Bifurcations in 3D MHD equilibria, Influence of increasing pressure on islands Stuart R. Hudson...

Bifurcations in 3D MHD equilibria,Influence of increasing pressure on islands

Stuart R. Hudson

Outline of talk

1. Some new numerical results2. Some old theoretical predictions3. Some future research plans

1. Partition the plasma into N nested volumes (Taylor-relaxed, “resistive” volumes separated by ideal interfaces)

2. Define Energy and Helicity integrals

3. Construct multi-region, relaxed MHD energy functional, called MRxMHD

4. The extremizing solutions satisfy the Euler-Lagrange equations

5. Numerical implementation, Stepped Pressure Equilibrium Code (SPEC)

Computation of multi-region relaxed magnetohydrodynamic equilibriaS. R. Hudson, R. L. Dewar, G. Dennis, M. J. Hole, M. McGann, G. von Nessi and S. Lazerson

Physics of Plasmas, 19:112502, 2012

,2l l l ol

F W H H

The 3D equilibrium model is Multi-Region, relaxed MHD,

a generalization of ideal MHD and Taylor relaxation

2

where

energy helicity

and ., , 1 2

l l

l l pV constV V

p BW Hdv dv

B × AA B

2 [[p+B /2]] 0l l l B B

4

1.0 cos ,

boundary deformation induce

r sin

0.3 cos(2 ) cos(3 )

s islands

10

R r Z

r

Consider a circular-cross-section, axisymmetric tokmak with a small, resonant perturbation . . .

rotational-transform (constrained only at interfaces)

An equilibrium is defined by1.a given fixed boundary, 2.a pressure profile, and 3.a transform profile.

toroidal flux toroidal flux

As pressure is increased, the island may grow dramatically,and birfucated solutions may exist

Greene’s residue, R, serves as an indicator ofisland width and phase

ideal stabilityboundary(PEST)

increasing pressure

As pressure is increased, the island may grow dramatically,and birfucated solutions may exist

The following movies show a sequence of high resolution (M=10, N=5) SPEC equilibriafrom low to high pressure, and . . . from high to low pressure

Tokamak magnetic islands in the presence of nonaxisymmetric perturbationsA. H. ReimanPhysics of Fluids B, 3, 2167 (1991)

This is an old, important, problem.

increasing pressure


Theory of pressure‐induced islands and self‐healing in three‐dimensional toroidal magnetohydrodynamic equilibriaA. Bhattacharjee, T. Hayashi, C.C. Hegna, N. Nakajima & T. SatoPhysics of Plasmas 2, 883 (1995)

increasing pressure


Eliminating islands in high-pressure free-boundary stellarator magnetohydrodynamic equilibrium solutionsS. R. Hudson, D. A. Monticello, A. H. Reiman, A. H. Boozer, D. J. Strickler, S. P. Hirshman and M. C. ZarnstorffPhysical Review Letters, 89:275003-1 (2002)

It is time to re-visit this problem with a new computational tool.

The SPEC code is based on the multi-region, relaxed MHD energy functional,

SPEC is the first code to be based on a well-defined energy functional, so a self-consistent equilibrim and stability analysis is possible

As N, MRxMHD ideal MHD

Research Plan.1. Elucidate the nature of singular currents at rational surfaces,2. Understand the numerical results in context of previous theoretical treatments, e.g. Reiman et al., Bhattacharjee et. al,3. Extend SPEC to compute stability of partially relaxed MHD equilibria,c.f. axisymmetric stability, TERPSICHORE4. Consider importance of flow shear . . .

Ideal magnetohydrodynamic stability of configurations without nested flux surfacesP. Helander and S. L. Newton Physics of Plasmas, 20, 062504 (2013)

islands & chaos emerge at every rational about each rational / , introduce excluded region, width /

flux surface can survive if | / | / ,

k

k

n m r m

n m r m

KAM Theorem (Kolmogorov, Arnold, Moser) - " "

for all ,

Greene's residue criterion the most robust flux s

stronglywe say that is iirrati f avoids all exclona uded r ionsl eg

n m

urfaces are associated with alternating paths

0 1 1 2 3 5 8 13 21 Fibonacci ratios , , , , , , , , , . .

1 1 2 3 5 8 13 21 34

1 2

1 2

1 2

1 2

p p

q q

p p

q q

WHERE TO START? START WITH CHAOS

The fractal structure of chaos is related to the structure of numbers

0 1

Farey Tree1

1(excluded region) alternating path

alternating path

is it pathological?

