cid:12) − 𝜂. (26)To get the 𝛼 & 𝛽 , we need the simulation or observation data of 𝐸 𝑀 ( 𝑡 ) and 𝐻 𝑀 ( 𝑡 ) in each time ‘ 𝑡 𝑛 ’. For example, 𝑑𝐸 𝑀 / 𝑑𝑡 isapproximately ∼ ( 𝐸 𝑀 ( 𝑡 𝑛 ) − 𝐸 𝑀 ( 𝑡 𝑛 − ))/( 𝑡 𝑛 − 𝑡 𝑛 − ) . We compared ∇ × (cid:104) u × b (cid:105) with ∇ × ( 𝛼 B − 𝛽 ∇ × B ) in Fig.4(a), 4(b). In theearly time regime, they are quite close to each other. But, oscillation increases as the field becomes saturated (2 𝐸 𝑀 ∼ 𝐻 𝑀 ).And, for the anisotropic system, we need data for 𝐸 ⊥ , 𝑀 ( 𝑡 ) & 𝐻 ⊥ , 𝑀 ( 𝑡 ) , 𝐸 | | , 𝑀 ( 𝑡 ) & 𝐻 | | , 𝑀 ( 𝑡 ) , and the anisotropic solution ofEq.(25), (26). MNRAS , 1–14 () Kiwan Park et al. 𝛽 Now, we check the possibility of negative 𝛽 using analytic method. u × (− u · ∇ B ) → − 𝜖 𝑖 𝑗𝑘 (cid:104) 𝑢 𝑗 ( 𝑟, 𝑡 ) 𝑢 𝑚 ( 𝑟 + 𝑙, 𝜏 )(cid:105) 𝜕𝐵 𝑘 𝜕𝑟 𝑚 (27) ∼ − 𝜖 𝑖 𝑗𝑘 (cid:104) 𝑢 𝑗 ( 𝑡 ) 𝑢 𝑚 ( 𝜏 )(cid:105) 𝜕𝐵 𝑘 𝜕𝑟 𝑚 − (cid:104) 𝑢 𝑗 ( 𝑡 ) 𝑙 𝑛 𝜕 𝑛 𝑢 𝑚 ( 𝜏 )(cid:105) 𝜖 𝑖 𝑗𝑘 𝜕𝐵 𝑘 𝜕𝑟 𝑚 (28) ∼ − (cid:104) 𝑢 (cid:105) 𝜖 𝑖 𝑗𝑘 𝜕𝐵 𝑘 𝜕𝑟 𝑚 𝛿 𝑗𝑚 (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) − (cid:10) 𝜖 𝑗𝑛𝑚 𝑙 | 𝐻 𝑉 | (cid:11) 𝜖 𝑖 𝑗𝑘 𝜕𝐵 𝑘 𝜕𝑟 𝑚 𝛿 𝑛𝑘 𝛿 𝑚𝑖 (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) , (29)For the nonhelical field, the result will be ((cid:104) 𝑢 / (cid:105) + 𝑙 · ∇(cid:104) 𝑢 / (cid:105))(−∇ × B ) , which is the conventional positive 𝛽 . If we considerthe helical fields of 𝑢 𝑗 , 𝑢 𝑚 , and 𝐵 𝑘 are mutually perpendicular. Without loss of generality, ‘ 𝑢 𝑗 ’ can be considered as ‘ 𝑢 𝑝𝑜𝑙 ’.Then, ‘ 𝑢 𝑚 ’ should be ‘ 𝑢 𝑡𝑜𝑟 ’ (‘ 𝑚 ’ → ‘ 𝑖 ’), and ‘ 𝑛 ’ should be ‘ 𝑘 ’. Since the turbulent velocity part is saturated earlier than the largescale eddy, it can be calculated separately: 𝑙 / (cid:104) u · ∇ × u (cid:105)(≡ 𝑙 / 𝐻 𝑉 ) . The factor ‘ 𝑙 𝑛 ’ or ’ 𝑙 ( = | 𝑙 𝑛 |) ’ indicates the correlation lengthbetween u 𝑝𝑜𝑙 and u 𝑡𝑜𝑟 . For the left handed helical structure, we first place a virtual mirror on the right side of u 𝑡𝑜𝑟, . Then, thetoroidal velocity eddies will be reflected on the other side, and the correlation length ‘ 𝑙 𝑛 ’ is toward ‘ − ˆ 𝑙 ’. The reflection makes ∇ 𝑛 𝑢 𝑚 negative, but u 𝑝𝑜𝑙 does not change. That is, ‘ 𝑙 𝑛 ’ should be sort of a pseudo scalar (refer to the difference between (cid:104) 𝑢 (cid:105) and (cid:104) u · 𝜔 (cid:105) ). Considering the statistical meaning of each component, we can make a more general form as follows: ∼ − (cid:104) 𝑢 (cid:105)∇ × B + 𝑙 | 𝐻 𝑉 |∇ × B → − 𝛽 ∇ × B . (30)The total diffusion effect becomes ( 𝛽 + 𝜂 )∇ B , whose Fourier transformed expression is −( 𝛽 + 𝜂 ) 𝑘 B regardless of chirality.Kraichnan derived the negative magnetic diffusion effect in Lagrangian formation with the assumption of strong helical field,(Kraichnan 1976): 𝜕 B 𝜕𝑡 = 𝛽 ∇ B + 𝜏 ∇ × (cid:104) 𝛼 ∇ × 𝛼 (cid:105) B → ( 𝛽 − 𝜏 𝐴 )∇ B . (31)(Here, 𝛽 = ∫ 𝜏 𝑢 𝑑𝑡 ∼ 𝜏 𝑢 , 𝛼 ( x , 𝑡 ) = (−) / (cid:104) u · 𝜔 (cid:105) 𝜏 , (cid:104) 𝛼 ( x , 𝑡 ) 𝛼 ( x (cid:48) , 𝑡 (cid:48) )(cid:105) = 𝐴 ( 𝑥 − 𝑥 (cid:48) ) 𝐷 ( 𝑡 − 𝑡 (cid:48) ) , 𝜏 = ∫ ∞ 𝐷 ( 𝑡 ) 𝑑𝑡 .) 