The heliosphere appears to be powered by coaxial helicity injection from a negative helicity injector in the northern hemisphere and a positive one in the southern hemisphere. The injector magnetic flux for both is the measured solar polar magnetic flux, and the injector voltage is generated by a simple dynamo effect due to the differential rotation of the solar surface with the polar magnetic flux. The dynamo current is estimated from the solar motion that it causes. This current also appears to sustain a thin, shallow global magnetic structure over most of the solar surface that has the form of a 1D minimum energy state. The current channel appears to be destroyed and reformed every 11 years. The currents and magnetic fields reverse in this solar cycle. A brief discussion of surface phenomena observed during this cycle is given. Plasma self-organization is briefly discussed and used in this analysis of solar data. The magnetic phenomena discussed include torsional oscillations, the heat source for the chromosphere and the corona, filaments, meridional flow, the solar cycle, sunspots, CMEs, and flares.
A century from the first observations1 of the solar magnetic field, consensus still has not been reached2 as to the specifics of the mechanism by which that field is generated. While Hale's pioneering discoveries include the magnetic field in sunspots and the twenty-two-year cycle, the solar magnetograph3 showed that the entire solar field reversals every eleven years were almost certainly due to the existence of a global magnetic field structure (GMS) within the Sun.
Developments since the 1980s have posed challenges to all dynamo models then in existence, from the reevaluation of magnetic buoyancy and diffusivity effects, to the simulation of existing models of the Sun, to the development of helioseismology, which allowed the interior of the Sun to be measured more directly.2 While these developments have afforded new opportunities (such as the discovery of the tachocline as a potential location for the solar dynamo), the field of solar physics has not yet settled into a standard model of the solar dynamo.4
Understanding of the solar dynamo process began a century ago with George Ellery Hale's observations of the magnetic field in sunspots.5 Motivated by the structure of fibrils around sunspots, which suggested the presence of strong magnetic field and electric currents in the region, Hale sought and found Zeeman splitting in the sunspots that confirmed their existence.1 Hale's continued observations led to three discoveries about the structure and evolution of the solar magnetic field. First, Hale's polarity law states that regions of strong magnetic fields on the Sun are grouped in pairs of opposite polarities (referred to as bipolar magnetic regions) and that the alignment with respect to the direction of rotation is the same in a given hemisphere but reversed between the northern and southern hemispheres. Next, the polarity of the sun's global magnetic field reverses between two successive 11-year sunspot cycles, such that together they form a 22-year magnetic (or “Hale”) cycle.6 Finally, Joy's law establishes that the leading sunspot—which has the same magnetic polarity as the polar magnetic flux in that hemisphere—is closer to the equator “on average” than the following sunspot. Together, these contributions form a cornerstone of solar dynamo theory that any complete theoretical model must reproduce.
Following Hale's observations came early attempts to characterize the self-generation of large-scale magnetic fields in planets and stars, forming the basis of dynamo theory. In his antidynamo theorem, Thomas Cowling demonstrates conclusively that an axisymmetric magnetic field cannot be maintained by dynamo action.7 Later, Horace W. Babcock posited that bipolar magnetic regions in the Sun are formed where buoyant magnetic flux loops of the submerged field are brought to the surface.8 Reconnection between a pair of bipolar magnetic regions across the equator neutralizes a portion of the polar magnetic flux. Multiple such reconnections would eventually generate a new dipolar field with reversed polarity from the original. Robert B. Leighton then constructed a semiempirical model that qualitatively reproduces the structure of the magnetic butterfly diagram.9 In the model, amplification of the azimuthally symmetric magnetic field to some critical value leads to the eruption of sunspots, which then diminishes the field strength. The radial field produced during eruption disperses through random-walk diffusion.
B. Contents of paper
In this paper, the dynamo activity is estimated to occur within a particularly thin (≈0.15 Mm) shallow (≈0.8 Mm) region of the sun and explores how the resulting global magnetic structure (GMS) is consistent with observations of solar features. As discussed later, the GMS may have only large, distantly spaced holes for convective heat transfer through the region, which would result in supergranulation. The GMS's fringe fields and polar magnetic flux are likely the sources of the observed magnetic field called the magnetic carpet10 and the chromospheric magnetic network.11 For reference, discussions of solar phenomena associated with the terms used in this paper and this paper's impact on the understanding of the phenomena are given in the Appendix.
As will be shown, the dynamo is the result of polar magnetic flux and solar differential rotation, driven by powerful thermal convection motion and gravitational effects (not MHD driven). Thermal convection distributes the MHD force down to 0.1 R⊙, and this magnetic driven dynamic motion is too small to produce significant MHD effects. This paper presents the first understanding of the driving of torsional oscillations that confirm this interpretation. Applying recent advances in understanding of helicity injection current drive and plasma self-organization, summarized in Sec. I C, to the measured values of the solar polar magnetic flux and the measured amplitude of torsional oscillations reveals the nature and source of solar magnetic phenomena.
The following phenomena substantiate these new and unorthodox conclusions: meridional flow; prominences and filaments; heating the chromosphere and the corona; the current sheet current measured at the radius of the earth's orbit; and synchronization for symmetric butterfly sunspot patterns. Thus, this paper may be a breakthrough in the area of solar dynamo physics and solar MHD physics.
When the polar flux leaves the GMS, the process of injecting helicity into the heliosphere begins by expanding toward the surface and transitions from thermal convection dominated to MHD dominated dynamics. This paper is focused on the voltage and current driving the injectors and the scaling laws of current and the rate that magnetic energy is delivered to the heliosphere as a function of distance from the GMS. The corresponding properties of the accelerated plasma are subjects of research and will not be discussed.