0, if ( , ) s.t. | / | / , where , are integer

'1, otherwise

km n n m r m m np

Diophantine Pressure Profile

pressure gradient at Diophantine irrationals flatten pressure at every rational

infinite fractal structure Riemann integrable is not

THEN, ADD PLASMA PHYSICS

Force balance means the pressure is a “fractal staircase” p = j × B, implies that Bp=0 i.e. pressure is constant along a field line

Pressure is flat across the rationals (assuming no “pressure” source inside the islands)

→ islands and chaos at every rational → chaotic field lines wander about over a volume

Pressure gradients supported on the “most-irrational” irrationals→ surviving “KAM” flux surfaces confine particles and pressure

11

( ) '( ) lim '( )( )N

n n nn

b

a Np x p x dx p x x x

infinitely discontinuous

Q) How do non-integrable fields confine field lines?A) Field line transport is restricted by KAM surfaces and cantori

r

adia

l coo

rdin

ate

510 iterations

“noble”cantori(black dots)

KAM surface

cantor set

complete barrier

partial barrier

KAM surfaces are closed, toroidal surfaces; and stop radial field line transport Cantori have many holes, but still cantori can severely “slow down” radial field line transport

Example, all flux surfaces destroyed by chaos, but even after 100 000 transits around torus the field lines cannot get past cantori

Calculation of cantori for Hamiltonian flowsS.R. Hudson, Physical Review E 74:056203, 2006

0 1 2 1

1 3 3 1delete middle third

510 iterations

gap

small changes to coil geometry remove resonant fields stable, high-pressure plasma quasi-symmetry satisfy engineering constraints

Non-axisymmetric (i.e. three-dimensional) experiments

designed to have “good-flux-surfaces”

2cm

“healed” equilibrium

• The construction of ghost-surfaces quadratic-flux minimizing surfaces provides an easy-to-calculate measure of the island size • Standard numerical optimization methods can be used to design non-axisymmetric experiments with “good flux surfaces”

• Example : In the design of the National Compact Stellarator Experiment (NCSX), small changes in the coil geometry were used to remove resonant error fields

“equilibrium” with chaos

Eliminating islands in high-pressure, free-boundary stellarator MHD equilibrium solutionsS.R. Hudson et al., Physical Review Letters 89:275003-1, 2002Invited Talk 19th IAEA Fusion Energy Conference, 2002

1958 An Energy Principle for hydromagnetic stability problems I.B. Bernstein, E.A. Freiman, M.D. Kruskal & R.M. Kulsrud

plasma displacement

ideal plasma response

Brief History of MHD equilibrium theory 2

1 2

( - )

W

= , 0,

W=

V

V

p B

p

dv

t

dv

j B ξ

x x ξ

BE E v B


plasma displacement


1954 KAM theorem1962 A.N. Kolmogorov (1954), J. Moser (1962), V.I. Arnold (1963)

1967 Toroidal confinement of plasma H. Grad “very pathological pressure distribution” islands and chaos not allowed by ideal variations dense set of singular currents plasma variations are over constrained


1 2

( - )

W

= , 0,

W=

V

V

p B

p

dv

t

dv

j B ξ

x x ξ

BE E v B


plasma displacement




1983 3D ideal equilibrium codes1984 BETA Garabedian et al., VMEC Hirshman et al. minimize W allowing ideal variations (other “codes” that are ill-posed, include non-ideal effects,…) do not allow islands & chaos cannot resolve singular currents, fail to converge

( If you don’t get the mathematics correct, the “numerics” won’t work. )


1 2

( - )

W

= , 0,

W=

V

V

p B

p

dv

t

dv

j B ξ

x x ξ

BE E v B


plasma displacement




1983 3D ideal equilibrium codes1984 BETA Garabedian et al., VMEC Hirshman et al. minimize W allowing ideal variations do not allow islands & chaos cannot resolve singular currents, fail to converge

IDEAL VARIATIONS OVERLY CONSTRAIN THE TOPOLOGY, LEAD TO A DENSE SET OF SINGULARTIES

HOWEVER, IF THE VARIATION IS UNCONSTRAINED, THEN THE MINIMIZING STATE IS TRIVIAL i.e. vacuum.