𝛽 is the conventional positive magnetic diffusion effect. But, − 𝜏 𝐴 , which is from the correlation of (cid:104) 𝛼𝛼 (cid:105) , actually plays therole of the negative magnetic diffusion 𝛽 effect. The detailed derivation of Eq.(31) is not the same as Eq.(27)-(29). However, theyboth are from the turbulent velocity u and shows its helical feature produces the negative magnetic diffusion. Besides, there aresome theoretical and experimental works associated with negative magnetic diffusivity ( ?? , references therein). They are basedon 𝛼 − 𝛼 correlation in the strong helical system and do not explain the coupling of 𝐻 𝑀 and 𝐸 𝑀 . As Fig.1(a), 1(b) show, we briefly discuss how the helical magnetic field constrains plasma with the negative 𝛽 effect (Park 2020).As Eq.(2), (4) imply, plasma and magnetic field are coupled through Lorentz force and EMF. If we take the scalar product of B or U on the curl of EMF or momentum equation respectively, we get B · ∇ × ( U × B ) or U · ( J × B ) . And they are practically thesame except the opposite sign. To make it clear, the field scales can be divided into U , B and turbulent u , b . Taking the averageand applying Reynolds rule, we get U · J × B = − B · ∇ × ( U × B ) − B · ∇ × (cid:104) u × b (cid:105)−(cid:104) b · ∇ × ( U × b )(cid:105) − (cid:104) b · ∇ × u × B (cid:105) (32)Considering Fig.2, we see the first term is negligible. The third term is not so significant because of the high helicity ratio in thesmall scale regime. And the fourth term can be dropped replacing j with 𝜌 u . The second term can be rewritten like − B · ∇ × (cid:104) u × b (cid:105) = − B · ∇ × ( 𝛼 B − 𝛽 ∇ × B ) = − 𝛼 B · ∇ × B − 𝛽 B · ∇ B . (33)The first term becomes negligible, but the second term is − 𝛽 B · ∇ B → 𝛽𝑘 𝐵 . Then, the negative 𝛽 suppresses the plasmamotion U while it amplifies B (see Fig.1(a), (b)).In comparison with helical large scale dynamo in Fig.1, nonhelical small scale dynamo in Fig.7 shows the supplementary roleof 𝛽 in the large scale plasma motion U . Although small scale kinetic energy ( ∼ (cid:104) 𝑢 (cid:105) ) is much larger than that of HMFD, there is MNRAS000
