Working in toroidal geometry: * magnetic flux coordinates,* neoclassical particle orbits

Bogdan Hnat

Confinement and topologyPoincare’s theorem (in my own words…)Let S be a smooth, closed surface and C(x) be well behaved vector field such that the component of Ctangent to S never vanishes.The surface S must then be a torus.

Confinement and topologyConsider the outmost bounding surface of

some confined plasma.• Particles can stream free along the magnetic field lines• An ideal confining magnetic field should have no component normal to the bounding surface• Magnetic field B must cover the entire surface and the tangent component can not vanish anywhere

Conclusion:The bounding surface must be a torus.

Flux surfacesRational surfaces: magnetic

field line on such surface closes on itself after n toroidaland m poloidal turns

Ergodic surface: magnetic field line never closes on itself, thus covering densly the surface

Stochastic regions (volums):magnetic field has radial component which allows for fast transport between different flux surfaces (can not support pressure gradient)

Covariant and contravariantrepresentation (1)

( , , ) with ( , , ); ;

( )

; ;

r r x y ze r e r e r

J e e e

e e e

ψ ψ ϑ ϑ ζ ζ

ψ ϑ ζ

ψ ϑ ζ

ψ ϑ ζ

ψ ϑ ζ

== ∂ = ∂ = ∂

= ×

= ∇ = ∇ = ∇

i

Two coordinate systems

Covariant basis

Jacobian; J>0 => right handed coord

Contravariant basis

Vector, A, can be written using covariant or contravariant basis:

i ii iA A e Ae= =

Covariant and contravariantrepresentation (2)

Useful relations

1

2 2 2 2

2 2

,

use the chainrule: ( , , )

det( )

e e e e I

e e J e e e J e

ds dx dy dz dr drdrdr dd

ds e e d d g d d g J

α α αβ β α

γ α β αβγα β αβγ γ

α β αβ αβ

δ

ε ε

α α ψ ϑ ζα

α β α β

−

⋅ = ⋅ =

× = × =

= + + = ⋅

⎛ ⎞= =⎜ ⎟⎝ ⎠

= ⋅ = =

Cylindrical coordinates (R,θ,z)

Covariant basis( , , ) (cos ,sin ,0)( , , ) ( sin , cos ,0)( , , ) (0,0,1)

R R R R

z z z z

e x y ze x y z R Re x y zθ θ θ θ

θ θθ θ

= ∂ ∂ ∂ == ∂ ∂ ∂ = −

= ∂ ∂ ∂ =

Contravariant basis 1 1

(cos ,sin ,0)( sin , cos ,0)

(0,0,1)

R

z

e Re R Re z

θ

θ θ

θ θ θ− −

= ∇ =

= ∇ = −

= ∇ =

2 2 2

cos sin 0 1 0 0det sin cos 0 0 0 det( )

0 0 1 0 0 1

RJ R R g R g R J

θ θθ θ

−⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= = = = =⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

Toroidal coordinates( ) ( )2 2

0 0; arctan /( ) ;p p pr R R z z R Rϑ ζ ϕ= − + = − = −

( ( ), ( ), ( )) such that ( ) 0 and finiteF r r r J Fθ ζ θ ζ= ∇ ⋅ ∇ ×∇ >These may be defined from (rp, θp, ζp), for example

( , 2 , 2 ) ( , , ) 2

( , 2 , 2 ) ( , , ) 2

( , 2 , 2 ) ( , , )

p p p p p p

p p p p p p

p p p p p p

r m n r m

r m n r n

F r m n F r

θ θ π ζ π θ θ ζ π

ζ θ π ζ π ζ θ ζ π

θ π ζ π θ ζ

+ + = +

+ + = +

+ + =

Flux surface labelsAngle coordinate: any multivalued function that

increases by certain value after each revolution around the circular path; natural selection to describe position on the torus (here angles as previously given: θ, ζ)

“Radial” coordinate, F, labels each surface, is constant on it and varies smoothly from one surface to another

0B F∇ =iPlasma pressure is a natural candidate for the flux surface label.