1 2

( - )

W

= , 0,

W=

V

V

p B

p

dv

t

dv

j B ξ

x x ξ

BE E v B

Q) What constrains a weakly resistive plasma?A) Plasmas cannot easily untangle themselves

Question 1. (topology) how many times do two closed curves loop through each other?

Question 2. (topology) how ‘knotted’ is a magnetic field?

Question 3. (physics) are there simple principles that govern plasma confinement?

HYPOTHESIS OF TAYLOR RELAXATION Weakly resistive plasmas will relax to minimize the energy (and some flux surfaces may break), but the plasma cannot easily “untangle” itself i.e. constraint of conserved helicity

Minimize W Energy, subject to constraint of conserved H Helicity = H0

Taylor relaxed state is a linear force free field, ×B= B

The Stepped Pressure Equilibrium Code (SPEC),has excellent convergence properties

convergence with respect to radial resolution, N

• First 3D equilibrium code to 1.allow islands & chaos,2.have a solid mathematical foundation,3.give excellent convergence,

convergence with respect to Fourier resolution, M

approximate solution exact solution

error decreases as N=h-1, M increase

Poincaré Plot (of convergence calculation)(axisymmetric equilibrium plus resonant perturbation)

1,error exp(- )n

h Mf f h M

cubic, n=3quintic, n=5

1if is small, error exp(- )nh M

h = 1/N radial resolutionM Fourier resolution

1if exp(- ) is small, error nM h

• Consider a DIIID experimental shot, with applied three-dimensional “error fields” (used to suppress edge instabilities)

• Vary the equilibrium parameters until “numerical” diagnostics match observations (i.e. plasma boundary, pressure and current profiles), (e.g. Thomson scattering, motional Stark effect polarimetry, magnetic diagnostics)

• SPEC is being incorporated into the “STELLOPT” equilibrium reconstruction code by Dr. S. Lazerson, a post-doctoral fellow at Princeton Plasma Physics Laboratory

Equilibrium reconstruction of 3D plasmasis where “theory meets experiment”

pres

sure

, p

radial coordinate

3D Equilibrium Effects due to RMP application on DIII-DS. Lazerson, E. Lazarus, S. Hudson, N. Pablant, D. Gates39th European Physical Society Conference on Plasma Physics, 2012

safe

ty fa

ctor

, q

Stepped-pressure approximation to smooth profileDIIID Poincaré Plotexperimental reconstructionusing SPEC

MRXMHD explains self-organization ofReversed Field Pinch into internal helical state

Excellent Qualitative agreement between numerical calculation and experiment this is first (and perhaps only?) equilibrium model able to explain internal helical state with two magnetic axes publication presently being prepared by Dr. G Dennis, a post-doctoral fellow at the Australian National University

NUMERICAL CALCULATION USING STEPPED PRESSURE EQUILIBRIUM CODETaylor relaxation and reversed field pinchesG. Dennis, R. Dewar, S. Hudson, M. Hole, 2012 20th Australian Institute of Physics Congress

Fig.6. Magnetic flux surfaces in the transition from a QSH state . . to a fully developed SHAx state . . The Poincaré plots are obtained considering only the axisymmetric field and dominant perturbation”

EXPERIMENTAL RESULTSOverview of RFX-mod resultsP. Martin et al., Nuclear Fusion, 49 (2009) 104019

August, 2009nature

physicsReversed-field pinchgets self-organized



1967 Toroidal confinement of plasma H. Grad “very pathological pressure distribution”

1974 Relaxation of Toroidal Plasma and Generation of Reverse Magnetic Fields J.B. Taylor relax the ideal constraints, include helicity constraint

1996 Existence of Three-Dimensional Toroidal MHD equilibria with Nonconstant Pressure O.P. Bruno & P. Laurence “. . . our theorems insure the existence of sharp boundary solutions . . .” i.e. stepped pressure equilibria are well defined

2012 Computation of Multi-Region, Relaxed Magnetohydrodynamic Equilibria S.R. Hudson, R.L. Dewar et al. chaotic equilibria with arbitrary pressure combines ideal MHD and Taylor relaxation

for N →, recover globally-constrained, ideal MHD

for N=1, recover globally-relaxed Taylor force-free state


1 2W

V

p Bdv

V

H dv A B

( / 2)l l llF W H

1996 Existence of Three-Dimensional Toroidal MHD equilibria with Nonconstant Pressure O.P. Bruno & P. Laurence “. . . our theorems insure the existence of sharp boundary solutions . . .” i.e. stepped pressure equilibria are well defined

2 [[ 2]] 0l l l p B B B

1. There are no magnetic monopoles . . .i.e. B = 0, magnetic field lines have “no end”

2. The hairy ball theorem (of algebraic topology)

there is no nonvanishing continuous tangent vector field on a sphere, . . . . . but there is on a torus!