22 year oscillation period. We solved it in 2D ( 𝑟, 𝜃 ) simulation in spherical coordinates. Fig.A1(b) shows that the modified 𝛼 reproduces the period of 21.7 without tuning any code variable. If additional physical effect exists, it can be applied to 𝛼 & 𝛽 rather than EMF. It is also possible to infer 𝛼 & 𝛽 from 𝐸 𝑀 ( 𝑡 ) and 𝐻 𝑀 ( 𝑡 ) . Test Field Method (TFM) was suggested to extract 𝛼 & 𝛽 from the simulation data (Schrinner et al. 2005). If the simulation with theartificial test field 𝐵 𝑇 is repeated, the data set for u & b can be obtained. Then, from 𝜉 𝑖 = (cid:104) u × b (cid:105) 𝑖 = 𝛼 𝑖 𝑗 𝐵 𝑇𝑗 + 𝛽 𝑖 𝑗𝑘 𝜕𝐵 𝑇𝑗 / 𝜕𝑥 𝑘 + 𝛾 𝑖 𝑗𝑙𝑘 ... ,the coefficients can be calculated. TFM provides detailed information on 𝛼 𝑖 𝑗 & 𝛽 𝑖 𝑗𝑘 depending on the component and posi-tion. Especially, TFM also shows that time averaged magnetic diffusion effect, i.e., 𝛽 𝑟 𝜃 , 𝛽 𝑟 𝜙 is practically negative like ourresult(Käpylä et al. 2009; ? ; ? ).However, there are a couple of things to be checked for the validity of 𝐵 𝑇 . If the test magnetic field is applied, the chargedparticle motion parallel to 𝐵 𝑇 is not influenced and can move in a free way. But, the motion perpendicular to 𝐵 𝑇 is constrainedby the field. Larmor radii of the particles decrease as 𝐵 𝑇 increases, and the electric Coulomb effect with binding energy amongcharged particles changes, too. Consequently, the geometry of collected particles will be like a needle. Then, distribution function 𝑓 becomes anisotropic so that 𝑓 should be divided into 𝑓 | | parallel to 𝐵 𝑇 and 𝑓 ⊥ perpendicular to 𝐵 𝑇 . And, if ‘ 𝑓 ’ is anisotropicwith 𝐵 𝑇 , Eq.(1), (2) should be divided into two parts, parallel and perpendicular direction. The simple replacement of EMF with 𝛾 is neglected for simplicityMNRAS000 , 1–14 () egative Magnetic Diffusivity 𝛽 effect in a Magnetically Dominant System 𝛼 , 𝛽 , and B is not valid anymore. Of course, compared to hydrodynamics, MHD is not free from the anisotropic issue due to theinternally produced magnetic field. But, the generated magnetic fields are coupled with the plasma particles that transfer theirovergrown momentum through frequent collisions. But, if external 𝐵 𝑇 field is applied to the system, the anisotropy issue cannotbe ignored anymore.Moreover, it is not clear if 𝛼 & 𝛽 with 𝐵 𝑇 really describe the system. Even if the system shows a similar result to that of TFMsimulation, there is no guarantee that they are the same.Instead of modifying the system with the artificial application, we can find 𝛼 & 𝛽 using the data of magnetic helicity andmagnetic energy. We pointed out that Eq. (17) is formally consistent with the statistical relation. Then, from Eq.