CHINESE ASTRONOMY AND ASTROPHYSICS
ELSEVIER
ChineseAstronomy Astronomyand and Astrophysics Astrophysics 34 Chinese 34 (2010) (2010)288–304 288–304
The SHASTA Code Modified by Self-adaptive Mesh and Numerical Experiment of Magnetic Reconnections SHEN Cheng-cai1,2
LIN Jun1,3
1
National Astronomical Observatories/ Yunnan Astronomical Observatory, Kunming 650011 2 Graduate University of Chinese Academy of Sciences, Beijing 100049 3 Harvard-Smithsonian Center for Astrophysics, Cambridge MA 02138,USA
Abstract SHASTA (Sharp and Smooth Transport Algorithm) is a code with single mesh to solve the 2-dimensional magnetohydrodynamic (MHD) equations. When SHASTA is used to the numerical simulation of magnetic reconnection problem, it is modified to be the code which adopts the method of the selfadaptive mesh. The modified code can carry out refined calculations in diffusion regions. In the process of the self-adaptive calculations with SHASTA, a “plugand-play” strategy is adopted and the original algorithm to solve 2-dimensional MHD partial differential equations is treated as an independent cell. In addition, the hierarchical data structure is used in this modification and parameters in each refined level are described by a 2-dimensional variable array. The regions where the distributions of magnetic field and pressure exhibit steep variations are marked as the refined regions. Then, the distributions of physical quantities and the boundary conditions in the grid points of refined levels are deduced via interpolation method. Finally, the refined calculated results of refined regions are assigned to the previous level of mesh and the existing results are updated. The numerical experiment of magnetic reconnections which adopts refined calculations indicates that compared with the code with single mesh, the resolution of details is improved and the corresponding increment of computing time is related to the selection of parameters in the simulation. The calculation accuracy and effect on instability, which are caused by a part of the self-adaptive code, † Supported by 973 Project of Ministry of Science and Technology, National Natural Science Foundation and Chinese Academy of Science Received 2008–11–13; revised version 2009–06–03 A translation of Acta Astron. Sin. Vol. 50, No. 4, pp. 391–405, 2009
[email protected]
0275-1062/01/$-see front matter © 2010 Elsevier B.V. All rights reserved. c 2010 Elsevier Science B. V. All rights reserved. 0275-1062/01/$-see front matter doi:10.1016/j.chinastron.2010.07.008 PII:
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depend on the boundary settings, push strategy over each single step as well as the interpolation algorithm. Key words: sun: flares, MHD — methods: numerical
1. INTRODUCTION Magnetic reconnections play important roles in a lot of astrophysical processes. In eruptive activities, such as solar flares and CMEs (coronal mass ejections), the energy of magnetic field is rapidly released through the magnetic reconnection processes, transformed to the kinetic energy and thermal energy of plasma and high-energy particles are accelerated[1]. In these solar eruptive processes, quite long current sheets are usually formed. Generally, the large-scale structure of these current sheets extends from solar flares up to the high level of corona. Theoretical researches have pointed out the existence and developing characteristic of such structures[2−3] . The observations and numerical experiments in recent years also confirm the existence of similar structures[4−8] . As the current sheet becomes thinner and longer and the ratio of its length to thickness exceeds 2π, the resistance unstable mode in the current sheet, such as the tearing mode, develops[9] . The development of this unstable mode has a tendency of tearing the current sheet into separate filaments and the corresponding magnetic islands[10−12] . In this process, the topological structure of the magnetic field is changed, the abnormal reconnection induced by instability is intensified and unsteady magnetic reconnection process is formed. With the growth of the tearing mode, the structure of magnetic islands may merge because of the current merging instability. This process is nonlinear, in which the current may become very strong and an eruptive magnetic reconnection may appear. Researches on the instability of the tearing mode have an important significance to understand the reconnections in eruptive activities like flares. The nonlinear evolution of the structure of magnetic islands produced by the instability of the tearing mode, such as the dynamical processes like the merging instability may enlarge the magnetic flux of reconnections[13−15] and lead to a higher occurrence rate of reconnections. Observations also find that in the corresponding diffusion regions of flares (or CMEs), there are plasmoid structure and its evolution, which