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Nonlinearly perturbed MHD equilibria, with or without magnetic islands Stuart Hudson, and R.L. Dewar, M.J. Hole & M. McGann PPPL Australian National University → The simplest model of approximating global, macroscopic force-balance in toroidal plasma confinement with arbitrary geometry is magnetohydrodynamics (MHD). → Non-axisymmetric magnetic fields generally do not have a nested family of smooth flux surfaces, unless ideal surface currents are allowed at the rational surfaces. → If the field is non-integrable (chaotic, fractal phase space), then any continuous pressure that satisfies B∙p=0 must have an infinitely discontinuous gradient, p. → Instead, solutions with stepped-pressure profiles are guaranteed to exist. A partially-relaxed, topologically-constrained, MHD energy principle is described. → Equilibrium solutions are calculated numerically. Results demonstrating convergence tests, benchmarks, and non-trivial solutions are presented. → The constraints of ideal MHD may be applied at the rational surfaces, in which case surface currents prevent the formation of islands. Or, these constraints may be relaxed in the vicinity of the rational surfaces, in which case magnetic islands will open if resonant perturbations are applied. An ideal equilibrium with non-integrable (chaotic) field and continuous pressure, is infinitely discontinous ideal MHD theory p j B, gives B p 0 → transport of pressure along field is “infinitely” fast → no scale length in ideal MHD → pressure adapts to fractal structure of phase space chaos theory nowhere are flux surfaces continuously nested *for non-symmetric systems, nested family of flux surfaces is destroyed *islands & irregular field lines appear where transform is rational ( n / m); rationals are dense in space Poincare-Birkhoff theorem periodic orbits, (e.g. stable and unstable) guaranteed to survive into chaos *some irrational surfaces survive if there exists an r , k s.t. for all rationals, |ι - n / m | r m -k i.e. rotational-transform, , is poorly approximated by rationals, Diophantine Condition Kolmogorov, Arnold and Moser ideal MHD chaos infinitely discontinuous equilibrium 0, if ( m, n) s.t. | x n | r m k p' m *iterative method for calculating equilibria is ill-posed; 1, otherwise 1) B n p 0 p is everywhere discontinuous, or zero; flatten pressure at every rational infinite fractal structure 2) j B n p Bn 2 pressure not Riemann integrable j everywhere discontinuous or zero ; 3) B n j B is densely and irregularly singular; is single valued if and only if j dl / B 0 C pressure must be flat across every closed field line, or parallel current is not single-valued; 4) B n 1 j B n +j solution only if B +j 0 radial coordinate≡transform To have a well-posed equilibrium with chaotic B need to introduce non-ideal terms, such as resistivity, , perpendicular diffusion, , [ HINT , M 3D, NIMROD, ..], or return to an energy principle, but relax infinity of ideal MHD constraints Instead, a multi-region, relaxed energy principle for MHD equilibria with non-trivial pressure and chaotic fields Energy, helicity and mass integrals (defined in nested annular volumes) p B2 Wl Vl 1 2 dv , Hl Vl A B dv , Ml Vl p1/ dv energy helicity mass F l 1Wl l H l / 2 l M l N Seek constrained, minimum-energy state 1st variation due to unconstrained variations p, A, and interface geometry, ξ, except ideal "topological" constraint B × ξ × B imposed discretely at interfaces 1 l p1/ 1 F l 1 { Vl 1 p dv A B l B dv [[ p B 2 / 2]] ξ.dS } N Vl Vl continuity of total pressure B l B in each annulus p1/ p /( 1) const . across interfaces in each annulus Equilibrium solutions when B l B in annuli, [[p+B2 /2]]=0 across interfaces partial Taylor relaxation allowed in each annulus; allows for topological variations/islands/chaos; global relaxation prevented by ideal constraints; non-trivial stepped pressure solutions; B l B is a linear equation for B; depends on interface geometry; solved in parallel in each annulus; solving force balance adjusting interface geometry to satisfy [[p+B2 /2]]=0; ideal interfaces that support pressure generally have irrational rotational-transform; standard numerical problem finding zero of multi-dimensional function; call NAG routine; → this was a strong motivation for pursuing the stepped-pressure equilibrium model → how large the “sufficiently small” departure from axisymmetry can be