(6) we get B · 𝜕 B 𝜕𝑡 = 𝛼 B · ∇ × B + ( 𝛽 + 𝜂 ) B · ∇ B = 𝛼 J · B − ( 𝛽 + 𝜂 ) B · B ( 𝑘 = )→ 𝜕𝜕𝑡 𝐸 𝑀 = − ( 𝛽 + 𝜂 ) 𝐸 𝑀 + 𝛼𝐻 𝑀 . (20)We can also derive the evolving magnetic helicity as follows. → 𝑑𝑑𝑡 𝐻 𝑀 = 𝛼𝐸 𝑀 − ( 𝛽 + 𝜂 ) 𝐻 𝑀 . (21)These two equations are functions of actual data 𝐸 𝑀 and 𝐻 𝑀 resulting from all internal and external effects. One of a simplemethod to solve this coupled equation set is to diagonalize the matrix using an invertible matrix 𝑃 , [ 𝐻 𝑀 , 𝐸 𝑀 ] ≡ 𝑃 [ 𝐻, 𝐸 ] .Then, (cid:20) 𝜕𝐻𝜕𝑡𝜕𝐸𝜕𝑡 (cid:21) = 𝑃 − (cid:20) − ( 𝛽 + 𝜂 ) 𝛼𝛼 − ( 𝛽 + 𝜂 ) (cid:21) 𝑃 (cid:20) 𝐻𝐸 (cid:21) = (cid:20) 𝜆 𝜆 (cid:21) (cid:20) 𝐻𝐸 (cid:21) . (22) [ 𝐻 𝑀 , 𝐸 𝑀 ] can be found from 𝑃 − [ 𝐻, 𝑀 ] , where the column vector of 𝑃 forms the basis of eigenvectors. The result is,2 𝐻 𝑀 ( 𝑡 𝑛 ) = ( 𝐸 𝑀 ( 𝑡 𝑛 − ) + 𝐻 𝑀 ( 𝑡 𝑛 − )) 𝑒 ∫ 𝑡𝑛 ( 𝛼 − 𝛽 − 𝜂 ) 𝑑𝜏 −( 𝐸 𝑀 ( 𝑡 𝑛 − ) − 𝐻 𝑀 ( 𝑡 𝑛 − )) 𝑒 ∫ 𝑡𝑛 (− 𝛼 − 𝛽 − 𝜂 ) 𝑑𝜏 , (23)4 𝐸 𝑀 ( 𝑡 𝑛 ) = ( 𝐸 𝑀 ( 𝑡 𝑛 − ) + 𝐻 𝑀 ( 𝑡 𝑛 − )) 𝑒 ∫ 𝑡𝑛 ( 𝛼 − 𝛽 − 𝜂 ) 𝑑𝜏 +( 𝐸 𝑀 ( 𝑡 𝑛 − ) − 𝐻 𝑀 ( 𝑡 𝑛 − )) 𝑒 ∫ 𝑡𝑛 (− 𝛼 − 𝛽 − 𝜂 ) 𝑑𝜏 . (24) 𝐻 𝑀 is always smaller than 2 𝐸 𝑀 , which satisfies realizability condition. But, 𝐻 𝑀 → 𝐸 𝑀 as the system is getting saturated. Incase of right handed HMFD, clearly 𝛼 > 𝐻 𝑀 ( 𝑡 𝑛 ) as well as 𝐸 𝑀 ( 𝑡 𝑛 ) is positive. But in case of left handed HMFD, the second term is dominant. This indicates that 𝐻 𝑀 ( 𝑡 𝑛 ) is negative, but 𝐸 𝑀 ( 𝑡 𝑛 ) is positive. On the contrary, in case of positively forced HKFD, 𝛼 is negative so that the second term in each equationis dominant leading to negative 𝐻 𝑀 . Still, 𝐸 𝑀 is not influenced by the chirality of forcing. These inferences are well consistentwith the simulation result of HKFD or HMFD. 𝛼 & 𝛽 from above results are (Park 2020) 𝛼 ( 𝑡 ) = 𝑑𝑑𝑡 𝑙𝑜𝑔 𝑒 (cid:12)(cid:12)(cid:12)(cid:12) 𝐸 𝑀 ( 𝑡 ) + 𝐻 𝑀 ( 𝑡 ) 𝐸 𝑀 ( 𝑡 ) − 𝐻 𝑀 ( 𝑡 ) (cid:12)(cid:12)(cid:12)(cid:12) , (25) 𝛽 ( 𝑡 ) = − 𝑑𝑑𝑡 𝑙𝑜𝑔 𝑒 (cid:12)(cid:12)(cid:0) 𝐸 𝑀 ( 𝑡 ) − 𝐻 𝑀 ( 𝑡 ) (cid:1) (cid:0) 𝐸 𝑀 ( 𝑡 ) + 𝐻 𝑀 ( 𝑡 ) (cid:1)(cid:12)(cid:12) − 𝜂. (26)To get the 𝛼 & 𝛽 , we need the simulation or observation data of 𝐸 𝑀 ( 𝑡 ) and 𝐻 𝑀 ( 𝑡 ) in each time ‘ 𝑡 𝑛 ’. For example, 𝑑𝐸 𝑀 / 𝑑𝑡 isapproximately ∼ ( 𝐸 𝑀 ( 𝑡 𝑛 ) − 𝐸 𝑀 ( 𝑡 𝑛 − ))/( 𝑡 𝑛 − 𝑡 𝑛 − ) . We compared ∇ × (cid:104) u × b (cid:105) with ∇ × ( 𝛼 B − 𝛽 ∇ × B ) in Fig.4(a), 4(b). In theearly time regime, they are quite close to each other. But, oscillation increases as the field becomes saturated (2 𝐸 𝑀 ∼ 𝐻 𝑀 ).And, for the anisotropic system, we need data for 𝐸 ⊥ , 𝑀 ( 𝑡 ) & 𝐻 ⊥ , 𝑀 ( 𝑡 ) , 𝐸 | | , 𝑀 ( 𝑡 ) & 𝐻 | | , 𝑀 ( 𝑡 ) , and the anisotropic solution ofEq.(25), (26). MNRAS , 1–14 () Kiwan Park et al. 