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In this work, we systematically investigate the scale-dependent angular momentum flux by analysing high-resolution three-dimensional magnetohydrodynamic simulations in which the solar-like differential rotation is reproduced without using any manipulations. More specifically, the magnetic angular momentum transport (AMT) plays a dominant role in the calculations. We examine the important spatial scales for the magnetic AMT. The main conclusions of our approach can be summarized as follows: 1. Turbulence transports the angular momentum radially inward. This effect is more pronounced in the highest resolution calculation. 2. The dominant scale for the magnetic AMT is the smallest spatial scale. 3. The dimensionless magnetic correlation is low in the high-resolution simulation. Thus, chaotic but strong small-scale magnetic fields achieve efficient magnetic AMT.

The physical connection between thermal convection in the solar interior and the solar wind remains unclear due to their significant scale separation. Using an extended version of the three-dimensional radiative magnetohydrodynamic code RAMENS, we perform the first comprehensive simulation of the solar wind formation, starting from the wave excitation and the small-scale dynamo below the photosphere. The simulation satisfies various observational constraints as a slow solar wind emanating from the coronal hole boundary. The magnetic energy is persistently released in the simulated corona, showing a hot upward flow at the interface between open and closed fields. To evaluate the energetic contributions from Alfv\'en wave and interchange reconnection, we develop a new method to quantify the cross-field energy transport in the simulated atmosphere. The measured energy transport from closed coronal loops to open field accounts for approximately half of the total. These findings suggest a significant role of the supergranular-scale interchange reconnection in solar wind formation., Comment: 10 pages, 5 figures, accepted for publication in ApJL

We investigate the dependence of the angular momentum transport (AMT) on the spatial scales with numerical simulation of solar-like stars. It is thought that turbulence has an essential role in constructing solar differential rotation (DR). In a widely used method to analyse the construction mechanism of DR, the flow is divided into two components, `mean flow' and `turbulence', where `turbulence' includes a broad spectrum of spatial scales. The features of the AMT are expected to depend on the scale. In this study, we decompose the angular momentum flux (AMF) to investigate the dependence of the AMF on the spatial scale. We compare the results with anti-solar- (fast pole) and solar-type (fast equator) DR. Our conclusions are summarized as 1. Radially outward AMT is seen on a large scale (60 Mm < L <120 Mm) in rotationally constrained systems. 2. Even when the scale-integrated AMF is negative, we sometimes observe positive AMF on certain scales. 3. Small-scale turbulence tends to transport the angular momentum radially inward and causes the anti-solar DR, indicating that high-resolution simulation is a negative factor for solar-like DR. Our method to decompose the AMF provides a deep understanding of the angular momentum and construction mechanism of DR.

We analyze the simulation result shown in Hotta & Kusano, 2021 in which the solar-like differential rotation is reproduced. The Sun is rotating differentially with the fast equator and the slow pole. It is widely thought that the thermal convection maintains the differential rotation, but recent high-resolution simulations tend to fail to reproduce the fast equator. This fact is an aspect of one of the biggest problems in solar physics called the convective conundrum. Hotta & Kusano, 2021 succeed in reproducing the solar-like differential rotation without using any manipulation with unprecedentedly high-resolution simulation. In this study, we analyze the simulation data to understand the maintenance mechanism of the fast equator. Our analyses lead to conclusions that are summarized as follows. 1. Superequipatition magnetic field is generated by the compression, which can indirectly convert the massive internal energy to magnetic energy. 2. The efficient small-scale energy transport suppresses large-scale convection energy. 3. Non-Taylor--Proudman differential rotation is maintained by the entropy gradient caused by the anisotropic latitudinal energy transport enhanced by the magnetic field. 4. The fast equator is maintained by the meridional flow mainly caused by the Maxwell stress. The Maxwell stress itself also has a role in the angular momentum transport for fast near-surface equator (we call it the Punching ball effect). The fast equator in the simulation is reproduced not due to the low Rossby number regime but due to the strong magnetic field. This study newly finds the role of the magnetic field in the maintenance of differential rotation., Comment: 39 pages, 37 figures, 1 table, published in ApJ