In regions of solar plasma just below the photosphere, the plasma is well described by the resistive MHD Ohm's law because the drift speed of the electrons with respect to the fluid is ≈2 × 10−4 ms−1, which is much less than the lowest fluid velocities of interest, ≈1 ms−1. Thus, ve ≈ v and the Hall term is negligible.12 Ohm's law is
where E is the electric field, v is the plasma velocity, B is the magnetic field, η is the resistivity, and j is the current density. The Woltjer-Taylor minimum energy principle13,14 states that magnetic plasmas relax toward a state of minimum energy while conserving helicity (the MECH state15). Magnetic helicity, , where is the magnetic vector potential, is the linkage of magnetic flux with magnetic flux, and for two linked closed flux tubes (see Fig. 1) of φ1 and φ2, K = 2φ1φ2.16 For a closed conducting boundary, with the boundary conditions of only the magnetic field that penetrates or links the boundary, the MECH state satisfies∇ ×B = λeqB, where is a global constant,13 and this MECH state is called the “Taylor state.”17 Helicity dissipated per unit volume is given by 18 and, therefore, since E from velocity turbulence is perpendicular to B, it cannot dissipate helicity.12 Roughly, flux is conserved on the reconnection time scale and the minimum energy is what remains when all energetically favorable reconnections are done.14 Furthermore, helicity decays on the resistive time scale (the longest characteristic time scale in the system) and the MECH state is stable.12 Ampère's law yields, λeq = μojll/B, where is the inverse magnetic gradient length. The dissipation of energy per unit of helicity dissipation is j•E/2E•B = j/2B = λ/2μo. For similarly shaped MECH states, λeq is proportional to 1/characteristic length scale. The sign of λeq is the sign of the helicity.
The experimental results show that grossly stable MECH states can be sustained but nonaxisymmetric perturbations must be imposed.19 The SSPX experiment used axisymmetric helicity injection and instability was required to make the necessary perturbations.20 Sustained plasmas have a λinj value as part of the boundary conditions which is larger than the λ value of the rest of the state and the MECH state does not have uniform λ.15 λinj is defined by an external “injector” circuit and equals μoIinj/ψinj, where Iinj is the current driven parallel to the magnetic flux, ψinj. Imposed perturbations dissipate the excess magnetic energy by causing the transfer of energy from the λinj region to the rest of the state, sustaining the MECH state, which is a Global Magnetic Structure.21,22 The perturbations also break the symmetry of the magnetic axis and the sustainment is consistent with Cowling's Theorem.7
When no magnetic flux or current links or penetrates the boundary, the solution to ∇ × B = λB becomes an eigenvalue problem with discrete λegn.14 Since the solution only depends on the ratio of B to j, an arbitrarily large energy or helicity can exist in an eigenstate. Since λegn is proportional to the energy per unit helicity, λself never exceeds the lowest eigenvalue, λeg1; otherwise, the helicity can relax into the lowest eigenvalue and release energy. The injector state is defined as the Taylor state having λ equal to λinj and having the conducting boundary of the geometry. Of course, the injector state can relax if λinj is higher than the lowest eigenvalue of the geometry.
The following assumptions and their justification are given:
Assumption on self-organization. The self-organization properties of magnetized plasma, as justified above, are assumed valid. While helicity conservation is very robust, reaching the MECH state takes longer than an Alfven time and therefore requires an equilibrium. This “relaxation time”14 is often assumed to be comparable to the Sweet/Parker time.
Assumption on axisymmetry. Solar dynamics are assumed to be dominated by toroidal symmetry. The torsional oscillations are symmetric, the Sun is symmetric, and at a given latitude, the probability of a magnetic event does not appear to depend on longitude. Thus, this simplifying assumption of axisymmetry is used.
Assumption on dynamo driving. The solar dynamo is assumed to cause the torsional oscillation. The solar magnetic activity and the torsional oscillations (speed-up and slow-down bands of the azimuthal flow) are well correlated23 and both are very likely powered by the solar dynamo. This will be shown to lead to approximately the correct power to the chromosphere and corona.
Assumption on diffusion as the clock. The resistive magnetic diffusion time (≈μo/ηλ2)15 may control the time of the solar cycle. The solar cycle period is nearly independent of the strength of the cycle,24 and the magnetic diffusion time is independent of the field strength and depends on the temperature, density, and characteristic length scale of the plasma.25 The temperature and density depend on solar convection since the solar convection power is over three orders of magnitude larger26 (≈3 × 1026 W) than the dynamo power (≈1.4 × 1023 W as discussed below) and the solar power convected to the surface and then radiated appears to be constant.26 Thus, the temperature and density are probably constant. The magnetic gradient length of the plasma is the inverse of λinj and depends on the ratio of current to magnetic flux (see Sec. I C), and the dynamo voltage will be shown to be proportional to the magnetic flux. Thus, a constant injector impedance will make λinj constant. For a coaxial injector with electrode spacing that is a small fraction of the plasma radius, the plasma acceleration distance is most likely proportional to the electrode spacing. This leads to an injector impedance depending only on the inverse of the constant solar radius.27 Thus, the magnetic diffusion time is constant, and if the cycle period is proportional to diffusion time, it will also be constant.
Assumption on uniform λinj. The minimum energy principle holds that λ is uniform for closed, perfectly conducting magnetic boundary conditions.14 This is not true when an external circuit enforces a λ boundary condition λinj,15 which is the case for the Sun and the helicity injected torus with steady inductive helicity injection (HIT-SI) experiment.19 On HIT-SI, the simplifying assumption was made that a uniform λinj is on field lines that are driven by the injector when only the average value is enforced and a uniform λself was used for the other part of the equilibrium, giving a step profile. This approximation had fewer fitting parameters than other functions tried and fit the data better.28 Therefore, the step profile approximation can be used for the order of magnitude estimates given in this paper.
Assumption on torque distribution. Assume that thermal convection mass flow distributes the torque from the dynamo down to the depth that motion is observed because the convective turnover times are much shorter than the 11-year period.29 This depth is ≈0.1 R⊙ for both the torsional oscillations23 and the meridional flow,30 where R⊙ is the solar radius. This convective coupling of a large mass for the dynamo current to accelerate prevents the resultant velocity from producing significant back EMF. (The back EMF ≈0.9 MV compared to 27 MV from differential rotation). While the low back EMF makes the simple circuit analysis presented valid, the velocity of the accelerated mass is measured and allows the estimation of the dynamo current that is given in Sec. II C. This effect of large convective flows also prevents radial gradients of the horizontal velocity and justifies the assumption that the rotation velocity is approximated as uniform across the radial direction in the injector magnetic flux of the GMS. This uniform-rotation assumption is used in deriving Eq. (5).