needs to be explored numerically By definition, an equilibrium code must constrain topology; Definition: Equilibrium Code (fixed boundary) given (1) boundary (2) pressure (3) rotational-transform inverse q-profile (or current profile) calculate B that is consistent with force-balance; pressure profile is not changed ! c.f. " coupled equilibrium - transport " approach, that evolves pressure while evolving field Cannot apriori specify pressure without apriori constraining topology of the field the constraint B p =0 means the structure of B and p are intimately connected; if p is given and B that satisfies force balance is to be constructed, then flux surfaces must coincide with pressure gradients; (e.g. if p is smooth, B must have nested surfaces). specifying the profiles discretely is a practical means of retaining some control over the profiles, whilst making minimal assumptions regarding the topology; pressure gradients are assumed to coincide with a set of strongly-irrational " noble " flux surfaces Farey tree noble irrational limit of alternating path down Farey-tree Fibonacci sequence p1 p2 p1 p2 p1 p2 , , ,. . . , golden mean q1 q2 q1 q2 q1 q2 Extrema of energy functional obtained numerically; introducing the Stepped Pressure Equilibrium Code (SPEC) The vector-potential is discretized * toroidal coordinates ( s , , ), *interface geometry R l m,n Rl , m , n cos( m n ), Z l Z l , m , n sin( m n ) m,n * exploit gauge freedom A A ( s , , ) A ( s , , ) * Fourier A m , n a ( s ) cos( m n ) * Finite-element a ( s ) i a , i ( s ) ( s ) piecewise cubic or quintic basis polynomials and inserted into constrained-energy functional F l 1 Wl l H l / 2 l M l N * derivatives w.r.t. vector-potential linear equation for Beltrami field B B solved using sparse linear solver * field in each annulus computed independently, distributed across multiple cpus * field in each annulus depends on enclosed toroidal flux (boundary condition) and poloidal flux, P , and helicity-multiplier, adjusted so interface transform is strongly irrational geometry of interfaces, ξ Rm , n , Z m , n Force balance solved using multi-dimensional Newton method. * interface geometry is adjusted to satisfy force F[ξ] [[p+ B 2 ]]m , n =0 2 * angle freedom constrained by spectral-condensation, adjust angle freedom to minimize ( m 2 n 2 ) Rmn Z mn 2 2 * derivative matrix, F[ξ], computed in parallel using finite-differences minimal spectral width [Hirshman, VMEC] * call NAG routine: quadratic-convergence w.r.t. Newton iterations; robust convex-gradient method; Numerical error in Beltrami field scales as expected Scaling of numerical error with radial resolution depends on finite-element basis A =A A , B = A, j= B, need to quantify error = j - B A , A O( h n ) h = radial grid size = 1 / N j - B s O ( h n 1 ) j - B O ( h n 2 ) j - B n = order of polynomial n2 O(h ) g B s A A O (h n ) error (logscale) g B s A O(h n 1 ) g B s A O(h n 1 ) g j s O (h n 1 ) g j O (h n 2 ) N g j O(h n 2 ) (logscale) Poincaré plot, =0 Poincaré plot, =π Example of chaotic Beltrami field in single given annulus; R 1.0 r ( , ) cos , Z r ( , ) sin , (m,n)=(3,1) island + (m,n)=(2,1) island inner surface = chaos r 0.1 outer interface r 0.2 cos(2 ) cos(3 ) Stepped-pressure equilibria accurately approximate smooth-pressure axisymmetric equilibria upper half = SPEC & VMEC in axisymmetric geometry . . . magnetic fields have family of nested flux surfaces cylindrical Z equilibria with smooth profiles exist, may perform benchmarks (e.g. with VMEC) increasing pressure lower half = SPEC interfaces (arbitrarily approximate smooth-profile with stepped-profile) approximation improves as number of interfaces increases location of magnetic axis converges w.r.t radial resolution magnetic axis vs. radial resolution using quintic-radial finite-element basis (for high pressure equilibrium) (dotted line indicates VMEC result) cylindrical Z cylindrical R cylindrical R increasing pressure resolution ≡ number of interfaces N ≡ finite-element resolution stepped-profile approximation to smooth profile Pressure, p transform toroidal flux toroidal flux Equilibria with (i) perturbed boundary & chaotic fields, and (ii) pressure are computed . Poincaré plot (cylindrical) Poincaré plot (cylindrical) β = 0% β 4% boundary deformation induces islands R 1.0 r cos , Z r sin r 0.3 cos(2 ) cos(3 ) (2,1) island 104 Demonstrated Convergence of high-pressure equilibrium with