are directly related to the growth rate of the tearing mode[6,8,16−17] . The structure of magnetic islands caused by the instability of tearing mode also provides a new form of accelerating the electric field by particles. Its dynamical characteristics and the corresponding movement of plasmoid may have an obvious influence on the particle acceleration and radio bursts in flares[18] . For such an unsteady reconnection problem, the numerical experiments become an important method to investigate magnetic reconnections because of the difficulties in solving nonlinear MHD equations. The finite difference method is a common method in MHD numerical simulations and its differential accuracy is enhanced with the decrease of mesh interval. With the improvement of computing ability, high-resolution mesh can reduce some numerical instability and nonphysical results, and make the numerical experiments be closer to real physics. However, due to the constraint of the conditions of numerical calculations, mainly the limits of computing ability of computers, the simulations of magnetic reconnection problems in many numerical experiments can only be performed in a narrow range of
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parameters. For unsteady magnetic reconnection problems, the high-resolution evolutional picture in diffusion regions needs a further investigation. The self-adaptive mesh is the mesh which can automatically converge to the regions with large gradients in flow fields during calculation processes. It makes use of characters of the flow field to be solved to define the positions of grid points in physical plane. The advantages of self-adaptive mesh are that the computing accuracy can be improved when the number of meshes is fixed and less grid points can be used to reach an accuracy when this accuracy is fixed[19] . Based on the principle of generating self-adaptive mesh revealed by Berger et al.[20−22] and the valid application of self-adaptive mesh to MHD problems[23−26] , a relatively concise method can be used to carry out calculations with self-adaptive mesh based on the existing explicit scheme code[27−28] . SHASTA is a kind of high accuracy and reliable algorithm to solve MHD partial differential equations[29−30] . After the modification of Webb et al.[31] and the addition of the part of calculations of magnetic diffusions, SHASTA can well solve the nonideal MHD equations. It can well distinguish the propagation of different kinds of nonlinear waves and reach a high resolution of several grid points for shock waves[31]. In this work, we use the magnetofluid simulation code based on the SHASTA algorithm and modify it to get a refined algorithm of self-adaptive mesh to meet the high accuracy required by calculations in diffusion regions of magnetic reconnections. The modified self-adaptive scheme tries to increase the computing time as less as possible to get simulation results with a higher resolution. The structure of this paper is arranged as follows: the 2-dimensional MHD equations to be solved, the solving methods used in SHASTA, the adopted self-adaptive strategy to modify the code as well as the implementing method of several basic structures in SHASTA are introduced in Section 2; in Section 3, several MHD problems are used to testify the modified code and the preliminary results of these experiments of unsteady magnetic reconnections are presented; in the end, we summarize the modified results and discuss some issues which need to be improved.
2. THE TREATMENT OF MHD EQUATIONS AND MODIFICATION OF THE ORIGINAL SHASTA CODE 2.1 MHD Equations The 2-dimensional MHD equations to be solved include the mass equation, momentum equation, magnetic field equation and energy equation. Meanwhile, the gas equation of state and Ohm’s law are also used. ∂ρ + ∇ · ρv = 0 , (1) ∂t ∂v + (v · ∇)v) = −∇p + j × B + F , (2) ρ( ∂t ∂B = ∇ × (v × B) + ∇ · (ηm ∇B) , (3) ∂t ∂ p ργ ( + v · ∇) γ = −ψ , (4) γ − 1 ∂t ρ 1˜ p = RρT , (5) μ ˜
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j=
1 (∇ × B) . μ0
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(6)
The variables ρ, v, B, p, j and T indicate the density, fluid velocity, magnetic induction ˜ indicate intensity, pressure, current density and temperature, respectively. ηm , γ, μ0 and R the magnetic diffusion coefficient, gas specific heat ratio, vacuum magnetic permeability and gas constant. The term ψ on the right side of the energy equation represents the contributions from different kinds of energy transferring processes, which may be Joule heating, coronal heating, radiation loss or contributions from other effects. The dimensionless form is described hereinafter. Through introducing several characteristic quantities, we define some dimensionless physical quantities as follows. B∗ =
B ∗ ρ p v , ρ = , p∗ = 2 , v∗ = , B0 ρ0 vA B 0 /μ0
where
√ v A = B 0 / μ0 ρ0 .
