Reconnection of magnetic field lines on ion scale

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1 POSTER 2015, PRAGUE MAY 14 1 Reconnection of magnetic field lines on ion scale Matus CVENGROS 1 1 Dept. of Physics, Czech Technical University in Prague, Břehová 7, Praha 1, Czech Republic cvengmat@fjfi.cvut.cz Abstract. Observation of reconnection of magnetic field lines is made with the use of Particle-In-Cell method in 3D spatial grid, where turbulent flow of electrons and ions is studied. space. For reconnection to occur, theory behind must either include finite conductivity, e. g. based on resistive magnetohydrodynamic theory, or statistical processes in velocity phase space. Keywords Reconnection, ion, magnetic field line, plasma, particlein-cell. 1. Introduction It is common in nature that a magnetic fields of various sources tend to connect and interact, generating some kind of a complex magnetic web. It is also common for the fields that they reconnect their field lines, changing the topology and effectively transition over to a state with lesser energy. Released magnetic energy is then very rapidly deposited to the surrounding plasma, effectively heating the plasma. This process basically creates a magnetic explosion, which can be seen for instance as massive solar flares, which consists of plasma gushes that carry their own magnetic fields. Another interesting phenomenon is magnetic reconnection at Earth s magnetosphere, where change of topology may create closed-loop of magnetic field lines that accelerate particles into Earth s atmosphere, ionizing its outer layers and creating beautiful results known as aurora. However, the space weather which is partially connected to magnetic reconnection can also damage Earth s satellites or electrical infrastructure, which is one of the reason for its further study. In such case, plasma no longer stays on one magnetic field line, but rather moves across many of them due to diffusion principles. This diffusion of particles is the bearer of energy associated with them, which can be released in the change of topology of magnetic field lines. The main occurence of reconnection is in a region called diffusion region. It is there, where diffusion happens and field tends to move into energetically more suitable state. In this region, if plasmas which are carrying oppositely oriented magnetic field lines are brought together, strong current sheet is established, in the presence of which even a vanishingly small amount of resistivity can become important, effectively allowing diffusion of particles across magnetic field lines to happen. Particles are pushed inside the diffusion region because of j B force, where j is charge density of plasma, resulting in plasma streaks as shown in Figure 1. Such change in topology of the magnetic field lines creates so 2. Principle of reconnections Magnetic reconnection cannot happen in ideal magnetohydrodynamics, as in that case conductivity of plasma σ is reviewed as infinite. Reason for this is the second component on the RHS of the equation (1) B t = 1 B + rot (u B), (1) µσ where B is magnetic field, u is velocity of plasma and µ 0 is permeability of space. With σ, the magnetic field lines become freezed into the plasma, copying its movement with dependency on the change of position of velocity vector in Figure 1: Magnetic reconnection inside of a difussion region. Source: Online. called X-points and O-points, where reconnection may create plasmoids, which are plasma packets with frozen magnetic field that they carry. For instance, plasmoids can be seen during eruptions on the surface of the Sun, resulting in sun flares. Example of X-points and O-points is shown in Figure 2.