𝛽 Now, we check the possibility of negative 𝛽 using analytic method. u × (− u · ∇ B ) → − 𝜖 𝑖 𝑗𝑘 (cid:104) 𝑢 𝑗 ( 𝑟, 𝑡 ) 𝑢 𝑚 ( 𝑟 + 𝑙, 𝜏 )(cid:105) 𝜕𝐵 𝑘 𝜕𝑟 𝑚 (27) ∼ − 𝜖 𝑖 𝑗𝑘 (cid:104) 𝑢 𝑗 ( 𝑡 ) 𝑢 𝑚 ( 𝜏 )(cid:105) 𝜕𝐵 𝑘 𝜕𝑟 𝑚 − (cid:104) 𝑢 𝑗 ( 𝑡 ) 𝑙 𝑛 𝜕 𝑛 𝑢 𝑚 ( 𝜏 )(cid:105) 𝜖 𝑖 𝑗𝑘 𝜕𝐵 𝑘 𝜕𝑟 𝑚 (28) ∼ − (cid:104) 𝑢 (cid:105) 𝜖 𝑖 𝑗𝑘 𝜕𝐵 𝑘 𝜕𝑟 𝑚 𝛿 𝑗𝑚 (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) − (cid:10) 𝜖 𝑗𝑛𝑚 𝑙 | 𝐻 𝑉 | (cid:11) 𝜖 𝑖 𝑗𝑘 𝜕𝐵 𝑘 𝜕𝑟 𝑚 𝛿 𝑛𝑘 𝛿 𝑚𝑖 (cid:124) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) , (29)For the nonhelical field, the result will be ((cid:104) 𝑢 / (cid:105) + 𝑙 · ∇(cid:104) 𝑢 / (cid:105))(−∇ × B ) , which is the conventional positive 𝛽 . If we considerthe helical fields of 𝑢 𝑗 , 𝑢 𝑚 , and 𝐵 𝑘 are mutually perpendicular. Without loss of generality, ‘ 𝑢 𝑗 ’ can be considered as ‘ 𝑢 𝑝𝑜𝑙 ’.Then, ‘ 𝑢 𝑚 ’ should be ‘ 𝑢 𝑡𝑜𝑟 ’ (‘ 𝑚 ’ → ‘ 𝑖 ’), and ‘ 𝑛 ’ should be ‘ 𝑘 ’. Since the turbulent velocity part is saturated earlier than the largescale eddy, it can be calculated separately: 𝑙 / (cid:104) u · ∇ × u (cid:105)(≡ 𝑙 / 𝐻 𝑉 ) . The factor ‘ 𝑙 𝑛 ’ or ’ 𝑙 ( = | 𝑙 𝑛 |) ’ indicates the correlation lengthbetween u 𝑝𝑜𝑙 and u 𝑡𝑜𝑟 . For the left handed helical structure, we first place a virtual mirror on the right side of u 𝑡𝑜𝑟, . Then, thetoroidal velocity eddies will be reflected on the other side, and the correlation length ‘ 𝑙 𝑛 ’ is toward ‘ − ˆ 𝑙 ’. The reflection makes ∇ 𝑛 𝑢 𝑚 negative, but u 𝑝𝑜𝑙 does not change. That is, ‘ 𝑙 𝑛 ’ should be sort of a pseudo scalar (refer to the difference between (cid:104) 𝑢 (cid:105) and (cid:104) u · 𝜔 (cid:105) ). Considering the statistical meaning of each component, we can make a more general form as follows: ∼ − (cid:104) 𝑢 (cid:105)∇ × B + 𝑙 | 𝐻 𝑉 |∇ × B → − 𝛽 ∇ × B . (30)The total diffusion effect becomes ( 𝛽 + 𝜂 )∇ B , whose Fourier transformed expression is −( 𝛽 + 𝜂 ) 𝑘 B regardless of chirality.Kraichnan derived the negative magnetic diffusion effect in Lagrangian formation with the assumption of strong helical field,(Kraichnan 1976): 𝜕 B 𝜕𝑡 = 𝛽 ∇ B + 𝜏 ∇ × (cid:104) 𝛼 ∇ × 𝛼 (cid:105) B → ( 𝛽 − 𝜏 𝐴 )∇ B . (31)(Here, 𝛽 = ∫ 𝜏 𝑢 𝑑𝑡 ∼ 𝜏 𝑢 , 𝛼 ( x , 𝑡 ) = (−) / (cid:104) u · 𝜔 (cid:105) 𝜏 , (cid:104) 𝛼 ( x , 𝑡 ) 𝛼 ( x (cid:48) , 𝑡 (cid:48) )(cid:105) = 𝐴 ( 𝑥 − 𝑥 (cid:48) ) 𝐷 ( 𝑡 − 𝑡 (cid:48) ) , 𝜏 = ∫ ∞ 𝐷 ( 𝑡 ) 𝑑𝑡 .) 𝛽 is the conventional positive magnetic diffusion effect. But, − 𝜏 𝐴 , which is from the correlation of (cid:104) 𝛼𝛼 (cid:105) , actually plays therole of the negative magnetic diffusion 𝛽 effect. The detailed derivation of Eq.(31) is not the same as Eq.