islands, with Fourier Resolution, (3,1) island Convergence of (2,1) & (3,1) island widths .. Poincaré plot (toroidal) with Fourier resolution, β 4% case β 4% poloidal resolution 0mM toroidal resolution -N n N Pressure profile W radial coordinate poloidal angle Sequence of equilibria with increasing pressure shows plasma can have significant response to external perturbation. axisymmetric plus small perturbation R 1.00 0.30 cos( ) 0.05 cos(2 ) [ 21 cos(2 ) 31 cos(3 )]cos( ) Z 1.00 0.40 sin( ) [ 21 cos(2 ) 31 cos(3 )]sin( ) tot 0.000 tot 0.018 100s resonant error field TVMEC TF 30s TF 1.5h pressure If ideal constraint applied at rational surfaces, then shielding currents prevent island formation. axisymmetric boundary, plus perturbation ( =104 ) R 1.0 0.3cos 0.05 cos 2 , R cos(2 ) cos Z 0.4 sin Z cos(2 ) sin rotational transform with rational ideal interface non-linear IPEC cylindrical Z pressure gradients coincide with without rational ideal interface irrational interfaces q=1/2 island opens pressure cylindrical R Summary → A partially-relaxed, topologically-constrained energy principle has been described and the equilibrium solutions constructed numerically * using a high-order (piecewise quintic) radial discretization, and a spectrally condensed Fourier representation * workload distrubuted across multiple cpus, * extrema located using standard numerical methods (NAG): modified Newton’s method, with quadratic-convergence * non-axisymmetric solutions with chaotic fields and non-trivial pressure guaranteed to exist (under certain conditions) → Specifying the profiles discretely is a practical means of retaining some control over the profiles, while making minimal assumptions regarding the topology of the field * it is only assumed that some flux surfaces exist * pressure gradients coincide with strongly irrational flux surfaces → Convergence studies have been performed * expected error scaling with radial resolution confirmed * detailed benchmark with axisymmetric equilibria (with smooth profiles) * demonstrated convergence of island widths with Fourier resolution → By enforcing the ideal constraint at the rational surfaces, the formation of magnetic islands is prohibited by the formation of surface “shielding” currents * similar to non-linear generalization of IPEC * relaxing ideal constraint at rational surfaces allows islands to open Sequence of equilibria with increasing pressure shows plasma can have significant response to external perturbation. axisymmetric plus small perturbation R 1.00 0.30 cos( ) 0.05 cos(2 ) [ 21 cos(2 ) 31 cos(3 )]cos( ) Z 1.00 0.40 sin( ) [ 21 cos(2 ) 31 cos(3 )]sin( ) tot 0.00 tot 0.05 resonant error field pressure Sequence of equilibria with increasing pressure shows plasma can have significant response to external perturbation. axisymmetric plus perturbation 21 = 31 =10 4 R 1.00 [0.30 21 cos(2 ) 31 cos(3 )]cos( ) Z 1.00 [0.30 21 cos(2 ) 31 cos(3 )]sin( ) Resonant radial field at rational surface; n=1,2,3 stability from PEST; Sequence of equilibria with slowly increasing pressure axisymmetric : R 1.00 0.30 cos( ) 0.05 cos(2 ) plus Z 1.00 0.40 sin( ) perturbation : R [ 21 cos(2 ) 31 cos(3 )]cos( ) Z [ 21 cos(2 ) 31 cos(3 )]sin( ) tot 0.000 tot 0.018 resonant error field TF 20s TF 60 m pressure Toroidal magnetic confinement depends on flux surfaces Transport in magnetized plasma dominately parallel to B if the field lines are not confined (e.g. by flux surfaces), then the plasma is poorly confined Axisymmetric magnetic fields possess a continuously nested family of flux surfaces nested family of flux surfaces is guaranteed if the system has an ignorable coordinate magnetic field is called integrable rational field-line periodic trajectory family of periodic orbits ≡ rational flux surface rational field-line = 0.3333… irrational field-lines cover irrational flux surface periodic poloidal angle magnetic field lines wrap around toroidal “flux” surfaces irrational field-line = 0.3819… periodic poloidal angle straight-field-line flux coordinates, B 0 B g B magnetic differential equation, B s, is singular at rational surfaces, m n m, n i ( g s ) m , n periodic toroidal angle Ideal MHD equilibria are extrema of energy functional The energy functional is W ( p B 2 / 2) dv V ≡ global plasma volume V ideal variations mass conservation t v 0 state equation dt ( p ) 0 Faraday's law, ideal Ohm's law B × ξ × B →ideal variations don’t allow field