3. Noether’s theorem (of theoretical physics) e.g. if the system does not depend on the angle, the angular momentum is constant

each symmetry (i.e. ignorable coordinate) of a Hamiltonian system has a integral invariant

4. Toroidal magnetic fields are Hamiltonian can use all the methods of Hamiltonian and Lagrangian mechanics!

Action Integral ( , , )

C

S L t dt

gauge

B A

B

Hamilton's Equations , .

C

d dfield line satisfies S

d d

canonical momentum

Hamiltonian

A dl

action integral , "integrable" Hamiltonian, ( )

0, frequency the action is constant and the angle increases linearly with time

pdqJ H JdJ H H

dt J

Several theorems of mathematics & theoretical physics,prove that . . .

Trajectories are extremal curves of action-integral

1. Poincaré-Birkhoff Theorem

For every rational, ω = n / m, where n, m are integers,

• a periodic field-line that is a minimum of the action integral will exist• a . . . . . . . . . . . . . . saddle . . . . . . . . . . . will exist

2. Aubry-Mather Theorem

For every ω ≠ n / m , • there exists an “irrational” field-line that is a minimum of the action integral

3. Kolmogorov-Arnold-Moser Theorem

4. Greene’s residue criterion• the existence of a KAM surface is related to the stability of the nearby Poincaré-Birkhoff periodic orbits

Theorems from Hamiltonian chaos theoryprovide a solid foundation

is very irrational if there exist an , such that | , for all integers ,

if is irrational then the Aubry-Mather field line will cover a surface, called a KAM surface

-kr k - n / m | r m n m

very

if not, the Aubry-Mather field line will cover a Cantor set, called a cantorus

(delete middle third)

magnetic field-line action =

curves that extremize the action integral are field linescurve

A dl

Diophantine condition

→ transport of pressure along field is “infinitely” fast→ no scale length in ideal MHD→ pressure adapts to fractal structure of phase space

&

, gives 0

*for non-symmetric systems, nested family of flux surfaces is destroyed*islands irregular field l

in

MHD theory

chaos theory nowhere are flux surfaces continuously nes

p p

ted

ideal

j B B

rationals are dense in space

Poincare-Birkhoff theorem periodic orbits, (e.g. stable and unstable) guaranteed to survive into chaoses appear where transform is rational ( / );

*some irration

n m

i.e. rotational-transform, , is by rationals,al surfaces survive if there exists an , s.t. for all rationals, |

ideal MHD chaos infinitely discontinuous equ

-k

poorly approximatedr k ι - n / m| r m

2

is , or zero;

everywhere discontinuous or zero

*iterative method for calculating equilibria is ill-posed;

1) 0

2) ;

3)

everywhere

i

discontinuou

librium

sn

n n

n

pp

p B

j

B

j B

B j

pressure must be flat across every closed field line, or parallel current is not single-valued;

is single valued if and only if / 0

is singular;

Cdl B

densely and irregularly

j

B

1 solution only if + 0

introduce non-ideal terms, such as resistivity, , perpendicular diffusion, , [ ,To have a well-posed equilibrium with chaoti

4) +

c need

t

o

n n

HINT

B jB j B

B

j

3 , , ..], or return to an energy principle, but relax infinity of ideal MHD constraints

M D NIMROD

flatten pressure at every rational

infinite fractal structure

not Riemann

0, if ( , ) s.t. | |

1, otherwise'

km n x r mp

nm

integrable

An ideal equilibrium with non-integrable (chaotic) field and continuous pressure, is infinitely discontinous

radial coordinate≡transform

pres

sure

Diophantine ConditionKolmogorov, Arnold and Moser

Extrema of energy functional obtained numerically;introducing the Stepped Pressure Equilibrium Code (SPEC)

, ,, , , ,R cos( ), sin( )* toroidal coordinates ( , , ), *interface geometry

* exploit gauge freedom ( , , ) ( , , )

* Fourier

The vector-potential is discretized

m n m nl l m n l l m nR m n Z Z m ns

A s A s

A

,

,

1

( ) cos( )

( )

* Finite-element ( ) ( )