(7)
(8)
Here B 0 and ρ0 indicate the characteristic values of the magnetic field and density, respectively. v A is the corresponding Alfven velocity of background. The corresponding dimensionless temperature and current density are T∗ =
p∗ β0 T = , ρ∗ 2T0
Here
j ∗ = ∇∗ × B ∗ .
β0 = 2μ0 p0 /B 20 ,
(9)
(10)
T0 and p0 represent the characteristic values of the temperature and pressure, respectively. When defining the initial conditions, we make the initial values of physical quantities equal to their characteristic values. This may make the initial values be distributed around 1 as much as possible. Meanwhile, the initial definitions of the temperature and pressure of background can be related to β0 in the process of selecting dimensionless quantities[32] . The dimensionless length x∗ and z ∗ and time t∗ can be defined as x∗ =
x , L
z∗ =
z , L
t∗ =
t , τA
(11)
where τA = L/vA is the characteristic time of simulated regions, L is the characteristic length of simulated region. In this way, the dimensionless nonideal MHD equations can be obtained (the asterisk “ ∗ ” is omitted in the dimensionless expressions in the following text). Dρ + ρ∇ · v = 0 , Dt Dv = −∇p + (∇ × B) × B + F˜ , Dt 1 2 ∂B = ∇ × (v × B) + ∇ B, ∂t Rm 1 ∂p + ∇ · (vp) = −(γ − 1)p∇ · v + (γ − 1) (∇ × B)2 + (γ − 1)ψ˜ , ∂t Rm ρ
(12) (13) (14) (15)
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where the magnetic Reynolds number Rm = LvA /ηm is introduced; F˜ is the dimensionless forces in other forms, such as gravity; ψ˜ indicates the other dimensionless energy contributions other than Joule heating. In the SHASTA code, we apply the finite difference method to the Nx × Nz plane mesh and solve the above equations. In SHASTA, each differential equation is considered as a combination of two parts: convection part and diffusion part. We use the explicit scheme algorithm to deal with the former part, and the tridiagonal matrix method of implicit scheme to deal with the latter one. Therefore, we adopt the method which respectively solves the convection and diffusion parts of equations in each time step in SHASTA. First, equations are considered as ideal MHD equations when solving the convection part of equations in the time step dt. Then, when solving the diffusion part, the diffusion and dissipation processes of correlative physical quantities are solved to get the modification of ideal MHD results. If the time step is smaller enough, for instance, much less than the characteristic time scale of the variation of physical quantities, such separation in solving equations will not lead to an obvious decrease in the accuracy of solutions[31] . Webb et al. previously finished such calculations and then Forbes et al.[32] partly modified the code. Here we use the code modified by Forbes et al. and use the equation of magnetic vector potential A ∂A + ∇ · (Av) = A∇ · v + ηm ∇2 A ∂t
(16)
to replace the magnetic field equation of B. If B is directly used, some minor deviations in diffusion calculations of Bx and By from the passive condition of magnetic field ∇·B = 0
(17)
may be continually enlarged, while the satisfaction of this condition can always be assured if using the magnetic vector A to calculate the diffusion part[32] . After the above treatment of MHD equations, we may come to the next step to solve these equations. The convection parts of equations are expressed as
where
Df = ∇ · G + S∇H + Q , Dt
(18)
Df ∂f = + ∇(f · v) . Dt ∂t
(19)
Now f corresponds to the various components in Eqs.(12) and (15), i.e., ρ, ρvx , ρvz , Bx , Bz and p. For instance, for Eq.(12), f represents ρ, and G = ρv, S = 0, H = 0 and Q = 0 in the corresponding Eq.(18). Note that the diffusion term Rm −1 ∇2 B on the right side of Eq.(14) is not included here. The diffusion parts of equations can be expressed as ∂ ∂ ∂f = F1 (F2 F3 f ) + SF , ∂t ∂x ∂x
(20)
where f represents B, corresponding to the diffusion part of Eq.(14), and accordingly F1 = 1, F2 = Rm −1 , F3 = 1 and Sf = 0. In our current calculations, the treatment of diffusion part is only made to the magnetic field because the temperature diffusion is not considered.
