Reconstructing Force-Free Fields by a Lagrange Multiplier Technique S. Nasiri In collaboration with T. Wiegelmann and B. Inhester MPS Solar Group Seminar June 18, 2013 Contents Force free modeling for solar coronal magnetic field Numerical methods for nonlinear force-free fields The Wheatland et al optimization method Constraint optimization (the method of Lagrange multiplier) Results and discussions Conclusions Force free modeling for solar coronal magnetic field For negligible non-magnetic forces, the force free field is defined as vanishing the Lorentz force J ×B = 0, or (1) (∇×B)×B = 0. (2) Equation (2) is non-linear, in general, and could be rewritten as (∇×B) = α(r)B. (3) Taking divergence of Eq. (3) and by Maxwell's equation ∇·B = 0, (4) equation (2) becomes B·∇α(r) = 0, , which means that α(r) remains constant along any field line. By the coronal conditions it seems that the force free model is applicable To the coronal magnetic field. To get a force free magnetic field inside the computational box in the corona, it has to fulfill the vanishing of net force and torque on the boundary (Molodenskii, 1969; 1974 and Aly, 1989). In the corona, the plasma flow which is mostly governed by the magnetic field, is allowed only in along the field lines and the cross- field flow is strongly prohibited. This means that for negligible pressure and gravity forces, the magnetic field is force free in low order approximation ( <<1). Plasma Beta Model over active regions (Gary, 2001) (Aschwanden, M. J, 2005) In Eq. (3) gives the potential fields (PF), could be used as the simplest assumption for coronal magnetic field (Schatten et al 1969). +LOS of mag. field gives the linear force free fields (LFFF), fits the observations better than PFs (Seehafer, 1978). +LOS of mag. field gives the NLFFF,still more compatible with coronal events: flaring, return current,… (Aly, 1988). For non-linear force free fields (NLFFF) it is difficult to solve the resultant equation analytically, specially for 3D. Reduction to Grad- Shafranov equation and its applications to corona, are relatively simple cases that has been solved as boundary value problem (Low and Lou, 1990). Numerical methods for NLFFFs PFs and LFFFs are inadequate to represent the observed coronal structures in images and the magnetic events happening in the corona. Due to nonlinear behavior and complicated nature of the equations the analytic solutions are limited. To obtain numerical solutions for the nonlinear coronal magnetic fields the computational codes use different methods such as the MHD relaxation (Chodura and Schluter, 1981), Grad-Rubin (Grad and Rubin, 1958; Sakurai, 1981), upward integration (Nakagawa, 1974), boundary-element (Yan and Sakurai, 2000), Euler potentials (Stern, 1966; Parker, 1979) and optimization (Wheatland et al, 2000), methods. It seems that important factors such as reliability of the solutions, fulfillment of ∇·B and ∇×B conditions, computation and converging speed, field connectivity, magnetic energy, alignment of field lines with currents, etc., are better fulfilled with optimization and the Grad – Rubin methods (see Schrijver et al., 2006 and 2008) The Wheatland optimization method The optimization method proposed by Wheatland, Sturrock and Roumeliotis (WRS), 2000, solves an iterative equation obtained by optimizing the following functional Force free conditions: where If we define The Wheatland et al optimization method (continued) Assuming · where is a positive constant, one gets The magnetic field evolves in such a way that L decreases. It is clear that if L goes to zero, then we will have a force free and divergence free field everywhere in the computational box volume. Iteration history of L (left up), force freeness (middle up). Distribution of fractional flux (left down). Comparison of filed lines calculated by Low&Lou (right up)with those reconstructed by WRS model (right down). ( fromWheatland et al, 2000) Constraint optimization (the method of Lagrange multiplier) According to Eq. (2) as the force free field condition, one may optimize the functional subject to the constraint equation (4) which preserves the solenoidal nature of the magnetic field Force free condition where Solenoidal condition Or one may optimize By some vector identities and integration by parts it ca n be shown that where (5) The term in Eq.(5) plays the role of divergence cleaning term which comes out from discretisation scheme for numerical purposes. By assuming (6) One gets (7) Results and discussions We are looking for those that satisfies equations (5) and (7). It satisfies the Poisson equation (8) with the Neumann Boundary condition Solving the Poisson’s equation with above boundary condition during the iteration is possible but requires substantial additional computing time. One possible solution is where is a time independent function As a special case one gets and and noting that it can be easily shown that Other forms of f(r) satisfying Eq. (8) give a class of iteration equation which are different from the Wheatland et al solutions. An example is optimization method using the weighting functions for making the boundary conditions compatible to the flux preserving force free fields. (Wiegelmann, 2004), which uses the following functional Different profiles for w(x, y, z), they are equel to 1 on physical boundary and dccrease to zero across the boundary layer. Another solution for is as follows (9) is a scalar function of space and time with the dimension of magnetic field over length. The solution (9) satisfies where (10) and Starting by a potential field it must be used to Obtain in Eq.(10) which updates B through an iterative scheme and by making L stationary provides a force free field in the volume. This is done for 5000 iteration steps and the results are tested by Lou & Lou analytic solutions as well as with WRS solutions for the same number of iteration steps. In WRS method, since it minimizes both term in the functional integral, simultaneously, it is not necessary to correct for solenoidal deviation due to descretising effect. Here we must do divergence cleaning to get better physical results. This is shown in lower left slide. Relative Magnetic Energy Vector Correlation Mean Vector Error The values of and tells how J ×B = 0 and ∇·B = 0 are fulfilled, respectively. Quality comparison of NLFFF field obtained by WRS and our method for a computational box: nx=32, ny=32,nz=32 and average magnetic field of 184.34 G. Conclusions 1) is considered as a constraint on the force free field condition. 2) The WRS method emerges as a special case of a more general solution for the Lagrange multiplier. 3) plays the role of divergence cleaning term for corrections of the deviation of the magnetic field from solenoidal condition due to descretising effects. 4) The Lagrange multiplier function could be obtained in 3 ways: a) by solving the Poisson equation for b) by solving the following equation c) by writing the optimization equation as Thanks a lot for your attention .