(27)-(29). However, theyboth are from the turbulent velocity u and shows its helical feature produces the negative magnetic diffusion. Besides, there aresome theoretical and experimental works associated with negative magnetic diffusivity ( ?? , references therein). They are basedon 𝛼 − 𝛼 correlation in the strong helical system and do not explain the coupling of 𝐻 𝑀 and 𝐸 𝑀 . As Fig.1(a), 1(b) show, we briefly discuss how the helical magnetic field constrains plasma with the negative 𝛽 effect (Park 2020).As Eq.(2), (4) imply, plasma and magnetic field are coupled through Lorentz force and EMF. If we take the scalar product of B or U on the curl of EMF or momentum equation respectively, we get B · ∇ × ( U × B ) or U · ( J × B ) . And they are practically thesame except the opposite sign. To make it clear, the field scales can be divided into U , B and turbulent u , b . Taking the averageand applying Reynolds rule, we get U · J × B = − B · ∇ × ( U × B ) − B · ∇ × (cid:104) u × b (cid:105)−(cid:104) b · ∇ × ( U × b )(cid:105) − (cid:104) b · ∇ × u × B (cid:105) (32)Considering Fig.2, we see the first term is negligible. The third term is not so significant because of the high helicity ratio in thesmall scale regime. And the fourth term can be dropped replacing j with 𝜌 u . The second term can be rewritten like − B · ∇ × (cid:104) u × b (cid:105) = − B · ∇ × ( 𝛼 B − 𝛽 ∇ × B ) = − 𝛼 B · ∇ × B − 𝛽 B · ∇ B . (33)The first term becomes negligible, but the second term is − 𝛽 B · ∇ B → 𝛽𝑘 𝐵 . Then, the negative 𝛽 suppresses the plasmamotion U while it amplifies B (see Fig.1(a), (b)).In comparison with helical large scale dynamo in Fig.1, nonhelical small scale dynamo in Fig.7 shows the supplementary roleof 𝛽 in the large scale plasma motion U . Although small scale kinetic energy ( ∼ (cid:104) 𝑢 (cid:105) ) is much larger than that of HMFD, there is MNRAS000 , 1–14 () egative Magnetic Diffusivity 𝛽 effect in a Magnetically Dominant System no significant decrease in 𝐸 𝑉 (∼ 𝑈 ). This indicates that the positive 𝛽 provides magnetic energy to the large scale plasma motion. We have pointed out the possibility of helical magnetic forcing dynamo (HMFD). Although HMFD is highly probable in themagnetically dominant plasma system, HMFD has several unique features that distinguish it from helical kinetic forcing dynamo(HKFD). Being different from conventional dynamo theory, plasma kinetic energy 𝐸 𝑉 is not or little converted into magneticenergy 𝐸 𝑀 . Rather, externally given 𝐸 𝑀 is converted into 𝐸 𝑉 to transport 𝐸 𝑀 into the large and small scale region. Moreover,fully helical magnetic energy produces helical kinetic motion with the same chirality as 𝐸 𝑀 through Lorentz force. 