2 2 M. CVENGROS, Reconnection of magnetic field lines on ion scale Figure 2: X and O points. Source: [1]. 3. Particle-In-Cell in 3D This method is used for simulations in such a case, where simulated plasma is not assessed as continuos fluid, but rather as individual particles. For successful results, many particles are needed, where plasma must fulfill the principle of quasineutrality, represented as Qα n α = 0 = ρ Q = 0, (2) where Q p is charge of a particle, Ω γ is the volume of a cell derived by the position of a given particle at position x and positions of all eight vertexes defining the volume and Γ is any finite set of i, j and k labeling the vertexes of the grid. For illustration purposes, situation is portrayed in Figure 3 for spatial grid of two dimensions, where γ = 1,..., 4. For current density, the process is where ρ Q is charge density, Q α is charge of particles, n α is their density in space and α stands for specific kind of particles, for instance electrons or ions. In this case, it is ensured that the particles are not accelerated beyond all bounds. The cycle of particle-in-cell process is following: 1. Weight the charge density and the current density onto a spatial grid. 2. Solve Maxwell s equations for the fields. 3. Weight the fields to the particle positions. 4. Push the particles using the Lorentz equation Weighting of charges and currents onto spatial grid Each particle has associated its charge and velocity vector. Spatial grid consists of vertexes, which divide it into rows, columns and layers, vertexes of which are marked with the use of symbols i, j and k. These vertexes carry the information about the charge density and current density associated with them, which are later used for finding the solution to Maxwell s equations, such as electric potential φ and vector potential A. Weighting is a process where the charge of a particle Q p and its charge density j p is mapped onto vertexes of a spatial grid. Mapping is dependent on position of each particle in space, which determines its position to the grid, where the volume of grid is any chosen set Ω and its sub-cell volume Ω represents any volume that is surrounded by its eight vertexes in 3D cartesian space. For charge density in vertex i, j, k then follows ρ ijk = Ω γ Q p i, j, k Γ, γ = 1,..., 8, (3) Figure 3: Weighting of charge density in 2D spatial grid. similar, written as j ijk = Ω γ Q p u p i, j, k Γ, γ = 1,..., 8, (4) where u p is velocity vector of any given particle. Reader can obtain details for weighting in [6] or [7] Solution of Maxwell s equations To obtain electric and magnetic field that is mapped on the spatial grid, we first need to obtain scalar potential and vector potential associated with it. Effectively, we are solving Poisson s equation for electric potential φ ijk = ρ ijk ɛ 0 i, j, k Γ, (5) and equation for vector potential A ijk = µ 0 j ijk i, j, k Γ, (6) where ɛ 0 is electric permitivity of vacuum, µ 0 is permeability of vacuum and is the Laplace operator. Numerical integration of these partial differential equations is done with Gauss-Seidel [3] method, where [3] also explains the discretization process Interpolation of fields to particles Electric field is generated from potential as E ijk = φ ijk, (7)

3 POSTER 2015, PRAGUE MAY 14 3 and magnetic field from vector potential as B ijk = A ijk. (8) For interpolation, mid-point differential method [1] is used, which means that the derivation of electric potential in x axis would be φ i,j,k x φ i,j+1,k φ i,j 1,k, (9) 2 x with the same for y and z axis with change of indexes i and k respectively, while j remains unchanged. Rotation is obtained with the same discretization principle Plasma characteristics Plasma consists of electrons and ions, which are generated with Maxwell-Boltzmann distribution. The most probable speed is set as 2kB T v =, (14) m α where k B is Boltzmann s constant, T is temperature that is set to value of T = 1000 K for simulation purposes and m α is the mass of electrons or ions. Electrons and ions are generated uniformly in such a way that they are separated at beginning, and only after simulation start they react to one another. Situation is illustrated at Figure 4. After this process, fields are weighted by the same process as before into separated particles which carry the information about electric and magnetic field acting on them. Assignment of electric field from vertexes of spatial grid to individual particles is then E p = Ω γ E ijk i, j, k Γ, γ = 1,..., 8, (10) and magnetic field similarly as B p = Ω γ B ijk i, j, k Γ, γ = 1,..., 8. (11) 3.4. Particle push Particles are pushed by Lorentz equation F p = Q p ( Ep + u p B p ), (12) where F p is force acting on a particle. For simulation purposes, Boris-Buneman [1] numerical method is used. Figure 4: Starting positions of electrons and ions. 4. Numerical simulation 4.1. Grid geometry For purposes of this paper, spatial grid with periodic boundary conditions and volume Ω = { } x R 3 : x ( a, a) ( b, b) ( c, c), (13) is generated and filled with electrons and ions, where a = b = c = 0.02 m. Grid is then divided into 40 rows, columns and layers, therefore creating a matrix with vertexes. These are the vertexes that the charge density ρ ijk, current density j ijk, electric potential φ ijk, vector potential A ijk, electric field E ijk and magnetic field B ijk are computed at Fields and magnetic reconnection Main advantage of PIC method is the native possibility to track turbulences of magnetic fields generated by particles, since every particle is represented inside a 3D cartesian grid. Simplest scenario for tracking magnetic reconnection is to use current sheet and create so called neutral layer, where magnetic field is generated with opposing directions above and under the sheet. Current density j ijk is j ijk = ρ ijk u ijk i, j, k Γ. (15) Since u ijk is velocity vector of sum of velocities of particles inside a vertex of the grid with the coordinates i, j and k, then generation of actual current sheet is not needed, because current is generated by particles themselves. Instead, external magnetic field can be generated, which forces particles to