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Through a pair of cross-transfer algorithm between A and B, calculations of the magnetic vector which is discussed above come true. For the convection part, SHASTA uses the flux-corrected transport (FCT) method to solve equations. It assures the stability in the regions with steep variations as well as a favorable resolution of shock waves in computing processes. A multiple explicit scheme method in the form of “pre-estimated step added by corrected step” is adopted in calculations, making results have a 2-order accuracy. For the diffusion part, a tridiagonal matrix method which uses implicit scheme with 2-order differential accuracy is used in SHASTA to solve equations. Eq.(20), corresponding to the magnetic diffusions, is expressed as the following differential form to be solved, ∂ fi = F1i fi−1 + F2i fi + F3i fi+1 + SF . ∂t
(21)
The advantage of doing so is to bring the character of the stability in solving the diffusion equations via tridiagonal matrix method into play. The calculated results have a 2-order accuracy. 2.2 Refinement of Self-adaptive Mesh Based on the SHASTA algorithm, one kind of “plug-and-play” self-adaptive method is adopted and the SHASTA algorithm is separately inherited as a basic computing unit, which serves the overall solution domain as a relatively independent part via the partitioned and hierarchical data structure. The advantage of doing so is to retain the characters of effectiveness and stability of SHASTA in solving equations. Simultaneously it focuses on some interesting regions and carries out calculations in solving the overall solution domain. This is the basic idea of refined self-adaptive mesh and its implementation method is: the calculation with temporal evolution is carried out starting from a basic mesh and one time step is calculated by using the difference algorithm of explicit scheme. First, the region which needs refined calculations is found. Then, this region is assigned to several rectangular regions, which are called as “blocks”[22,27−28,33]. According to a certain rate of refinement, the mesh for which refined calculations will be performed is automatically generated at these “blocks”. Then at the level of new mesh, different physical quantities in this region are assigned to new grid points and calculations are carried out at the mesh of this level. If finished, the calculated results update the existing results at the previous level. Otherwise, more refined meshes are generated and calculations are performed to the next level. The detailed process includes the strategy of promoting calculations with time and the method of marking refined regions (please refer the authors’ another paper [34] for the detailed description). 2.3 Definition of Boundary Conditions The boundaries of different blocks at refined levels can be divided into two kinds as the internal boundary and external boundary. The internal boundary lies in the overall solution domain, and its conditions should be defined by the connecting regions between blocks or the value at parent level. The external boundary lies at the boundary of the overall solution domain, and its conditions are defined by the computing conditions of the overall solution domain or physical requirements. For the internal boundary, we use the interpolation methods to interpolate with space
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and time from the grid points of the parent level of present block and derive the values at the boundary. Among these methods, the linear interpolation has a better stability. When the time step length of refined level is in integral times of that of its parent level, the interpolation with time is simplified, and the consistency between the boundary values of the refined level and its parent level is assured after integration of k steps with time. In this way, we make the flux in the refined block consist with that in the same region of the coarse mesh of its parent level. This method simply deals with the flux conservation problem of the connecting surface of the refined blocks and the coarse mesh of its parent level. However, as a matter of fact, due to factors such as the interpolation error, the assurance of the flux conservation is still needed to be improved. The external boundaries mainly include the physical boundary, free boundary, symmetric boundary and periodic boundary. Because the boundary condition in SHASTA is used as a separate subprogram, the original boundary algorithm can still be used in the calculations of refined blocks. However, because the convection part and diffusion part of equations are separately solved in SHASTA and the multiple scheme is used to solve the convection part, the uses of boundaries of refined blocks also respectively occur in these different sections. In some special symmetric settings, the symmetric boundaries of refined blocks need to be disposed specially. When the symmetric axis of solution domain lies at internal points of mesh, the position index of the symmetric axis at the mesh may be different from that at its parent mesh in a deeper refined level as the defined process goes in depth. For instance, the symmetric axis lies at the the second point of mesh for the coarse mesh of the first level. And when the regions including this symmetric axis are continually refined, this symmetric axis can lie at other points of refined mesh in a deeper refined level. 