The Sun rotates differentially with a fast equator and slow pole. Convection in the solar interior is thought to maintain the differential rotation. However, although many numerical simulations have been conducted to reproduce the solar differential rotation, previous high-resolution calculations with solar parameters fall into the anti-solar (fast pole) differential rotation regime. Consequently, we still do not know the true reason why the Sun has a fast-rotating equator. While the construction of the fast equator requires a strong rotational influence on the convection, the previous calculations have not been able to achieve the situation without any manipulations. The problem is called convective conundrum. The convection and the differential rotation in numerical simulations were different from the observations. Here, we show that a high-resolution calculation succeeds in reproducing the solar-like differential rotation. Our calculations indicate that the strong magnetic field generated by a small-scale dynamo has a significant impact on thermal convection. The successful reproduction of the differential rotation, convection, and magnetic field achieved in our calculation is an essential step to understanding the cause of the most basic nature of solar activity, specifically, the 11-year cycle of sunspot activity., Comment: 11 pages, 4 figures, published in Nature Astronomy (2021)

The nuclear spin-lattice relaxation time ($T_1$) of lanthanum and aluminum nuclei in a single crystal of lanthanum aluminate doped with neodymium ions is studied to estimate the feasibility of the dynamically polarized lanthanum target applicable to beam experiments. The application of our interest is the study of fundamental discrete symmetries in the spin optics of epithermal neutrons. This study requires a highly flexible choice of the applied magnetic field for neutron spin control and favors longer $T_1$ under lower magnetic field and at higher temperature. The $T_1$ of $^{139}{\rm La}$ and ${}^{27}{\rm Al}$ was measured under magnetic fields of $0.5$-$2.5$ T and at temperatures of $0.1$-$1.5$ K and found widely distributed up to 100 h. The result suggests that the $T_1$ can be as long as $T_1 \sim$ 1 h at $0.1$ K with a magnetic field of $0.1$ T, which partially fulfills the requirement of the neutron beam experiment. Possible improvements to achieve a longer $T_1$ are discussed., Comment: 10 pages, 8 figures

We perform radiative magnetohydrodynamic calculations for the solar quiet region to investigate the dependence of statistical flow on magnetic properties and the three-dimensional (3D) structure of magnetic patches in the presence of large-scale flow that mimics differential rotation. It has been confirmed that strong magnetic field patches move faster in the longitudinal direction at the solar surface. Consequently, strong magnetic patches penetrate deeper into the solar interior. The motion of the deep-rooted magnetic patches is influenced by the faster differential rotation in the deeper layer. In this study, we perform realistic radiative magnetohydrodynamic calculations using R2D2 code to validate that stronger patches have deeper roots. We also add large-scale flow to mimic the differential rotation. The magnetic patches are automatically detected and tracked, and we evaluate the depth of 30,000 magnetic patches. The velocities of 2.9 million magnetic patches are then measured at the photosphere. We obtain the dependence of these values on the magnetic properties, such as field strength and flux. Our results confirm that strong magnetic patches tend to show deeper roots and faster movement, and we compare our results with observations using the point spread function of instruments at the Hinode and Solar Dynamics Observatory (SDO). Our result is quantitatively consistent with previous observational results of the SDO., Comment: 7 pages, 9 figures, accepted for publication in MNRAS

We perform a series of radiative magnetohydrodynamic simulations to understand the amplification mechanism of the exceptionally strong horizontal magnetic field in delta-type sunspots. In the simulations, we succeed in reproducing the delta-type sunspot and resulting strong magnetic field exceeding 6000 G in a light bridge between the positive and negative polarities. Our conclusions in this study are summarized as follows: 1. The essential amplification mechanism of the strong horizontal magnetic field is the shear motion caused by the rotation of two spots. 2. The strong horizontal magnetic field remains the force-free state. 3. The peak strength of the magnetic fields does not depend on the spatial resolution, top boundary condition, or Alfven speed limit. The origin of the rotating motion is rooted in the deep convection zone. Therefore, the magnetic field in the delta-spot light bridge can be amplified to the superequipartition values in the photosphere., Comment: 12 pages, 17 figures, 1 table, accepted for publication by MNRAS

We investigate the rising flux tube and the formation of sunspots in an unprecedentedly deep computational domain that covers the whole convection zone with a radiative magnetohydrodynamics simulation. Previous calculations had shallow computational boxes (< 30 Mm) and convection zones at a depth of 200 Mm. By using our new numerical code R2D2, we succeed in covering the whole convection zone and reproduce the formation of the sunspot from a simple horizontal flux tube because of the turbulent thermal convection. The main findings are (1) The rising speed of the flux tube is larger than the upward convection velocity because of the low density caused by the magnetic pressure and the suppression of the mixing. (2) The rising speed of the flux tube exceeds 250 m/s at a depth of 18 Mm, while we do not see any clear evidence of the divergent flow 3 hr before the emergence at the solar surface. (3) Initially, the root of the flux tube is filled with the downflows and then the upflow fills the center of the flux tube during the formation of the sunspot. (4) The essential mechanisms for the formation of the sunspot are the coherent inflow and the turbulent transport. (5) The low-temperature region is extended to a depth of at least 40 Mm in the matured sunspot, with the high-temperature region in the center of the flux tube. Some of the findings indicate the importance of the deep computational domain for the flux emergence simulations., Comment: 17 pages, 24 figures. Accepted for publication by MNRAS