* derivatives w.r.t. vector-potential

and inserted into constrained-energy functiona / 2l

i

m n

i

N

l l l l ll

s m n

s

A

s s

a

a a

F W H M

(boundary condition)

linear equation for Beltrami field

* field in each annulus computed independently, distributed across multiple cpus

* field in each annulus depends on enclosed toroidal flux

B B

, ,

and

poloidal flux, , and helicity-multiplier,

geometry of interfaces, ,

* interf

Force balance solved using multi-dimensional Newton method.m n m n

P

R Z

ξ

2 2 2 2

2,

adjust angle freedom to minimize (

ace geometry is adjusted to satisfy force [[p+ 2 ]] =0

* angle freedom constrained by spectral-condensation,

* derivative matrix, , compute

) mn mn

m n

m n R Z

B

F[ξ]

F[ξ] d in parallel using finite-differences

* call NAG routine: quadratic-convergence w.r.t. Newton iterations; robust convex-gradient method;

piecewise cubic or quintic basis polynomials

minimal spectral width [Hirshman, VMEC]

solved using sparse linear solver

adjusted so interface transform is strongly irrational

depends on finite-element basis

=A A , = , = , need to quantify = - A ,A ( )

( ) (

Scaling of numerical error with radial resolution

n

s n

s

O h

gB A A O hgB A O h

A B A j B error j B

1

1

1

2

2

) ( )

( )( )( )

n

ns

s n

n

n

gB A O h

g j O hg j O hg j O h

Numerical error in Beltrami field scales as expected

h = radial grid size = 1 / Nn = order of polynomial

Poincaré plot, =0

(m,n)=(3,1) island+ (m,n)=(2,1) island= chaos

cos(2 ) cos(3 )

( , )( , ) ,

inner surface0.1

outer interface0.2

1.0 cos , sin

r

r

R rZ r

quintic, n=5

cubic, n=3

(logscale) N

(logscale) error

1 2 2- ( - ( - () ) )n n n

s h h hO O O j B j B j B

Example of chaotic Beltrami fieldin single given annulus;

sub-radial grid, N=16

cylin

dric

al Z

cylindrical R

Stepped-pressure equilibria accurately approximate smooth-pressure axisymmetric equilibria

magnetic fields have family of nested flux surfaces

equilibria with smooth profiles exist,

may perform benchmarks (e.g. with VMEC)

(arbitrarily approximate smooth-prof

in axisymmetric geometry . . .

ile with stepped-profile)

approximation improves as number of interfaces increases

location of magnetic axis converges w.r.t radial resolution

cylindrical R

cylin

dric

al Z

lower half = SPEC interfaces

upper half = SPEC & VMEC

incr

easi

ng p

ress

ure

increasing pressure resolution ≡ number of interfacesN ≡ finite-element resolution

magnetic axis vs. radial resolutionusing quintic-radial finite-element basis

(for high pressure equilibrium)(dotted line indicates VMEC result)

toroidal flux

Pre

ssur

e, p

toroidal flux

stepped-profile approximation to smooth profiletr

ansf

orm

Force balance condition at interfaces gives rise to auxilliary pressure-jump Hamiltonian system.

2

Beltrami condition, , and interface constraint, 0, gives 0,

suggests surface potential, , , so that 0,

( 2 ) ( ) ,

s

B f B f B B

B g f f g f f g f f g g g g

B B B n B

2 2 11 2 2 2

1 1

metric elements g

Force balance condition, [[ / 2]] 0, introduce 2( - ) .

Let tangential field on "inner-side" of interface be given, , ,

tangential fie

p B H p p B B const

B f B f

x x

2 2

, ,

ld on "outer-side", , , determined by characteristics

( , , , ) = , = - , = , = -

2 d.o.f. Hamiltonian system, and invariant surfaces only exi

p p

B p B p

H p p H H Hp p

p p

st if "frequency" is irrational

ideal interfaces that support pressure must have irrational transform

Hamilton-Jacobi theory for continuation of magnetic field across a toroidal surface supporting a plasma pressure discontinuityM. McGann, S.R.Hudson, R.L. Dewar and G. von Nessi, Physics Letters A, 374(33):3308, 2010

Bifurcations in 3D MHD equilibria, Influence of increasing pressure on islands Stuart R. Hudson...

Documents

Transcript of Bifurcations in 3D MHD equilibria, Influence of increasing pressure on islands Stuart R. Hudson...