Assumption on the GMS boundary: the unmagnetized plasma can act as a resistive magnetic flux conserver for the GMS magnetic fields which expand into the unmagnetized plasma by resistive diffusion. From analysis of torsional oscillations and meridional flow, given below, the vertical convection stops essentially all horizontal MHD flow. Thus, the massive mass flows from thermal convection prevent plasma motion that is orthogonal to the convective flow. The measured horizontal flow in the supergranulation is considerable31,32 and can provide the vertical equilibrium forces. Thus, it appears that the region in the supergranulation below the watershed region31 stably confines the GMS with high-speed plasma providing the boundary force for a grossly stable MECH state. The supergranulations have a similar topology31 to a smoke ring fluid and to a spheromak magnetic field. The unmagnetized plasma flow and the stable internal MECH state seem to form a resistive boundary for the GMS. In addition to these thermal convection flow effects on the boundary, the internal magnetic structure may also play a role in shaping the GMS as discussed in Sec. II H below.
B. Solar dynamo
Figure 2 shows a poloidal cross section of the injector for the northern hemisphere and it is not to scale. All of the polar magnetic flux (polar magnetic flux is also called injector magnetic flux) is between the coaxial surfaces of the gray lines that pass through the solar surface. The solar dynamo generates voltage between these surfaces that causes helicity and power to be injected into the heliosphere. The distance between these surfaces will be derived to be only ≈105 m. The solar radius is 7 × 108 m. The magnetic flux on the right stretches to beyond the solar system (the earth is 1.5 × 1011 m away).
At this solar minimum, based on the magnetogram in Fig. 3(a),33 the injector magnetic flux enters the GMS in the polar region of the southern hemisphere and leaves the GMS in the polar region of the northern hemisphere. The magnetic flux seems to reach from the pole region to pole region below the solar surface and in the heliosphere. From the dynamo drive geometry, the current is parallel to the magnetic flux in the southern GMS and antiparallel in the northern. Thus, the minimum in 1997 has the current entering the surface at the polar regions and leaving at the equator region.
The measured polar magnetic flux and the amplitude of the torsional oscillations plus the “assumption on axisymmetry,” “uniform λinj,” and “dynamo driving” lead to the characteristic length scale and current of the dynamo. The measured unsigned magnetic flux at the solar minimum in 1997 is given in Ref. 34, and the polar magnetic flux, ψinj (≈3 × 1014 Wb), is one-half of the unsigned magnetic flux given. This “injector magnetic flux” forms poloidal magnetic flux surfaces that connect the differentially rotating plasmas under the solar surface to the heliosphere. This is the solar dynamo and is shown in Fig. 2. The zero magnetic flux surface and the highest magnetic flux surface with a value of ψinj are labeled in Fig. 2. The net solar dynamo EMF exists when the open polar magnetic flux passes through the Sun as in Fig. 2. Using Eq. (1) and Fig. 2 and a nonrotating rest frame at the center of the Sun, the EMF voltage is
The bottom equations are for evaluating the EMF along paths of constant z and constant R directions. ηd includes additional possible dynamic impedance. All of the physics not in the first two terms are lumped into ηd j. The magnetic flux, ψinj, is
Below the solar surface, the plasma velocity, vϕ, is independent of R in the thin injector magnetic flux region by the “assumption on torque distribution” (see Sec. II A). The integral is across the injector magnetic flux region between gray lines in Fig. 2 that penetrate the solar surface. The distance Raxis of the solar axis to this region depends on its z position and at a given z-position is approximated as independent of R since the dynamo region is thin. Since Bz ≈ Bϕ and vz ≪ vϕ, the term with vz is the meridional flow speed. Therefore, at constant z,
where ω is vϕ/Raxis.
Thus, the magnetic flux simply wraps on itself (like early dynamo models35) with little MHD effects, which is made valid because of the “torque distribution.” Therefore, outside the GMS, the plasma velocity is defined by thermal convection. The convection drives a large amount of mass past the GMS before its flow pattern can be changed significantly by MHD. This unmagnetized plasma defines a flux conserver, which contains the GMS. The GMS is probably a stable MECH state because the imposed perturbation causes the excess energy above the minimum to dissipate. The GMS also has little dynamic impedance because stable equilibria tend to be quiet. The GMS can be formed and sustained by helicity injection current drive, which, as discussed in Sec. I C, does not require instability when perturbations are imposed. The GMS stability contributes to the magnetic flux conserver stability because by definition, a stable equilibrium resists deformation. Thus, there is likely little MHD driven dynamic impedance and ηd may be from classical resistivity.
Since ω varies over the solar surfaces, the EMF varies over the surface and this variation drives circulating current in the conducting fluid containing this thin shell of injector magnetic flux. The current flows across injector magnetic flux in the direction of the EMF (taking power from the solar rotation there) where the EMF is higher and against the EMF where it is lower (giving less power back to the rotation here). In the steady-state, on any closed path where E is given in Eq. (1). For example, from Fig. 2(a), it is found that the current path crosses the injector magnetic flux, in the direction of EMF, from the lowest magnetic flux (zero) to the highest magnetic flux (ψinj), in the equator area, where the EMF is the highest. The current arrows cross the entire flux to emphasize that the currents are distributed to keep λ uniform. The current path then leads to the lower EMF places on this high magnetic flux and then crosses the injector magnetic flux, against the EMF, to the low magnetic flux side. The path then leads back to the equatorial area to complete the loop. The net power comes from the solar rotation.