The morphological asymmetry of leading and following sunspots is a well-known characteristic of the solar surface. In the context of large-scale evolution of the surface magnetic field, the asymmetry has been assumed to have only a negligible effect. Using the surface flux transport model, we show that the morphological asymmetry of leading and following sunspots has a significant impact on the evolution of the large-scale magnetic field on the solar surface. By evaluating the effect of the morphological asymmetry of each bipolar magnetic region (BMR), we observe that the introduction of the asymmetry in the BMR model significantly reduces its contribution to the polar magnetic field, especially for large and high-latitude BMRs. Strongly asymmetric BMRs can even reverse the regular polar field formation. The surface flux transport simulations based on the observed sunspot record shows that the introduction of the morphological asymmetry reduces the root-mean-square difference from the observed axial dipole strength by 30--40 percent. These results indicate that the morphological asymmetry of leading and following sunspots has a significant effect on the solar cycle prediction., Comment: 21 pages, 11 figures, accepted for publication in ApJ

The reduced speed of sound technique (RSST) has been used for efficient simulation of low Mach number flows in solar and stellar convection zones. The basic RSST equations are hyperbolic, and are suitable for parallel computation by domain decomposition. The application of RSST is limited to cases where density perturbations are much smaller than the background density. In addition, non-conservative variables are required to be evolved using this method, which is not suitable in cases where discontinuities like shock waves co-exist in a single numerical domain. In this study, we suggest a new semi-conservative formulation of the RSST that can be applied to low Mach number flows with large density variations. We derive the wave speed of the original and newly suggested methods to clarify that these methods can reduce the speed of sound without affecting the entropy wave. The equations are implemented using the finite volume method. Several numerical tests are carried out to verify the suggested methods. The analysis and numerical results show that the original RSST is not applicable when mass density variations are large. In contrast, the newly suggested methods are found to be efficient in such cases. We also suggest variants of the RSST that conserve momentum in the machine precision. The newly suggested variants are formulated as semi-conservative equations, which reduce to the conservative form of the Euler equations when the speed of sound is not reduced. This property is advantageous when both high and low Mach number regions are included in the numerical domain. The newly suggested forms of RSST can be applied to a wider range of low Mach number flows., Comment: 12 pages, 10 figures, accepted for publication in Astronomy & Astrophysics

The differentially rotating outer layers of stars are thought to play a role in driving their magnetic activity, but the underlying mechanisms that generate and sustain differential rotation are poorly understood. We report the measurement of latitudinal differential rotation in the convection zones of 40 Sun-like stars using asteroseismology. For the most significant detections, the stars' equators rotate approximately twice as fast as their mid-latitudes. The latitudinal shear inferred from asteroseismology is much larger than predictions from numerical simulations., Comment: 45 pages, 11 figures, 4 tables, published in Science

We carry out high-resolution calculations for the stellar convection zone. The main purpose of this study is to investigate the effect of a small-scale dynamo on the differential rotation. The solar differential rotation deviates from the Taylor-Proudman state in which the angular velocity does not change along the rotational axis. To break the Taylor-Proudman state deep in the convection zone, it is thought that a latitudinal entropy gradient is required. In this study, we find that the small-scale dynamo has three roles in the deviation of the stellar differential rotation from the Taylor-Proudman state. 1. The shear of the angular velocity is suppressed. This leads to a situation where the latitudinal entropy gradient efficiently breaks the Taylor-Proudman state. 2. The perturbation of the entropy is increased with suppressing the turbulent velocity between up- and downflows. 3. The convection velocity is reduced. This increases the effect of the rotation on the convection. The second and third factors increase the latitudinal entropy gradient and break the Taylor-Proudman state. We find that the efficient small-scale dynamo has a significant impact on the stellar differential rotation., Comment: 18 pages, 5 figures, accepted for publication in The Astrophysical Journal Letters