3. NUMERICAL TESTS In this part of work, the SHASTA code modified by the self-adaptive mesh is used to accomplish the calculations of a set of magnetic reconnection problems, and the results are compared with those of SHASTA without using the self-adaptive mesh technology. These problems of reconnection tests come from the two-ribbon flare model, which is commonly accepted at present. There have been many results of numerical researches which can be referred to for us [32,35−38] . The relatively simplified situation of this model is set to be tested. A complete dynamical simulation of two-ribbon flare was first accomplished by Forbes et al.[35] . The configuration of magnetic field shown in Fig.1 is taken as the initial configuration of simulation to begin calculations. In such a configuration, a neutral current sheet separates two parts of magnetic field with reverse polarities. Such a configuration is the structure of magnetic field at the beginning of two-ribbon flare. Such a structure is unstable in the low solar atmosphere and is generally formed at the initial time of explosive process (refer to the discussion in Lin et al.[2,39] ). Due to the instability of the structure of magnetic field, a part of magnetic field and magnetic flux is thrown into interplanetary space. The induced extreme extension of the closed magnetic field leads to two parts of magnetic field with reverse polarities close to each other, forming current sheet. In general, the conductivity of current sheet is not zero. Besides, the length of current sheet is far larger than its thickness. So in current sheet, the instability of resistance tear-
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Fig. 1
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The initial configuration of magnetic field and the distribution of density on the x-z plane
ing mode easily occurs[1,9]. Then quick magnetic reconnections and continually expanding systems of flare loops are produced[40] . In situations with high magnetic Reynolds number, many X and O points are formed. We can see that magnetic islands are formed in current sheet and flow along it. In the following calculations, the initial conditions are set as: the current sheet with a thickness of 2ω is distributed along the z axis, the air pressure and magnetic pressure are in equilibrium both inside and outside the current sheet, and the initial structure of current sheet is in magnetostatic equilibrium. Here, the effect of gravity is not considered. The initial distribution of magnetic field is shown in Fig.1 and the current sheet lies in the z axis along the vertical direction. x/|x| , |x| > w Bz = . (22) sin(πx/2w), |x| ≤ w Meanwhile, Bx = 0,
v = 0,
ρ = (β + 1 − B 2 )/β,
p = ρT = ρβ/2 .
(23)
The simulated region is set as the first quadrant region with 0 ≤ x ≤ 1 and 0 ≤ z ≤ 2 in the rectangular coordinate system. Because the initial distribution of this system and the electric resistance are symmetric with respect to z axis and the evolution should be symmetric, the z axis is taken as the symmetric axis in calculations, which are only carried out in the right half plane.
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The four boundary conditions of the simulated region are: the left boundary x = 0 is a symmetric boundary, in which the tangential component of magnetic field and the normal component of velocity are equal to zero; the right boundary and top boundary are free boundaries. The tangential component of magnetic field and the normal components of velocity, pressure and density are set to zero; the bottom boundary z = 0 is a physical boundary, which satisfies the frozen-in condition of magnetic lines as ∂A = 0. ∂t
(24)
The passive condition of magnetic field is accordingly used to deal with Bx and Bz [32] . It is assumed that the resistivity of the simulated region is uniformly distributed and the magnetic Reynolds number of the whole region is used to describe the freezing intensity of magnetic field. The settings of correlative simulated parameters are presented in Table 1, in which Case 1, Case 3 and Case 6 are simulated by using single mesh with uniform grid points. As stated above, simulation is only carried out in the right half plane, while the left boundary is considered as a symmetric boundary. So in the direction of x axis, the distribution of grid points of simulated region lies from −Δx, 0, Δx to 1.0, where Δx represents the spatial step length. A grid point is set at x = 0 and then the non-physical results such as the pseudo reconnections which happen due to using symmetric boundary can be avoided[41]. The distribution of grid points in the direction of the z axis lies from 0.0 to 2.0. Case 2, Case 4 and Case 5 correspond to simulations by using the self-adaptive mesh and their settings of the initial mesh (or called as the first-level mesh) are the same with Case 1, Case 3 and Case 6, respectively. Small disturbances of velocity are introduced in the settings of initial conditions in Case 5 and Case 6, and their effects will be described in the following text. Table 1 Case No. Grid Refining ratio Rm β0
The simulation parameters in different situations
1 49×97 0 500 0.1