topology to change “frozen-flux” the first variation in plasma energy is globally ideally-constrained W p - j× B ξ dv Euler Lagrange equation for variations V ideal-force-balance p j× B → two surface functions, e.g. the pressure, p(s) , and rotational-transform ≡ inverse-safety-factor, (s) , and → a boundary surface ( . . for fixed boundary equilibria . . . ) , constitute “boundary-conditions” that must be provided to uniquely define an equilibrium solution . . . . . . The computational task is to compute the magnetic field that is consistent with the given boundary conditions . . . nested flux surface topology maintained by singular currents at rational surfaces from ( B j ) 0, parallel current must satisfy B - j , 2 where j B × p / B i ( g j ) m , n → magnetic differential equations are singular at rational surfaces (periodic orbits) m,n (m n) → pressure-driven “Pfirsch-Schlüter currents” have 1/ x type singularity ( m n) → - function singular currents shield out islands Topological constraints : pressure gradients coincide with flux surfaces The ideal interfaces are chosen to coincide with pressure gradients parallel transport dominates perpendicular transport, → structure of B and structure of the pressure are intimately connected; simplest approximation is B p 0 → cannot apriori specify pressure without pressure gradients must coincide with KAM surfaces ideal interfaces apriori constraining structure of the field; 10 p is small, e.g. t p p p 0, with , e.g. 2 2 [next order of approximation, B 10 *pressure gradients coincide with KAM surfaces, cantori . . → where there are significant pressure gradients, 1/ 4 *pressure flattened across islands, chaos with width > w C there can be no islands or chaotic regions with width > ∆wc * anisotropic diffusion equation solved analytically, p' 1 / 2 G , 2 is quadratic-flux across cantori, G is metric term ] A fixed boundary equilibrium is defined by : (i) given pressure, p( ), and rotational-transform profile, ( ) (ii) geometry of boundary; (a) only stepped pressure profiles are consistent (numerically tractable) with chaos and B p 0 (b) the computed equilibrium magnetic field must be consistent with the input profiles (a) + (b) = where the pressure has gradients, the magnetic field must have flux surfaces. non-trivial stepped pressure equilibrium solutions are guaranteed to exist Taylor relaxation: a weakly resistive plasma will relax, subject to single constraint of conserved helicity Taylor relaxation, [Taylor, 1974] W V pB 2 / 2 dv , H V A B dv plasma energy helicity, B =A Constrained energy functional F W H / 2, Lagrange multiplier Euler-Lagrange equation, for unconstrained variations in magnetic field, B B linear force-free field ≡ Beltrami field But, . . .Taylor relaxed fields have no pressure gradients Ideal MHD equilibria and Taylor-relaxed equilibria are at opposite extremes . . . . Ideal-MHD → imposition of infinity of ideal MHD constraints non-trivial pressure profiles, but structure of field is over-constrained Taylor relaxation → imposition of single constraint of conserved global helicity structure of field is not-constrained, but pressure profile is trivial, i.e. under-constrained We need something in between . . . . . . perhaps an equilibrium model with finitely many ideal constraints, and partial Taylor relaxation? Introducing the multi-volume, partially-relaxed model of MHD equilibria with topological constraints Energy, helicity and mass integrals p B2 Wl Vl 1 2 dv , Hl Vl A B dv , Ml Vl p1/ dv V4 V3 plasma energy helicity mass V2 Multi-volume, partially-relaxed energy principle V1 * A set of N nested toroidal surfaces enclose N annulur volumes the interfaces are assumed to be ideal, B × ξ × B * The multi-volume energy functional is F l 1 Wl l H l / 2 l M l N → field remains tangential to interfaces, → a finite number of ideal constraints, Euler-Lagrange equation for unconstrained variations in A imposed topologically! In each annulus, the magnetic field satisfies Bl l Bl Euler-Lagrange equation for variations in interface geometry Across each interface, pressure jumps allowed, but total pressure is continuous [[p+ B 2 2 ]]=0 an analysis of the force-balance condition is that the interfaces must have strongly irrational transform ideal interfaces coincide with KAM surfaces

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posted: | 2/27/2013 |

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