It has recently been recognized that the convective velocities achieved in the current solar convection simulations might be over-estimated. The newly-revealed effects of the prevailing small-scale magnetic field within the convection zone may offer possible solutions to this problem. The small-scale magnetic fields can reduce the convective amplitude of small-scale motions through the Lorentz-force feedback, which concurrently inhibits the turbulent mixing of entropy between upflows and downflows. As a result, the effective Prandtl number may exceed unity inside the solar convection zone. In this paper, we propose and numerically confirm a possible suppression mechanism of convective velocity in the effectively high-Prandtl number regime. If the effective horizontal thermal diffusivity decreases (the Prandtl number accordingly increases), the subadiabatic layer which is formed near the base of the convection zone by continuous depositions of low entropy transported by adiabatically downflowing plumes is enhanced and extended. The global convective amplitude in the high-Prandtl thermal convection is thus reduced especially in the lower part of the convection zone via the change in the mean entropy profile which becomes more subadiabatic near the base and less superadiabatic in the bulk., Comment: 17 pages, 13 figures, accepted for publication in ApJ

Aims. We report the small temporal variation of the axial dipole moment near the solar minimum and its application to the solar cycle prediction by the surface flux transport (SFT) model. Methods. We measure the axial dipole moment using the photospheric synoptic magnetogram observed by the Wilcox Solar Observatory (WSO), the ESA/NASA Solar and Heliospheric Observatory Michelson Doppler Imager (MDI), and the NASA Solar Dynamics Observatory Helioseismic and Magnetic Imager (HMI). We also use the surface flux transport model for the interpretation and prediction of the observed axial dipole moment. Results. We find that the observed axial dipole moment becomes approximately constant during the period of several years before each cycle minimum, which we call the axial dipole moment plateau. The cross-equatorial magnetic flux transport is found to be small during the period, although the significant number of sunspots are still emerging. The results indicates that the newly emerged magnetic flux does not contributes to the build up of the axial dipole moment near the end of each cycle. This is confirmed by showing that the time variation of the observed axial dipole moment agrees well with that predicted by the SFT model without introducing new emergence of magnetic flux. These results allows us to predict the axial dipole moment in Cycle 24/25 minimum using the SFT model without introducing new flux emergence. The predicted axial dipole moment of Cycle 24/25 minimum is 60--80 percent of Cycle 23/24 minimum, which suggests the amplitude of Cycle 25 even weaker than the current Cycle 24. Conclusions. The plateau of the solar axial dipole moment is an important feature for the longer prediction of the solar cycle based on the SFT model., Comment: 5 pages, 3 figures, accepted for publication in A&A Letter

We investigate the inhomogeneity of kinetic and magnetic dissipations in thermal convection using high-resolution calculations. In statistically steady turbulence, the injected and dissipated energies are balanced. This means that a large amount of energy is continuously converted to internal energy via dissipation. As in thermal convection downflows are colder than upflows, the inhomogeneity of the dissipation potentially changes the convection structure. Our investigation on the inhomogeneity of the dissipation shows the following. 1. More dissipation is seen around the bottom of the calculation domain, and this tendency is promoted with the magnetic field. 2. The dissipation in the downflow is much larger than that in the upflow. The dissipation in the downflow is more than 80% of the total at maximum. This tendency is also promoted with the magnetic field. 3. Although 2D probability density functions of the kinetic and magnetic dissipations versus the vertical velocity are similar, the kinetic and magnetic dissipations are not well correlated. Our result suggests that the spatial inhomogeneity of the dissipation is significant and should be considered when modeling a small-scale strong magnetic field generated with an efficient small-scale dynamo for low-resolution calculations., Comment: 23 pages, 8 figures accepted for publication in The Astrophysical Journal

We carry out high resolution calculations of the solar overshoot region with unprecedentedly realistic parameters, especially the small energy flux compared with $\rho c^3_\mathrm{s}$, where $\rho$ and $c_\mathrm{s}$ are density and speed of sound. Our main purpose is to investigate behavior of the overshoot and the small-scale dynamo with parameters as close as possible to those of the Sun. Our calculations show that the bottom part of the convection zone becomes subadiabatic, which efficiently suppresses downflows. As a result we see a steep transition from the CZ to the RZ whose width is estimated 0.4% of the local pressure scale height. This result is consistent with a semi-analytic convection/overshoot model. We also find that the small-scale dynamo becomes efficient with a smaller energy flux. The sudden suppression of the downflows around the base of the convection zone increases the efficiency of the small-scale dynamo., Comment: 40 pages, 14 figures, 1 table, accepted by The Astrophysical Journal