The path of constant R in the z direction gives
When the injector magnetic flux leaves the GMS, it turns to the surface and it leaves the MHD-defeating convective turbulence and the CHI injector turns on. The −∫vR Bϕdz term is from the motion to the solar surface and the toroidal field in the injector magnetic flux. The CHI impedance is the dominant impedance. Thus, this term (−∫vR Bϕdz) is only slightly smaller than δωψinj/2π (the voltage from the differential rotation, δω, between the equator and ≈60° latitude) due to the helicity absorbed by the GMS. The −∫vR Bϕdz term also equals the rate of toroidal magnetic flux injection into the heliosphere. This toroidal magnetic flux links some of the polar magnetic flux, and the magnitude of the helicity injection rate into the heliosphere is ≈Vinjψinj, where Vinj is defined as δωψinj/2π. By inspection of Fig. 2, positive helicity is injected into the southern hemisphere and negative helicity into the northern hemisphere.
C. Torsional oscillations and the injector current
From the “assumption on uniform λinj” and “axisymmetry,” the torque about the solar axis is calculated. The radial current crossing net poloidal flux causes a torque, Γ, driving the torsional oscillations, and the radial current crossing the toroidal flux causes the force driving the meridional flow. The torque, Γ, is found by calculating the toroidal force and multiplying it times the radial distance from the axis,
where Raxis is the radial distance from the current to the axis and (IR × B)ϕ is the force per unit length in the toroidal direction. IR is the current as a function of R. IRo (= Iinj) is the current at the highest flux surface. IR drops as it crosses lower flux surfaces because it supplies current to the injector flux. From the assumption of axial symmetry and Kirchhoff's law, for a constant λinj,
With a change in integration variables, Eq. (7) becomes
The bands of the torsional oscillations have a depth of ≈0.1 R⊙ and solar activity is in the bands with positive acceleration.23 Although the bands oscillate out of phase, their amplitudes and frequencies are of the same order and so the time averaged magnitude of the torque for driving a band is proportional to the band's moment of inertia. Thus, the sum of the time averaged magnitudes of the torques is estimated to be equal to the magnitude of the torque needed to drive all of the mass in the bands to the common oscillation magnitude and frequency. Based on Fig. 3(b), the magnitude of the torque is estimated to be that required to oscillate the solar plasma mass in the volume between zero and 60° latitude and to a depth of 0.1 R⊙ with an amplitude of Ωo/2π equal to 3 nHz and the period of 11 years. Assume that the torque is equal to It.o.αt.o., where It.o. is the moment of inertia of the plasma mass defined (≈5.7 × 1044 kg m2) and αt.o. is the angular acceleration (≈10−16 s−2). Setting this torque equal to the dynamo torque gives an order of magnitude estimate of the dynamo current, Iinj ≈ 2.58 × 1015 A. Then, using μoIinj/ψinj = λinj, the inverse gradient scale length (≈10−5 m−1) of the dynamo is found.
D. The GMS
This very thin MECH structure has the dominate gradients in the radial direction, and so other gradients are ignored; then, using ∇ × B = λGMSB from Sec. I C gives
This leads to an equation for each component
which has the solution
in the GMS. λinj replaces λGMS in the injector magnetic flux.
Thus, B is parallel to the solar surface and rotates in space at the rate of λGMS (radians per unit length) in the radial direction. Ampère's law gives
Using Rflux = R⊙ and Iinj ≈ 2.58 × 1015 A yields Bo ≈ 0.73 T, and at this Bo, the dynamo magnetic flux is about π/2λinj thick.
To test the validity of this discovery of the nature of the dynamo, we must check that it is consistent with other solar observations. An algebra check is the injector magnetic flux,
The value of Iinj will be shown to be consistent with the magnitude of the current-sheet current measured at the Earth. When the injector magnetic flux leaves the GMS, it expands with distance from the GMS but the total polar magnetic flux into an expansion surface is conserved. For simplicity, assume that the injector magnetic flux flares out as it expands and the injector magnetic flux volume increases its characteristic length scale with its distance from the GMS. In the GMS, the injector current is parallel to the injector magnetic flux with λinj = μoIinj/ψinj. Since λ ∼ 1/characteristic length scale, λ drops with the increase in the characteristic length scale. Since the magnetic flux remains ψinj and λ = μoI/ψinj, a λ decrease causes a decrease in the total parallel current, and so
I‖ is the current flow parallel to the polar magnetic flux into the surface of expansion, which decreases with expansion. Kirchhoff's law requires the current returns, and this appears to be the current sheet. At Earth, the current sheet magnetic fields were measured36 and it carries 9 GA. Using λ = π/1 au, Iinj ≈ 2.58 × 1015 A, and λinj = 1 × 10−5 m−1, Eq. (17) gives 5 GA for one hemisphere, which agrees with 9 GA for both.
A dynamo of this current will be shown to have the power needed for the chromosphere, the corona, and the solar wind. The highest angular velocity on the solar surface is at the equator, where the angular velocity is almost uniform from 0 to 30° latitude.37,38 At the equator, the rotational frequency is given by ω/2π ≈ 460 nHz. The frequency difference between the equator and 60° latitude is ≈90 nHz. This gives the voltage across the polar magnetic flux of ≈138 MV and across the highest magnetic flux surface of ≈27 MV. Thus, the power available for solar activity across the highest magnetic flux surface is 27 MV × 2.58 × 1015 A ≈ 7 × 1022 W per hemisphere which will be shown to be enough to power the chromosphere, the corona, and the solar wind. It is believed that ≈2.65 × 1022 W is needed to heat the whole chromosphere.39 Heating the quiet corona takes ≈2 × 1021 W,40 and there is ≈1021 W in the solar wind.41,42
From helicity and magnetic flux conservation, the steady-state rate of helicity passing through surfaces of expansion is constant for all distances from the source. Thus, the power passing through the surface is proportional to λ. Using λ = π/s, where s is the distance from expansion point, gives
where Pinj is the power injected by the coaxial helicity injector, sinj = π/λinj, Pinjsinj = 2.2 × 1028 Wm, and Ps is the power through the expansion surface of one injector a distance s from the injector. Assuming the expansion enters the chromosphere at s = 1.3 Mm with 1.7 × 1022 W and leaves at s = 3.8 Mm with 5.8 × 1021 W gives 1.5 × 1022 W to the chromosphere and 5.8 × 1021 W to the corona and beyond, which are of the same order as given above. Two injectors will give 3 × 1022 W to the chromosphere compared to 2.65 × 1022 W and 1.16 × 1022 W to the corona and solar wind compared to 3 × 1021 W. They are of the right order.
