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

Abstract

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 ͑1985͔͒. 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 amplitudes of the spatial topological noise and magnetic errors in the DIII-D are used as magnetic perturbation. The discrete and continuous symplectic maps are used to calculate the magnetic footprint on the inboard collector plate of the DIII-D by inverting the natural coordinates to physical coordinates. The combination of highly accurate equilibrium generating function, natural canonical coordinates, symplecticity, and small step-size together gives a very accurate calculation of magnetic footprint. Radial variation of magnetic perturbation and the response of plasma to perturbation are not included. The inboard footprint from noise and errors are dominated by m =3, n = 1 mode. The footprint is in the form of a toroidally winding helical strip. The width of stochastic layer scales as 1 2 power of amplitude. The area of footprint scales as first power of amplitude. The physical parameters such as toroidal angle, length, and poloidal angle covered before striking, and the safety factor all have fractal structure. The average field diffusion near the X-point for lines that strike and that do not strike differs by about three to four orders of magnitude. The magnetic footprint gives the maximal bounds on size and heat flux density on collector plate. I. SYMPLECTIC INVARIANCE, MAGNETIC TOPOLOGY, MAGNETIC GEOMETRY, AND CANONICAL REPRESENTATIONSAny globally divergence-free vector field, such as the magnetic field B ជ in magnetic confinement schemes of tokamaks, stellarators, and reversed filed pinches for fusion, can be expressed asand are, respectively, any poloidal and toroidal angles; and and are the toroidal and poloidal magnetic fluxes, respectively. 1-3 From this expression for B ជ , we can show thatThis means that the trajectories of magnetic field lines are a Hamiltonian system, and the Hamiltonian function for this system is the poloidal magnetic flux, = ͑ , , ͒, with as its generalized position, as generalized momentum, and as generalized time. The trajectories of magnetic field lines are a single degree of freedom...

Symplectic calculation of the outboard magnetic footprint from noise and error fields in the DIII-D

“…The first equation of the forward map, i.e., (3), is an implicit equation in Ψ n+1 ; the second equation, (4), is an explicit equation in θ n . So, in advancing the forward map (ψ n , θ n ) → (ψ n+1 , θ n+1 ) in canonical time ϕ, we first solve Ψ n+1 equation implicitly, and then solve the θ n equation explicitly [1,2,7].…”

“…The results on the inboard footprint from noise and error using the forward DIII-D map are given in Ref. [7]. In this paper we study the stochastic layer and the outboard magnetic footprint created in the DIII-D from the statistical topological noise and field errors using the backward DIII-D map.…”

“…In this paper we study the stochastic layer and the outboard magnetic footprint created in the DIII-D from the statistical topological noise and field errors using the backward DIII-D map. The mode numbers for noise and error fields are (m,n) = {(3,1), (4,1), (6,2), (7,2), (8,2), (9,3), (10,3), (11,3)}. The amplitude δ is in the range 8×10 −6 6 δ 6 2×10 −5 .…”

“…The initial conditions for all the 100 000 lines are exactly the same as those for the forward map in Ref. [7]; now they become the initial conditions for the backward map. The scaling of the width of stochastic layer with the amplitude is calculated.…”

Divertor map with freedom of geometry and safety factor profile

“…Other divertor maps have been obtained by the mathematical construction of appropriate generating functions and canonical transformations [12][13][14][15][16]. Divertor maps using the geometry of the DIII-D tokamak have been obtained by fitting experimental data to find appropriate expressions for equilibrium Hamiltonians [17][18][19].…”