2 49×97 4 500 0.1
3 101×201 0 500 0.1
4 101×201 4 500 0.1
5 151×301 4 104 0.1
6 151×301 0 104 0.1
3.1 Tests with Low Magnetic Reynolds Number In the situation of low magnetic Reynolds number, the magnetic field is easier to decouple from the fluid field and the energy of magnetic field can be released more quickly. In this kind of situations, magnetic reconnections happen more rapidly. We compare the results by using the self-adaptive refined calculations with those by using the single mesh calculations. Fig.2 indicates the distributions of mass density and the configurations of magnetic field in Case 3 and Case 4 at the time when it is 17.5 times of characteristic time. Figs.2 (a) and (b) correspond to the results of Case 3, and (c) and (d) correspond to that of Case 4. From these four figures, it can be seen that the neutral point has risen to a height of z = 0.95, and there is an obvious formation of magnetic arcade at the bottom boundary. The configuration of magnetic field calculated by using single mesh is similar to that by using the self-adaptive mesh, and this is so for the distributions of density and current density. To reflect the varying situation of the occurrence rate of magnetic
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reconnections, the temporal variation of the electric field at neutral point is used in this paper to indicate the speed of magnetic reconnections. Fig.3 shows the temporal variation of the height of X point formed in the situations with different parameters and the temporal
Fig. 2
The distribution of the mass density and magnetic field at t = 17.5. Rm = 500 in this simulation. Panels (a) and (b) correspond to Case 3, (c) and (d) correspond to Case 4
variation of the strength of electric field around the X point. Figs.3 (a) and (b) correspond to Case 1 and Case 2, while (c) and (d) correspond to Case 3 and Case 4. Compared with the simulated results of single mesh, after using the refined self-adaptive mesh, the height of the formed X point and its ascending tendency are consistent. The variation of the strength of electric field exhibits fluctuations and deviations in comparison with calculations of single mesh. The deviations occurring in 49×97 mesh are larger than those in the denser 101×201 mesh. The results of Case 4 indicate that after using refined mesh calculations, the occurrence rate of magnetic reconnections in the early stage of evolution increases, but
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it is smaller than the results of single mesh when this rate quickly increases. Based on the aspect of the numerical stability, low magnetic Reynolds number corresponds to relatively
Fig. 3 The temporal variations of the height of the formed X point and the strength of electric field at X point in the situation of low magnetic Reynolds number. Rm = 500 in this simulation. Panels (a) and (b) correspond to Case 1 and Case 2, (c) and (d) correspond to Case 3 and Case 4.
rapid diffusion of magnetic field. In this way, the numerical instability in the process of solving equations comparatively relaxes the requirement of the mesh accuracy, i.e., a stable simulation can be carried out in the condition of the mesh accuracy which is not too small. With a single mesh, the mesh accuracy of reconnection simulation has already been reached and the evolution normally goes on. Therefore, the refined calculations here, in fact, get the results similar to those with single mesh. However, the refined calculations still reflect their
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ability of distinguishing details, and the interruptions from a part of structures occurring in single mesh can be improved now. 3.2 The Situation of High Magnetic Reynolds Number In the situation of high magnetic Reynolds number, the simulation of magnetic reconnections requires a higher resolution of mesh. To avoid the development of nonlinear numerical instability, mainly due to the computing instability caused by the instability of grid division [42] , the distribution scale of grid points should be less than the corresponding Kolomogoroff diffusion scale dεb λη = |( )Rm 3 |−1/4 , (25) dt where λη is the Kolomogoroff diffusion scale, dεb /dt is the dissipation rate of the magnetic field energy[36] . In this part of simulation, the simulated parameters are set as those of Case 5 and Case 6 in Table 1. At the initial stage, an inflow disturbance around the current sheet is artificially introduced and its magnitude is 5% of the characteristic velocity. The purpose of this introduction is to accelerate the reconnection process at the initial stage. With this set of parameters, the program is terminated around t=10 due to the numerical instability in the calculation process of Case 6 in which single mesh is used. So the two subsequent figures only show the calculated results of Case 5. Fig.4 shows the evolutions of the current density and magnetic field from t=23.4 to t=24.0 in Case 5. When t=23.4, an X point is formed at z=0.45. Then, as the magnetic lines
Fig. 4
The variation of the current density in the current sheet for Case 5 when the tearing mode
instability develops. The background and solid lines represent the distribution of current density and magnetic lines, respectively. Rm = 104 and β0 = 0.1 in this simulation.