𝐸 𝑉 in HMFDis subsidiary to the migration of 𝐸 𝑀 so that magnetic Reynolds number 𝑅𝑒 𝑀 ( = 𝑈 𝐿 / 𝜂 ) is negligibly small. Consequently, largescale magnetic energy 𝐸 𝑀 is amplified and saturated more efficiently than that of HKFD. Its nonlinear dynamo process can beexplained with 𝛼 & 𝛽 and large scale magnetic field B . Since the exact definitions of the pseudo scholars are not yet known, wecalculate them using Eq.(25), (26). This semi-analytic method does not require any artificial modification affecting the plasmasystem. Its solution is mathematically complete and needs only fundamental physical quantities 𝐸 𝑀 & 𝐻 𝑀 .The result shows that the role of 𝛼 as the 𝐸 𝑀 generator is not much. Rather, the negative 𝛽 effect plays the substantial roleof generating 𝐸 𝑀 with Laplacian ∇ → − 𝑘 . 𝛽 keeps negative and gets saturated along with 𝐸 𝑀 . Negative 𝛽 also explains howthe plasma motion is quenched in the helical dynamo system. We explained this dynamo process with field structure modeland analytic method in addition to numerical data. Analytically, HMFD occurs 𝐸 𝑀 𝐹 = ∫ 𝜕 u / 𝜕𝑡 𝑑𝜏 × b ∼ ∫ B · ∇ b 𝑑𝜏 × b ∼ / ∫ ( b ·∇× b ) 𝑑𝜏 B → (positive magnetic helicity); then, u × ∫ 𝜕 b / 𝜕𝑡 𝑑𝜏 ∼ u × ∫ B ·∇ u 𝑑𝜏 ∼ − / ∫ ( u ·∇× u ) 𝑑𝜏 B → (negativemagnetic helicity). 𝛼 quenching is the result of this series of interaction. The origin of negative 𝛽 is advective term − u · ∇ B . j (cid:48) from u 𝑝𝑜𝑙 × ∫ (− u · ∇ B ) 𝑑𝜏 generates ± (cid:0) j (cid:48) · ∫ B · ∇ u 𝑑𝜏 (cid:1) , which increases (decreases) magnetic helicity in the system. This isnamed as negative (positive) 𝛽 effect. And, to verify the half analytic 𝛼 & 𝛽 , we compared EMF with 𝛼 , 𝛽 , and B approximationin Fig.4. This analytic process is also compared with field structure model.Finally, we introduced Moffatt & Kraichnan’s work for the negative 𝛼 − 𝛼 correlation. The correlation effect is actually thesame as negative magnetic diffusivity 𝛽 effect. Assuming the helical field, they can show the amplification of magnetic field.However, 𝛼 − 𝛼 correlation cannot explain the coupling of 𝐸 𝑀 & 𝐻 𝑀 like Eq.(20), (21). The growth of 𝐸 𝑀 depends on thehelicity ratio as well as energy strength ( ? ). In this paper, we only used 𝑃𝑟 𝑀 = 𝑃𝑟 𝑀 ≠ ACKNOWLEDGEMENTS
The authors appreciate support from National Research Foundation of Korea: NRF-2020R1A2C3006177 and NRF-2013M7A1A1075764.
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Figure 1. (a) 2D(azimuthal angle 𝜙 independent) simulation of Solar Magnetic field with Eq. (18), (19). The simulation yields the period 𝜆 = .
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