E. Meridional flow
The torsional oscillations are driven by the radial current crossing the net polar magnetic flux [See Eq. (7)]. (The current can cross the magnetic flux because MHD is not active as discussed in “assumption 7.”) The meridional flow is driven by the radial current crossing net toroidal magnetic flux, which exists in the injector flux as shown in Fig. 2. With a thickness of π/2λinj, the magnetic energy in polar flux comes equally from the magnetic component in the toroidal direction and from the magnetic component in the poloidal direction. Therefore, the meridional flow speeds would be expected to be of the same order as torsional oscillations speeds, as is observed.43 The EMF for 0–30° latitude is nearly constant,38 and Fig. 3, year 1996, of Ref. 45 shows the meridional velocity increasing from 0 at 30° latitude. Therefore, positive radial current is shown in Fig. 2. Thus, Iinj parallel to the polar magnetic flux is maximum at 30°. The meridional flow acceleration from 0° to 30° can be estimated by assuming that the radial current density is uniform and the acceleration is uniform from 0° to 30°,
where M is It.o./(0.9R⊙)2 = 1.43 × 1027 kg (for comparing to torsional oscillations) and the force is 9.78 × 1019 N. The velocity as a function of latitude is
where s30 is the distance from 0 in the range from 0 to 30° (s30 = (degrees latitude)R⊙π/180). Eq. (20) gives a speed at 30° of ≈7 ms−1, which agrees within the resolution of this analysis, and the curve shape agrees with Fig. 3 of Ref. 43. Above 30°, the EMF drops and the negative radial current slows the flow. It is not surprising that the solar activity starts near the current maximum.
F. The 11 year period of the solar cycle and filaments and prominences
The solar cycle period is nearly independent of the strength of the cycle24 and the magnetic diffusion time is independent of the field strength.25 From the “assumption on diffusion as the clock,” the diffusion time needs to be ≈11 years. A 5-year diffusion time (=μo/ηλinj2) gives η ≈ 7 × 10−5 Ωm, and this is the value at ≈0.8 Mm below the surface. The pressure there is that of the magnetic pressure of ≈7.3 kG, which is a little more than sunspots. From the “assumption on GMS boundary,” at the end of a cycle, the GMS returns its initial conditions where only the dynamo-driven polar current and the polar magnetic flux are confined by the un-magnetized solar plasma. After this beginning, the confining eddy currents will diffuse into the unmagnetized solar plasma making the GMS thicker. The current in the dynamo-driven magnetic flux is determined by the external impedance and the current remains the same. The turbulence imposed by thermal convection keeps the growing equilibrium in the stable MECH state having a λ up to λinj. The lobe part of the GMS in Fig. 2 fills the space created by diffusion. If the initial GMS is π/2λinj thick, in ≈5 years, the GMS thickness will have grown to ≈2.57/λinj. When the region is less than π/λinj thick, the equilibrium is stable because π/λinj is the height of the lowest eigenmode. When the space gets thicker, from the “assumption on self-organization,” the MECH state will change topology to include the lowest eigenmode, perhaps causing a major reconnection event. [In the HIT-SI experiment, a similar event forms the spheromak.22] This probably triggers the solar cycle activity and all but the bottom of the GMS escapes to the solar surface each 11 years. As discussed in Sec. II E, the injector current appears to be highest at 30° latitude; thus, π/λinj is the smallest, resulting in the earliest activity trigger, as observed. The escaping thin sheets of the GMS plasma probably make filaments and prominences. The bottom is higher density and better conducting which may prevent the magnetic fields from escaping. Un-magnetized, high-speed, convection-driven, solar plasma moves in and surrounds this remaining open magnetic flux and the next cycle begins. This reversed magnetic flux will make all of the solar fields reversed in the next cycle as observed. The red magnetic flux now stores helicity in a growing gray lobe below it. [The plasma resistivity is estimated using Spitzer resistivity25 for the plasma plus a constant-cross-section () correction for electron collisions with neutrals.44 The Saha equation is used to find the percentage ionized. Estimates of solar properties are from Ref. 45.]
G. Source of helicity for the GMS and symmetric butterfly sunspot patterns
To build up the GMS during a solar minimum, the polar magnetic flux needs to reach from the polar region to polar region so that the negative and positive helicity can come from a common area below the equator where the polar magnetic flux crosses the equatorial plane. Since helicity cannot be injected through the conducting solar plasma and the CHI only ejects helicity out of the solar surface in the polar region into the heliosphere, this common area is the only source of helicity for each hemisphere during this time when the only open flux is in the polar region. Near the end of the cycle, the half of the red lobe nearest the surface in Fig. 2 may expand into the heliosphere in both hemispheres. They have less magnetic energy if they reconnect at the equator and expand as one flux surface. The north and south can reconnect above the equator, pull under the solar surface, and form pole to pole magnetic flux similar to earlier dynamo models.35 However, with a very shallow and very thin dynamo, this reconnection can occur in a resistive, small-characteristic-length-scale region near the solar surface and probably not requiring fast reconnection. However, the timing of the expansion must occur at the same time and toroidal location. This time, shown in Fig. 3, seems to exist about one half the time. About 160 solar revolutions occur in the cycle. If the rotation rate is different in the two hemispheres by 1 part in 320, there would be no common regeneration time at the end of the next cycle. The maximum regeneration occurs at the time of the average of the two cycles and is the same time in both hemispheres. This keeps the north and south synchronized for symmetric butterfly sunspot patterns.24 The maximum regeneration happens when the two cycle times are equal. The loss of this synchronization appears to have contributed to the recent decline in solar activity.