continually approach the middle X point and the initial O-type region is thrown upwards, a
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longer current sheet is newly formed around the X point. As the current sheet continually becomes thinner and longer, the tearing mode instability develops. After t=23.7, the first small structure of magnetic island gradually appears at the height of z=0.55. When t=24.0, this structure of magnetic island has developed into an obvious plasmoid. Fig.5 displays the variations of the mass density and magnetic field in the magnetic reconnection region after the development of the tearing mode instability. The background reflects the distribution
Fig. 5 The movements of plasmoids along current sheet for Case 5. The solid lines and background represent the magnetic lines and filled isogram of the density distribution, respectively. Rm = 104 and β0 = 0.1 in this simulation.
of density and the solid lines represent the distribution of magnetic lines. From Fig.5, it can be seen that plasmoids have complicated movements along the current sheet, including ascending, descending and merging movements[43] . When t=27.6, two small plasmoids at z=0.74 and z=0.92 tend to merge. These two plasmoids move in opposite directions with the approximately same velocity, which is about 15% of the characteristic velocity. After moving for 0.6 characteristic time, these two plasmoids merge into a little bigger plasmoid at the height of z=0.83 when t=28.2. A similar merging movement can also be seen slightly later from t = 28.8 to t=29.4. When t=29.4, two plasmoids at z=1.25 and z=1.50 merge
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into a bigger one at z=1.6. Being different from the former merge stated above, both plasmoids move upwards and the merge happens after the plasmoid with lower position but higher velocity catches up with another plasmoid. All the plasmoids are in the processes of dynamic upward and downward movements. From the last three panels of Fig.5, the upward and downward movements of plasmoids along the current sheet can be observed. When t=30.0, the plasmoid at z=1.55 rises to z=1.90 after 0.6 characteristic time. Later, this plasmoid is freely thrown from the top boundary and its velocity is about 0.6 of the characteristic velocity. The plasmoid at z=0.75 moves downwards when t=30.0. After 1.2 characteristic time, it finally runs into the closed magnetic arcade (flare loop) which has been already formed below, and stops moving at the end.
4. CONCLUSIONS For the numerical simulation of magnetic reconnection problem, we use the refined method of self-adaptive mesh. Through monitoring the magnetic fields and pressure of diffusion regions, the real-time region with the most violent change, which is also the region where the instability mode grows in the process of magnetic reconnection, is found and refined calculations are performed for it. In comparison with the method which increases the number of grid points to improve the resolution in the simulation of single mesh, when the computing time is approximately doubled, the calculation method which uses the refined self-adaptive mesh can make the minimal grid interval be reduced to 1/4 of the original value, benefiting from the refined calculations. Thereby, the ability of distinguishing the details of magnetic reconnections is improved. To mark and partition the region which needs to be refined is a critical part of the efficiency in the refined process. In the monitoring function used by us, the artificial setting of the general spatial error directly influences the size of the marked region, and a smaller estimated threshold of error will make the range of the marked region remarkably increase. Therefore, the selection of this value needs to be properly adjusted to aim at different experiments, leading to an equilibrium between the increase of the calculation amount and the refined requirement of the targeted region. For the present work, we adopt the error estimation of 0.05 to set. For the partitioning process, more small blocks help to include more marked points in each block and then increase the entire computing efficiency. However, excessive blocks generally make the lately and newly added blocks deviate from the physical continuity of the targeted region. If excessive blocks are not only for the purpose of separating several marked intervals of points, then partition is usually abandoned. It is more reasonable to consider this region as one block in the form of a continuum. In addition, excessive blocks make the treatment of boundaries become more complex. As stated above, the setting of boundary largely influences the calculated results. Therefore, considering the process of 2-dimensional magnetic reconnection, the targeted regions actually converge around the current sheet and the marked regions are relatively converged. So the upper limit number of blocks in each partition is set to be 4. The determination of boundary conditions and the design of the interpolation algorithm also have important influences on the results. At present, the linear interpolation method with time and space, which directly comes form the refined push strategy, is used to get the
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internal boundary. For the explicit scheme, there can be two basic structures to deal with the consistency problem of push time between refined levels and coarse mesh levels. Here, we adopt the method in which refined levels are simultaneously refined with time step. This directly accomplishes the same refinements with time step according to the refinement ratio. When solving the convection part of equations, for equations with the following form ∂U ∂U ∂2U +a =b 2 ∂t ∂x ∂x