We investigate small-scale dynamo action in the solar convection zone through a series of high resolution MHD simulations in a local Cartesian domain with 1$R_\odot$ (solar radius) of horizontal extent and a radial extent from 0.715 to 0.96$R_\odot$. The dependence of the solution on resolution and diffusivity is studied. For a grid spacing of less than 350 km, the root mean square magnetic field strength near the base of the convection zone reaches 95% of the equipartition field strength (i.e. magnetic and kinetic energy are comparable). For these solutions the Lorentz force feedback on the convection velocity is found to be significant. The velocity near the base of the convection zone is reduced to 50% of the hydrodynamic one. In spite of a significant decrease of the convection velocity, the reduction in the enthalpy flux is relatively small, since the magnetic field also suppresses the horizontal mixing of the entropy between up- and downflow regions. This effect increases the amplitude of the entropy perturbation and makes convective energy transport more efficient. We discuss potential implications of these results for solar global convection and dynamo simulations., Comment: 46 pages, 25 figures, 1 table, accepted by ApJ

We present a high-resolution, highly stratified numerical simulation of rotating thermal convection in a spherical shell. Our aim is to study in detail the processes that can maintain a near surface shear layer (NSSL) as inferred from helioseismology. Using the reduced speed of sound technique we can extend our global convection simulation to $0.99\,R_{\odot}$ and include near the top of our domain small-scale convection with short time scales that is only weakly influenced by rotation. We find the formation of a NSSL preferentially in high latitudes in the depth range $r=0.95-0.975R_\odot$. The maintenance mechanisms are summarized as follows. Convection under weak influence of rotation leads to Reynolds stresses that transport angular momentum radially inward in all latitudes. This leads to the formation of a strong poleward directed meridional flow and a NSSL, which is balanced in the meridional plane by forces resulting from the $\langle v'_r v'_\theta\rangle$ correlation of turbulent velocities. The origin of the required correlations depends to some degree on latitude. In high latitudes a positive correlation $\langle v'_rv'_\theta\rangle$ is induced in the NSSL by the poleward meridional flow whose amplitude increases with the radius, while a negative correlation is generated by the Coriolis force in bulk of the convection zone. In low latitudes a positive correlation $\langle v'_rv'_\theta\rangle$ results from rotationally aligned convection cells ("banana cells"). The force caused by these Reynolds stresses is in balance with the Coriolis force in the NSSL., Comment: 49 pages, 20 figures, accepted by ApJ

We carry out non-rotating high-resolution calculations of the solar global convection, which resolve convective scales of less than 10 Mm. To cope with the low Mach number conditions in the lower convection zone, we use the reduced speed of sound technique (RSST), which is simple to implement and requires only local communication in the parallel computation. In addition, the RSST allows us to expand the computational domain upward to about $0.99 R_{\odot}$ as it can also handle compressible flows. Using this approach, we study the solar convection zone on the global scale, including small-scale near-surface convection. In particular, we investigate the influence of the top boundary condition on the convective structure throughout the convection zone as well as on small-scale dynamo action. Our main conclusions are: 1. The small-scale downflows generated in the near-surface layer penetrate into deeper layers to some extent and excite small-scale turbulence in the region $>0.9R_\odot$, where $R_\odot$ is the solar radius. 2. In the deeper convection zone ($<0.9R_\odot$), the convection is not influenced by the location of the upper boundary. 3. Using an LES approach, we can achieve small-scale dynamo action and maintain a field of about $0.15-0.25 B_\mathrm{eq}$ throughout the convection zone, where $B_\mathrm{eq}$ is the equipartition magnetic field to the kinetic energy. 4. The overall dynamo efficiency varies significantly in the convection zone as a consequence of the downward directed Poynting flux and the depth variation of the intrinsic convective scales., Comment: 60 pages, 24 figures, accepted for publication in ApJ

Using two-dimensional magnetohydrodynamics calculations, we investigate a twist gen- eration mechanism on a magnetic flux tube at the base of the solar convection zone based on the idea of Choudhuri, 2003, Sol. Phys., 215, 31 in which a toroidal mag- netic field is wrapped by a surrounding mean poloidal field. During generation of the twist, the flux tube follows four phases. (1) It quickly splits into two parts with vortex motions rolling up the poloidal magnetic field. (2) Owing to the physical mechanism similar to that of the magneto-rotational instability, the rolled-up poloidal field is bent and amplified. (3) The magnetic tension of the disturbed poloidal magnetic field re- duces the vorticity, and the lifting force caused by vortical motion decreases. (4) The flux tube gets twisted and begins to rise again without splitting. Investigation of these processes is significant because it shows that a flux tube without any initial twist can rise to the surface in relatively weak poloidal fields., Comment: 10 pages, 6 figure

