Halima Ali

Symmetric simple map for a single-null divertor tokamak

A new map called the symmetric simple map is introduced to represent the chaotic trajectories of magnetic field lines in the scrape-off layer of a single-null divertor tokamak. Good surfaces of this map are very nearly axisymmetric. Therefore it gives a far better representation of the magnetic topology of a single-null divertor tokamak. The map is investigated in detail and used to analyze the generic features of the field line trajectories and their footprint on the divertor plate. The map is employed to calculate the variations in the fraction of magnetic flux from the stochastic layer diverted onto plate, in the footprint and in related parameters as the map parameter is varied. The Lyapunov exponents and the field diffusion coefficients are calculated. The low mode number map and the dipole map are introduced to include the effects of low and high mode number perturbations in the new map.

The low MN map for single-null divertor tokamaks

The low MN map is derived from the general theory of maps and the generating function for the low mn perturbation. The unperturbed magnetic topology of a single-null divertor tokamak is represented by the symmetric simple map. The perturbed topology is represented by the low MN map. The method of maps is applied to calculate the effects of low mn perturbation on the stochastic layer and the magnetic footprint. The low mn perturbation organizes the stochastic layer into large scale spatial structures. This is reflected in the phase portraits, safety factor, Liapunov exponents, magnetic footprints, and the semiconnection length. For the expected range of the amplitude of the low mn perturbation, the fraction of magnetic flux escaping the stochastic layer, the width of stochastic layer, the area of the magnetic footprints increase, while the number of hot spots and the fraction of flux going into the hot spots both decrease. The key features of the complex patterns in the heat deposition on the collector pla...

Symplectic approach to calculation of magnetic field line trajectories in physical space with realistic magnetic geometry in divertor tokamaks

A new approach to integration of magnetic field lines in divertor tokamaks is proposed. In this approach, an analytic equilibrium generating function (EGF) is constructed in natural canonical coordinates (ψ,θ) from experimental data from a Grad–Shafranov equilibrium solver for a tokamak. ψ is the toroidal magnetic flux and θ is the poloidal angle. Natural canonical coordinates (ψ,θ,φ) can be transformed to physical position (R,Z,φ) using a canonical transformation. (R,Z,φ) are cylindrical coordinates. Another canonical transformation is used to construct a symplectic map for integration of magnetic field lines. Trajectories of field lines calculated from this symplectic map in natural canonical coordinates can be transformed to trajectories in real physical space. Unlike in magnetic coordinates [O. Kerwin, A. Punjabi, and H. Ali, Phys. Plasmas 15, 072504 (2008)], the symplectic map in natural canonical coordinates can integrate trajectories across the separatrix surface, and at the same time, give traject...

Effects of dipole perturbation on the stochastic layer and magnetic footprint in single-null divertor tokamaks

In this paper, the method of maps is used to calculate the effects of high toroidal and poloidal mode number perturbation on the trajectories of magnetic field lines in a single-null divertor tokamak. First, a simplified derivation of the dipole map from the Hamiltonian mechanics of magnetic field is given. This map represents the effects of an externally located current carrying coil on the motion of field lines. The unperturbed magnetic field topology of a single-null divertor tokamak is represented by the symmetric simple map. The coil is placed across from the X-point on the line joining the X-point and the O-point at a fixed distance from the last good confining surface. The effects of coil on the stochastic layer and magnetic footprint are calculated using the symmetric simple map and the dipole map. Self-similarities, singularities, and topological equivalences in the pattern of physical parameters are found that characterize the stochastic layer and the magnetic footprint. The dipole perturbation ...

Derivation of the dipole map

In our method of maps [Punjabi et al., Phy. Rev. Lett. 69, 3322 (1992), and Punjabi et al., J. Plasma Phys. 52, 91 (1994)], symplectic maps are used to calculate the trajectories of magnetic field lines in divertor tokamaks. Effects of the magnetic perturbations are calculated using the low MN map [Ali et al., Phys. Plasmas 11, 1908 (2004)] and the dipole map [Punjabi et al., Phys. Plasmas 10, 3992 (2003)]. The dipole map is used to calculate the effects of externally located current carrying coils on the trajectories of the field lines, the stochastic layer, the magnetic footprint, and the heat load distribution on the collector plates in divertor tokamaks [Punjabi et al., Phys. Plasmas 10, 3992 (2003)]. Symplectic maps are general, efficient, and preserve and respect the Hamiltonian nature of the dynamics. In this brief communication, a rigorous mathematical derivation of the dipole map is given.