(26)
the explicit scheme algorithm requires the time step length to satisfy the following constraint condition a2 Δt ≤ 2b ≤ Δx2 /Δt . (27) Therefore, what assures the computing stability is to deduce the time step in detail in each refined level required by the CFL (Courant-Friedrichs-Lewy) condition, which satisfies both the stability and convergency of the explicit scheme. However, the deduced time step length may no longer meet our refined push strategy. So in a coarse mesh, an extra constraint factor of time step is used. This makes the time step Δt = AK × Δt of coarse mesh become much smaller than the value constrained by the stability condition with AK = 0.2 ∼ 0.5. In this way, the time step of refined level can still meet the stability condition after being refined with the same refinement ratio, but more time steps are needed to push the same evolutional time. As what corresponds to the setting of boundary condition stated above, the interpolation algorithm is related to the interpolation strategy of refined levels. If only using other interpolation calculation methods, such as the sectioned parabolic curve interpolation, N -points Lagrangian interpolation, cubic spline function interpolation[44−45] , the results with better accuracies can not be reached and reversely the computing instability appears together with some obvious non-physical results. Therefore, a unified design and discussion of the settings of boundaries, interpolation process and interpolation algorithm are needed to get a better calculation accuracy. The current data structure simplifies the allocation and management of data, and also saves computing resources. However, there are no connections among various blocks in each refined level, which makes the interpolation algorithm and updating process be excessively used. As a matter of fact, the relations which should exist among adjacent blocks in the same refined level are ignored. Meanwhile, the advantages of refined calculations can only be made full use of in parallel computing platform. Therefore, we will adopt a more perfect data structure and make calculations of blocks in one refined level be simultaneously carried out in the parallel platform. In addition, after being refined with the self-adaptive mesh, the velocity of magnetic reconnection has changed in comparison with the calculations of single mesh. In view of the numerical process, because the densified mesh has denser points, the dissipation introduced by the numerical discreteness is accordingly reduced. Then, the physical dissipation of current sheet becomes relatively outstanding and the effect of numerical simulation should increase the relative occurrence rate of magnetic reconnections. In view of the introduction of the refined self-adaptive mesh, the variation of the occurrence rate of magnetic reconnections after refinement partly results from the interpolation process of physical quantities. Especially for the physical quantities with relatively large gradients around the current sheet, the linear interpolation algorithm is obviously not ideal enough. The influence of the linear
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interpolation in Case 2 (49×97 grids) with relatively larger grid interval is greater than that in Case 4 (101×210 grids), so the deviation of magnetic reconnection is much larger. Assuming that an extra magnetic flux is introduced into the interpolation process, this will reduce the occurrence rate of reconnections. Otherwise, the link of blocks in refined levels is also possible to influence the occurrence rate of reconnections. Especially when intersection appears among several blocks, the treatment of boundary needs more considerations. Therefore, the factors which lead to the variation of the occurrence rate of reconnections after the refining process need a further analysis. Finally, we use the refined SHASTA code to carry out the simulation of the process of magnetic reconnection in a simple configuration of magnetic field, and this is compared with the calculated results by using the original SHASTA code. We find that these two methods have a similar evolution picture of magnetic reconnection in the situation with low magnetic Reynolds number. This is because the high dissipation of plasmoids with low magnetic Reynolds number erases many due details and makes the boundary of regions with high gradient or violent change become vague and diffuse. In the situation of high Reynolds number (Rm ∼ 104 ), this difference becomes very obvious. The original program is terminated due to the process of cumulating errors, while the refined code adequately manifests its power. Not only calculations can be carried on, but also many important physical results are fully revealed. These results are hid in the past because the magnetic Reynolds numbers are not high enough. Considering that the magnetic Reynolds number in actual corona is as high as 106 to 1012 , the refined mesh, which is more advanced theoretically and more feasible technically, is the problem needs to be faced in studying the physical processes of corona from now on. ACKNOWLEDGEMENT The authors thank the referee for careful inspection as well as precious suggestions and opinions to improve the quality of this paper. References 1
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