We investigate an intensification mechanism for the magnetic field near the base of the solar convection zone that does not rely on differential rotation. Such mechanism in addition to differential rotation has been suggested by studies of flux emergence, which typically require field strength in excess of those provided by differential rotation alone. We study here a process in which potential energy of the superadiabatically stratified convection zone is converted into magnetic energy. This mechanism, know as explosion of magnetic flux tubes, has been previously studied in the thin flux tube approximation as well as two-dimensional MHD simulations, we expand the investigation to three-dimensional MHD simulations. Our main result is that enough intensification can be achieved in a three-dimensional magnetic flux sheet as long as the spatial scale of the imposed perturbation normal to the magnetic field is sufficiently large. When this spatial scale is small, the flux sheet tends to rise toward the surface, resulting in a significant decrease of the magnetic field amplification., Comment: 14 pages. 5 figures accepted to publication in ApJ Letters

We investigate the value of horizontal turbulent diffusivity {\eta} by numerical calculation of thermal convection. In this study, we introduce a new method whereby the turbulent diffusivity is estimated by monitoring the time devel- opment of the passive scalar, which is initially distributed in a given Gaussian function with a spatial scale d0. Our conclusions are as follows: (1) Assuming the relation {\eta} = Lcvrms/3 where vrms is the RMS velocity, the characteristic length Lc is restricted by the shortest one among the pressure (density) scale height and the region depth. (2) The value of turbulent diffusivity becomes greater with the larger initial distribution scale d0. (3) The approximation of turbulent diffusion holds better when the ratio of the initial distribution scale d0 to the characteristic length Lc is larger., Comment: 18 pages, 4 figures, 1 table, accepted to ApJL

Context. The anelastic approximation is often adopted in numerical calculation with low Mach number, such as stellar internal convection. This approximation requires frequent global communication, because of an elliptic partial differential equation. Frequent global communication is negative factor for the parallel computing with a large number of CPUs. Aims. The main purpose of this paper is to test the validity of a method that artificially reduces the speed of sound for the compressible fluid equations in the context of stellar internal convection. The reduction of speed of sound allows for larger time steps in spite of low Mach number, while the numerical scheme remains fully explicit and the mathematical system is hyperbolic and thus does not require frequent global communication. Methods. Two and three dimensional compressible hydrodynamic equations are solved numerically. Some statistical quantities of solutions computed with different effective Mach numbers (due to reduction of speed of sound) are compared to test the validity of our approach. Results. Numerical simulations with artificially reduced speed of sound are a valid approach as long as the effective Mach number (based on the reduced speed of sound) remains less than 0.7., Comment: 16 pages, 10 figures, accepted to A&A

We investigated the dependence of the solar magnetic parity between the hemispheres on two important parameters, the turbulent diffusivity and the meridional flow, by means of axisymmetric kinematic dynamo simulations based on the flux-transport dynamo model. It is known that the coupling of the magnetic field between hemispheres due to turbulent diffusivity is an important factor for the solar parity issue, but the detailed criterion for the generation of the dipole field has not been investigated. Our conclusions are as follows. (1) The stronger diffusivity near the surface is more likely to cause the magnetic field to be a dipole. (2) The thinner layer of the strong diffusivity near the surface is also more apt to generate a dipolar magnetic field. (3) The faster meridional flow is more prone to cause the magnetic field to be a quadrupole, i.e., symmetric about the equator. These results show that turbulent diffusivity and meridional flow are crucial for the configuration of the solar global magnetic field., Comment: 19 pages, 5 figures

On the evaluation of $\tan{\beta}$ is revisited by the method of the renormalization group with the criteria of the bottom-tau unification in the SUSY SU(5) GUT model. Among the conventional supersymmetric theories, an energy-scale constant $M_{SUSY}$ is introduced by-hand to decouple the lower energy-scale region from the effects of the supersymmetric particles. Provided that $M_{SUSY}$ really exists, only small $\tan{\beta} \simeq 2$ is allowed theoretically. While, medium value as $\tan{\beta} = 6 \sim 25$ is also supported if $M_{SUSY}$ is vanished away from the model., Comment: 21 pages, 12 figures, 2 tabels