H. Supergranulation and the GMS
The inferred thickness of the GMS corresponds to the pressure scale-height in the shallow solar convection zone, or equivalently, the convective eddy width (to within some order unity factor) according to mixing length theory. Over time, the GMS tends toward a minimum energy state subject to several constraints, previously described in Sec. I C. To the lowest order, the radial force balance is described by the measured hydrostatic equilibrium. However, for any convection zone magnetic structure (not just the proposed GMS structure) to persist long enough to nontrivially affect solar dynamics, the magnetic field must be large enough to resist or stabilize the convective eddies. To satisfy these two additional constraints, the central region of the GMS contains a magnetic force-free equilibrium overlaid onto the hydrostatic equilibrium, surrounded by thin (≈the skin depth in the plasma for a frequency with a period equal the time the passing plasma contacts the GMS.) edge current sheets on the top and bottom.
Such a structure is wholly consistent with observations of magnetic self-organization in both laboratory and astrophysical plasmas, where a comparatively large region in magnetic force-free equilibrium is surrounded and supported by thin edge currents.28,46 By satisfying the magnetic and hydrostatic equilibria separately, we can ignore the transient, dynamic flows that would accompany changes in magnetic topology—particularly, the dissolution of such a magnetic structure—in more general equilibria. From our inferred thickness of the entire GMS, the edge current sheets would necessarily be much thinner than the convective eddy characteristic length scale, which again is only possible in a structure that persists for more than a few convective turnover times if the magnetic field is large enough to locally stabilize the convective turbulence.
The superficial discrepancies in the energy and force balances are reconciled by the GMS fragmentation, which is consistent with current understanding of supergranulation. The fast-moving large mass convective flow also prevents boundary motion. To allow the convective heat flux to pass through the GMS, a convective upflow punctures a hole through the thin magnetic structure. If the distance between holes is much larger than the thickness of the GMS, the hole will not significantly affect the characteristic scale length (1/λ) of the magnetic equilibrium, which is its energy per unit helicity. Thus, large spaces and holes do not take energy to form while small ones would. Therefore, the characteristic length scale of each hole must be much larger than the thickness of the GMS for sufficient energy flux through the layer. For the magnetic field to so influence the convection in this region suggests that beta (Beta is defined as the plasma pressure normalized by the magnetic pressure.) is of order one as it is, supporting this interpretation of the data.
Downflows balance the radial mass magnetic flux and establish horizontal flows across the top of the GMS. The GMS shields the solar plasma above from the convective heat magnetic flux below, causing it to preferentially transfer heat upward. Consequently, the plasma above the GMS cools slightly relative to the plasma in the upflow of the hole, which reduces its thermal buoyancy enough to counterbalance the magnetic buoyancy in the GMS. The temperature decreases with distance from the holes, and so the local downward force density is the greatest along curves between neighboring holes. For similarly characteristic length scaled holes that occur approximately equidistant from each other, this would form the characteristic honeycomb shape of the downflow lanes between supergranules. Thus, each supergranule would be a topologically toroidal piece of the overall GMS.
In summary, the data seem to be revealing the following: the momentum flow required by the solar thermal convection prevents MHD driven velocities in most of the convective zone. The differential flow of the solar surface is driven by gravitational and convective flow still produces an EMF in the presence of polar magnetic flux and the plasma is a resistive conductor (E = ηj). At the top of the convective zone, the convection drops as radiation begins increasing and a transition to MHD active plasma occurs. In this region, horizontal flows are comparable to the vertical flows and the up-across-down motions may produce stagnant volumes, which are MHD active. This transition allows the plasma flowing through and flowing outside of the supergranules to act as a resistive magnetic flux conserver while the plasma inside, which is below the watershed region, is MHD active and forms the GMS. The MHD activity is limited to the horizontal scale characteristic length scale of the supergranulation and smaller. Assumption six is still valid for the large-scale solar dynamo.
I. Dynamo action during active times
Dynamo action also occurs during times of solar activity. Dynamo drive occurs any time flux enters and leaves the Sun at locations of different rotational rates. (However, dynamos wholly in one hemisphere generate negative and positive helicity causing rogue pockets of helicity of the opposite sign of most of the helicity of the hemisphere.) In Fig. 2 at the point on the magnetic flux surface with the highest EMF, the dynamo drives current in the direction of the EMF, giving power out and slowing the solar rotation speed. At points of low EMF, the CHI current is driven against the EMF, absorbing power and accelerating the surface speed or accelerating plasma out the solar surface. The EMF might be different because the magnetic flux might be different. Thus, the near-equatorial opening areas in the magnetograms of Fig. 3(a) have high EMF and negative acceleration in the torsional oscillations of Fig. 3(b). White lines mark the dynamo-driven region at three different times. For example, the two single-hemisphere dynamos of Fig. 3(a) (≈year 2005) are shown by the gray areas in Fig. 3(a) and negative acceleration in Fig. 3(b). The active areas have low EMF, and blue and yellow areas of Fig. 3(a) have positive acceleration in Fig. 3(b).
J. Sunspots, CME (Coronal Mass Ejection), and flares
Sunspots have too much magnetic flux to be formed from the local magnetic fields of the thin GMS. They are more likely simply formed in place by helicity injection from the solar dynamo and the rest of the GMS. Figure 4(a) shows a cross section of a ribbon of the GMS separated from the GMS by reconnection. The ribbon is up to 50 Mm wide in the longitudinal direction and the same length as the dynamo, ≈R⊙, in the latitudinal direction. A longitudinal cross section is shown. At the beginning of the cycle, the polar magnetic flux can be thought of as the remaining part of the upper (gray) lobe that has partially escaped through the solar surface. The gray ⊙s represent this polar magnetic flux below the solar surface that will escape the Sun during this cycle. The red ⊙s represent the top of the lower lobe, which will become the new polar magnetic flux in the heliosphere. The red X symbols represent the lower part of the lower lobe, which will become the polar magnetic flux below the solar surface.