An accurate symplectic calculation of the inboard magnetic footprint from statistical topological noise and field errors in the DIII-D

Any canonical transformation of Hamiltonian equations is symplectic, and any area-preserving transformation in 2D is a symplectomorphism. Based on these, a discrete symplectic map and its continuous symplectic analog are derived for forward magnetic field line trajectories in natural canonical coordinates. The unperturbed axisymmetric Hamiltonian for magnetic field lines is constructed from the experimental data in the DIII-D [J. L. Luxon and L. E. Davis, Fusion Technol. 8, 441 (1985)]. The equilibrium Hamiltonian is a highly accurate, analytic, and realistic representation of the magnetic geometry of the DIII-D. These symplectic mathematical maps are used to calculate the magnetic footprint on the inboard collector plate in the DIII-D. Internal statistical topological noise and field errors are irreducible and ubiquitous in magnetic confinement schemes for fusion. It is important to know the stochasticity and magnetic footprint from noise and error fields. The estimates of the spectrum and mode amplitude...

Modeling of stochastic broadening in a poloidally diverted discharge with piecewise analytic symplectic mapping flux functions

A highly accurate calculation of the magnetic field line Hamiltonian in DIII-D [J. L. Luxon and L. E. Davis, Fusion Technol. 8, 441 (1985)] is made from piecewise analytic equilibrium fit data for shot 115467 3000ms. The safety factor calculated from this Hamiltonian has a logarithmic singularity at an ideal separatrix. The logarithmic region inside the ideal separatrix contains 2.5% of toroidal flux inside the separatrix. The logarithmic region is symmetric about the separatrix. An area-preserving map for the field line trajectories is obtained in magnetic coordinates from the Hamiltonian equations of motion for the lines and a canonical transformation. This map is used to calculate trajectories of magnetic field lines in DIII-D. The field line Hamiltonian in DIII-D is used as the generating function for the map and to calculate stochastic broadening from field-errors and spatial noise near the separatrix. A very negligible amount (0.03%) of magnetic flux is lost from inside the separatrix due to these n...

Effects of low and high mode number tearing modes in divertor tokamaks

The topological effects of magnetic perturbations on a divertor tokamak, such as DIII-D, are studied using field-line maps that were developed by Punjabi et al. [A. Punjabi, A. Verma, and A. Boozer, Phys. Rev. Lett. 69, 3322 (1992)]. The studies consider both long-wavelength perturbations, such as those of m=1, n=1 tearing modes, and localized perturbations, which are represented as a magnetic dipole. The parameters of the dipole map are set using DIII-D data from shot 115467 in which the C-coils were activated [J. L. Luxon and L. E. Davis, Fusion Technol. 8, 441 (1985)]. The long-wavelength perturbations alter the structure of the interception of magnetic field lines with the divertor plates, but the interception is in sharp lines. The dipole perturbations cause a spreading of the interception of the field lines with the divertor plates, which alleviates problems associated with heat deposition. Magnetic field lines are the trajectories of a one-and-a-half degree of freedom Hamiltonian, which strongly co...

The symmetric quartic map for trajectories of magnetic field lines in elongated divertor tokamak plasmas

The coordinates of the area-preserving map equations for integration of magnetic field line trajectories in divertor tokamaks can be any coordinates for which a transformation to (ψt,θ,φ) coordinates exists [A. Punjabi, H. Ali, T. Evans, and A. Boozer, Phys. Lett. A 364, 140 (2007)]. ψt is toroidal magnetic flux, θ is poloidal angle, and φ is toroidal angle. This freedom is exploited to construct the symmetric quartic map such that the only parameter that determines magnetic geometry is the elongation of the separatrix surface. The poloidal flux inside the separatrix, the safety factor as a function of normalized minor radius, and the magnetic perturbation from the symplectic discretization are all held constant, and only the elongation is κ varied. The width of stochastic layer, the area, and the fractal dimension of the magnetic footprint and the average radial diffusion coefficient of magnetic field lines from the stochastic layer; and how these quantities scale with κ is calculated. The symmetric quar...

Simple map in action-angle coordinates

A simple map [A. Punjabi, A. Verma, and A. Boozer, Phys. Rev. Lett. 69, 3322 (1992)] is the simplest map that has the topology of divertor tokamaks [A. Punjabi, H. Ali, T. Evans, and A. Boozer, Phys. Lett. A 364, 140 (2007)]. Here, action-angle coordinates, the safety factor, and the equilibrium generating function for the simple map are calculated analytically. The simple map in action-angle coordinates is derived from canonical transformations. This map cannot be integrated across the separatrix surface because of the singularity in the safety factor there. The stochastic broadening of the ideal separatrix surface in action-angle representation is calculated by adding a perturbation to the simple map equilibrium generating function. This perturbation represents the spatial noise and field errors typical of the DIII-D [J. L. Luxon and L. E. Davis, Fusion Technol. 8, 441 (1985)] tokamak. The stationary Fourier modes of the perturbation have poloidal and toroidal mode numbers (m,n,)={(3,1),(4,1),(6,2),(7,2...