4 4 M. CVENGROS, Reconnection of magnetic field lines on ion scale the neutral layer, since they feel the force density of f ijk as f ijk = jijk Bijk i, j, k Γ, (16) before their interpolation. This field is generated as Bxijk (zijk ) = B0 tanh zijk z0 i, j, k Γ, (17) where Bxijk represents that only x component of magnetic field mapped onto spatial grid is changed and is also dependent only on the position of vertexes of the grid at coordinates zijk, which could be also expressed as zijk = hk, where h is the distance between cells in z-axis and k is the number of any given vertex in z-axis. B0 is strength of the external field and z0 is the scale of the grid in z-axis. Such field in xz plane represents Figure 5, where B0 is set to B0 = T and z0 = 0.04 m. With the external magnetic field pushing Figure 6: Reconnection of magnetic field lines generated by particles. The background color corresponds to the field magnitude. Figure 5: External magnetic field generated by hyperbolic tangent function. The background color corresponds to the field magnitude. particles to the neutral layer, reconnection occurs, which can be seen in Figure 6. Figure 6 represents yz plain of magnetic field with slice being held at x coordinate x = 0 m, with highest magnitude of magnetic field generated by particles Bmax = ± T. For electric potential at this slice alongside with electric field incorporating it, results can be seen in Figure 7, where maximum electric potential held by particles is φmax = ± V and the lowest potential is in close proximity to this value but with minus sign due to symmetricity of the problem. Figure 7: Electric potential and orientation of electric field generated by particles. The background color corresponds to the field magnitude. 5. Conclusion Current research proved that presented 3D Particle-In-Cell code works and reconnection of magnetic field lines is indeed happening. Further research is focused on diagnostics of this phenomenon, for instance the typical reconnection

5 POSTER 2015, PRAGUE MAY 14 5 time and the velocity of plasma gushes extruded by magnetic field. Acknowledgements Research described in the paper was supervised by Prof. RNDr. Petr Kulhánek, CSc. and supported by the CTU grant No. SGS15/073/OHK3/1T/13 Waves and Instabilities in Plasmas Basic Research. Bibliography [1] KULHÁNEK, P. Úvod do teorie plazmatu. AGA, Praha ISBN [2] NEZBEDA, I., KOLAFA, J., KOTRLA, M. Úvod do počítačových simulací. Metody Monte Carlo a molekulární dynamiky. Nakladatelství Karolinum, Praha ISBN [3] PRESS, W., TEUKOLSKY, S., VATTERLING, W., FLANNERY, B. Numerical Recipes. The Art of Scientific Computing. Third Edition. Cambridge University Press. ISBN [4] CHEN, F. Introduction to Plasma Physics. Plenum Press, New York. ISBN [5] CHEN, F. Introduction to plasma physics and controlled fusion. Volume 1: Plasma physics. Springer. ISBN [6] REN, Ch. Introduction to Particle-in-Cell Methods in Plasma Simulations. University of Rochester. [Online] SummerSchool/lectures/Ren.pdf. [7] FILIPIČ, G. Principles of Particle in cell simulations. University of Ljubljana. Faculty for mathematics and physics. [Online] uni-lj.si/seminar/files/2007_2008/ Seminar2.pdf. About the Author... Matus CVENGROS was born in 16 November He currently studies at the Czech Technical University in Prague, Faculty of Nuclear Sciences and Physical Engineering with specialization in Physics and Techniques of Thermonuclear Fusion. He succumbed to the beauty of numerical simulations and is currently focusing his effort on plasma physics.