Sunspot formation can occur as follows: as the GMS expands upward the mass above, the GMS may become too small for stability or the convection-driven lateral mass flow of the un-magnetized plasma may become insufficient to act like a magnetic flux conserver to confine the GMS. The mass above is pushed away, and the GMS becomes locally thicker, causing a local reduction of λself which causes more helicity to flow into the region. The helicity flowing in releases energy from the decrease in the minimum energy required for the conserved helicity. The minimum magnetic energy moves in with the helicity and raises the magnetic pressure that speeds the thickening. Thus, it is unstable. This region of the GMS expands from this inflow of magnetic flux and current that links the polar magnetic flux channel, producing a “flat torus” that is threaded by the polar magnetic flux. See Fig. 4(b). This is very similar to the well-known sausage instability.
The gold lines in Fig. 4 represent the very resistive photosphere. Faculae could be arcing when the magnetic flux of the sunspot passes through the photosphere, which is represented by a jagged solar surface. The poloidal fields completely dissipate, and the toroidal field of the flat torus partially escapes but cannot leave because it links the polar magnetic flux. The trapped toroidal field enters and leaves the solar surface at the location of the resistive gap as in Fig. 4(c). All of the magnetic flux in the ribbon linking this polar magnetic flux may become available to the sunspot on the Alfvén time scale (≈a day). Thus, a large volume of high magnetic field can appear in a few days. The volume of the sausage is about 20 × 20 × 20 Mm3, and the ribbon volume is 700 × 0.15 × 50 Mm3 of the same order.
The linking magnetic flux is of order the polar magnetic flux, ≈3 × 1014 Wb, in agreement with the maximum of sunspot groups, ≈2 × 1014 Wb.47 Thus, in this interpretation, the sunspots are formed at the footprints of the escaped toroidal field of the flat torus. The helicity and plasma that flow to the low-λ region of the sunspot will be pushed downward by gravity giving the powerful converging (bringing in the helicity) and downward directed flows (making room for the thick sunspot) observed.48 The angle of sunspots can vary because the angle of the resistive gap can vary. While the shortest path is longitudinal, the gap will also tend to follow the magnetic field, which is that of the toroidal field of the flat torus. The dynamo-driven current, aligned with the polar magnetic flux, produces a magnetic field component in the latitudinal direction. As discussed above, this current increases from zero at the equator and reaches a maximum about where the sunspot cycle begins (≈30° latitude). Thus, the angle of the total field of the flat torus increases and the angle of the gap and sunspots increases up to 30° latitude as observed in Joy's law.
The toroidal field is not visible because it was stripped of its plasma as it passed through the insulating photosphere. The plasma escaping from the failing magnetic flux conserver may be observed as the Evershed clouds.49 They have a speed like the sound speed of the surface plasma; they vent at the edge of the magnetic flux and in the direction of the magnetic flux in the edge as might be expected. This venting would cause expansion cooling of the region viewed through the sunspot. The sunspot lasts until the old polar magnetic flux that is trapping the sunspot leaves the Sun, giving a large variation in the sunspot lifetime as observed.
If the polar magnetic flux is not strong enough to hold the flat torus in a sunspot, the flat torus will escape the surface perhaps as a CME as in Fig. 4(d). The magnetic fields in CMEs have this structure,50 and CMEs and flares tend to appear from the same event. The polar magnetic flux threads the sausage magnetic flux, and the magnetic flux going in and coming out make the two-ribbon flare commonly observed with CME.47 CMEs have produced up to 15 nT of magnetic field at the earth51 and have up to 1014 Wb of magnetic flux.47 The maximum magnetic flux of the sunspots and of the CMEs is of order the total solar polar magnetic flux, confirming this interpretation of sunspot and CME data.
Using the principles of self-organization of magnetized plasma, selected solar data are analyzed. The nature of the axisymmetric solar dynamo and its role of driving all other solar magnetic activity are revealed for the first time. The measured amount of polar magnetic flux and the torque required to drive the torsional oscillations show that the magnetic gradient scale length is ≈100 km and the current is ≈2.6 × 1015 A. Estimating the magnetic field strength from the current, the dynamo radial height required for the amount of polar magnetic flux is also ≈150 km, which is about 0.0002 of a solar radius. To test the validity of this discovery of the nature of the dynamo, consistency with the other solar observations is checked. From the polar magnetic flux and the rotation profile of the Sun, the dynamo voltage is estimated. From the voltage and current, the dynamo power is estimated to be enough to power the chromosphere, the corona, and the solar wind.
For this thin dynamo, the resistive diffusion time might be part of the 11-year clock. There are many ways this might work and more data are needed to determine the exact mechanism. The simple one discussed begins with the dynamo confined by unmagnetized solar plasma. Resistive diffusion causes the region to become thicker and the dynamo injects helicity which turbulence keeps in the stable minimum energy state. The minimum energy state is a uniform magnitude magnetic field parallel to the surface that rotates with position in the thin radial direction. The sheets have holes through them and slots through them, named lanes. The holes and lanes are probably pushed through the magnetic structure by solar thermal convective flow, allowing the solar heat through this supergranulation. When the sheet thickness exceeds π × 100 km, the minimum energy state changes topology perhaps triggering solar activity that releases most of the magnetic fields. The trigger should happen the earliest at 30° latitude as observed. When the sheets reach the surface, they become filaments and prominences. The bottom part of the original structure remains, and unmagnetized solar plasma flows in and confines it and the new cycle starts. However, the sign of the magnetic flux reverses causing the observed flipping of all magnetic fields every 11 years.
The sunspots are from some magnetic flux from sausage-like instabilities that are trapped under the surface and that stretch into the heliosphere. A CME occurs when the sausage escapes out of the surface and remains intact. This pulls out the dynamo-driven polar magnetic flux, causing a flare, and pulls out the mass in the sausage, causing a CME. These explanations are consistent with the observation that the maximum magnetic flux in sunspots and CMEs are the same order as the total solar polar magnetic flux and double ribbon flares often occur with CMEs.
Finally, discovery of the nature of the dynamo and the global magnetic structure has allowed a new paradigm for understanding of most solar magnetic activity, some of which has not been well understood. This is not surprising since, unlike other interpretation of the solar data, the dynamo is a closely coupled driver of solar magnetic activity. This may be a breakthrough for solar magnetic research.
IV. RECOMMENDATION FOR FUTURE RESEARCH
The following bullets are recommendations for future research:
Examine the Sun with the Hubble telescope at maximum resolution at the CIV (carbon four line) at 160 nm. This should have a resolution of ≈10 km. This wavelength might have enough power at the high densities of 1 Mm deep to tolerate a few e-foldings of attenuation. The present resolution of ≈2 Mm will not see magnetic fields of the GMS that changes sign every ≈0.3 Mm.
Take magneto-grams from the poles to nail down the polar magnetic flux values and patterns.
Re-examine all the solar data with the paradigm presented here. Now is a good time to re-examine the old data since the solar activity is low. The solar scientists who took the data are the best at this. The torsional oscillations and the magnetograms contain a great deal of detailed information about the dynamo.
Measure the autocorrelations of different depths of the torsional oscillations. The shallower should lead the deeper oscillations.
Try to model the Sun with computational fluid dynamics (CFD) and a diaphragm with the supergranulation hole pattern and depth.
Fit the rotation data of Ref. 38 to a function that allow steeper gradients in the 50°–60° region, which are the steepest now. If the CHI injector almost took all the power, the rotation would have a step function shape with the step at the injector location.
The authors wish to thank many solar scientists for the highest quality data and Dr. Greg Kopp for help in estimating the dynamo power from the total solar irradiance data. The authors thank the reviewers for constructive criticism. This work was supported by the U.S. Department of Energy Office of Science, Office of Fusion Energy Sciences under Award Nos. DE-FG02-96ER54361 and DE-SC0016256.
APPENDIX: NEW PHENOMENA AND NEW UNDERSTANDING
A short discussion of observed features of the solar phenomena discussed in this paper is given. The superscript N marks a new feature discovered in this study, and an F is for a known feature first explained by the process from this study. A new understanding is given to virtually all phenomena.
CHI injectorsN are coaxial helicity injectors that inject helicity, power, and plasma into the heliosphere. The voltage across the injector magnetic flux gives the helicity injection. Understanding the nature of the plasma from the injectors is a topic of research.
CMEF is a coronal mass ejection which is a blob of plasma and magnetic field that escapes the Sun. The magnetic structure is linked magnetic fields like a spheromak. The total magnetic flux of a CME can also be on the order of the total solar polar magnetic flux. It appears to be due to a sausage instability in the GMS that is propelled into the heliosphere.
The current sheetF is a symmetric sheet of current at the solar equator going out beyond the Earth's orbit. It appears to be the return current for some of the injector current.
Filaments and prominencesF are large thin sheets of plasma and magnetic field that come off the surface during active solar periods. Filaments have a solar background and prominences have the blackness of space as the background. They appear to be remnants of the layers of the thin GMS that lift off in the 11-year cycle.
Flares are electromagnetic explosions followed by plasma ejection from the solar surface. They appear to be from the surfacing of dynamo driven polar magnetic flux.
The global magnetic structure (GMS)N is a grossly stable thin sheet of plasma and magnetic field that covers most of the Sun. At the solar minimum, the sheet covering the northern hemisphere and the sheet covering the southern hemisphere are minimum energy states conserving helicity (MECH states) sustained by the solar dynamo which is part of the GMS. The helicity is negative in the northern hemisphere and positive in the southern hemisphere. The GMS topology depends on the thickness of the volume that bounds the GMS relative to the magnetic gradient scale length of the dynamo. They appear to be near the top of the convection zone where horizontal convection and vertical convection plus the stable magnetic structures make the GMS possible.
Injector Magnetic Flux is the magnetic flux that passes through a solar dynamo. At the solar minimum, it is assumed to also be the polar magnetic flux.
Meridional flowF is the flow in the meridian direction from the surface to about 0.1 R⊙ deep. The direction of the flow is from the equator to the poles. The acceleration is well correlated with solar magnetic activity. The speed varies with the solar cycle up to about 20 m/s.30 The radial current crossing the toroidal component of the polar the magnetic flux of the GMS confirms the structure of the GMS.
Polar Magnetic flux is the net magnetic flux that enters one solar hemisphere and exits the other hemisphere.
Sausage instabilityN is one the most common instabilities of a current channel where the magnetic flux produced by the channel accumulates in some locations and depletes in others. The channel becomes alternating fat and thin like a string of sausage. For the CME, there are one or two fat locations, and for the CME, the thin areas are the associated double-ribbon flares. They appear to be the first phase of CMEs and sunspots.
The solar 11-years cycleF starts and ends with low magnetic activity with a high activity between. The cycle is about eleven years, and all magnetic fields change sign from one cycle to the next. It appears that the GMS that has a magnetic diffusion time of about 11 years is independent of the magnetic field strength.
The solar dynamo generates power and helicity by wrapping the polar magnetic flux on itself because of the faster rotation of the equator. The wrapping generates positive helicity in the southern hemisphere and negative in the northern hemisphere. The wrapping also generates a voltage and current that appears to power all of the solar magnetic activity.
SunspotsF are the appearance of black spots on the Sun. They often come in pairs with the magnetic field going in one and out the other and bright lines appear in a direction of a line between the pair called faculae. The total magnetic flux of a sunspot cluster can be as high as the total polar magnetic flux. Previously, they were thought to be from the magnetic field generated near the tachocline. Magnetic flux ropes get to the surface as small structures that then stop and fast reconnection releases the field. They now appear to be sausages instabilities that occur in the GMS and grow through the solar surface.
SupergranulationF is a shallow region near the surface where the convection horizontal characteristic length scale changes from being ≈0.5 Mm to being between 5.0 Mm and 50 Mm and a large area called the watershed does not exhibit convection. They appear to be caused by the convective flow pattern through the GMS.
Torsional oscillationsF are azimuthally symmetric bands of azimuthal solar motion from the surface to about 0.1 R⊙ deep where R⊙ is the solar radius. The acceleration of the bands is correlated with solar magnetic activity. The speed varies with the solar cycle up to about 20 m/s. They appear to be caused by the radial current crossing the polar magnetic flux. They are possible because thermal convection distributes this force over a large mass that greatly limits the velocity and back EMF.