so it is evident that 0c p J B B p∇ = × ∇ =i

Toroidal coordinates1 ( ) ( ) 0 0

( , , ) and ( , , )

FB J JB JB B F B

JB G F JB G F

θ ζθ ζ

θ ζζ θθ ζ θ ζ

− ⎡ ⎤∇ = ∂ + ∂ = ∇ = =⎣ ⎦= −∂ = ∂

i i

But the field itself must be single valued, so G must be a linear combination of angles and some other function

1 2 1( ) ( ) ( , , )For any vector in contravariant notation:

( ) ( ) ( )

So for magnetic field:

Noticing that F

G F F G F

w J w F J w F J w F

B G F G F

G G F G G

B F G

ζ θ

θ ζ

ψ θ ψ ζ θ ζ

θ ζ θ ζ ζ θ

ζ θ

θ ζ

= − +

= ∇ ∇ ×∇ + ∇ ∇ ×∇ + ∇ ∇ ×∇

= ∂ ∇ ×∇ + ∂ ∇ ×∇

∇ = ∂ ∇ + ∂ ∇ + ∂ ∇

= ∇ ×∇

i i i

Magnetic flux coordinates( )

( ) ( )

f 1 1 2

1

2

d/ and replace F where dF

We then get: , where f f

G F

B q q

χθ θ θ ψ χ ψ

ψχ θ ζ χψ

→ = + → =

= ∇ ×∇ − =

( )1f f

f f

d Bd d d dB B d qB B Bθ ζ

θ θθ θ ζ ζζ χθ ζ ζ

∇= = = ⇒ = =

⋅∇ ⋅∇ ∇

ii

Magnetic field line is defined by:

The unrolled flux surface is deformed into a square such that the lines of constant θf and ζf form Cartesian grid. The field line on this grid is a straight line.

Physical meaning of χConsider toroidal flux, defined as:

( )

t

( )2 21

0 0 0

1

( )

p0

1 or by use of Gauss' theorem ( )2

1 ' ( ) ,2

1where( )

2 ( ') ', similarly 2 ( ) so that

tS

F

t

Ft

tp

dS B B dV

d d B J d

qJB

dq d F qd

B q

ζ

χπ π

χ

ψ ψ ζπ

ψ ζ χ ζ θπ

χ θ ζ ζ

ψψ π χ χ ψ πχψ

ζ χ θ ζ ζ

−

−

= = ∇

= ∇

= =∇ ∇ ×∇ ⋅∇

= = =

⋅∇ = ∇ ×∇ − ∇⎡ ⎤⎣ ⎦

∫ ∫

∫ ∫ ∫

∫

i i

i

i

i ( )q χ θ ζ= ∇ ∇ ×∇i

Guiding centre motionEstimation of confinement time based on particle

trajectories requires long time observations of trajectoriesEquations of motion (or Lagrangian) is averaged over

fast scales – gyro motion in the plane perp. to BIt is convinient to work with normalised quantities

( ) ( )0 0

2 2 20 0

/( ); / ;

/ / 2 2 /

t t eB mc r r R

E E E E mv R

ω ω

ρ

→ = →

→ = ⋅

Adiabatic invariants can be obtained if magnetic field varies on scales larger than gyro radius

11max , 1 | ln | | ln / |B BB B

L B B tLρδ τ

τ−⎛ ⎞

≡ ∇ ∂ ∂⎜ ⎟Ω⎝ ⎠∼ ∼

Gyro-averaged Lagrangian1 , / , /R r r b v b B B ZeB mρ −≡ − = −Ω × = Ω =

ˆ ˆ ˆ ˆ ˆ ˆ( cos sin ) and x y z x y ze e e e e B Beρ ρ ϑ ϑ= + = × =

The gyro-average Lagrangian

( ) ( )22 2

( , )2 2m BL b R ZeA R Ze Ze R tρ ϑρϑ⎡ ⎤= ⋅ + + ⋅ + − Φ⎢ ⎥⎣ ⎦

( , , , , , , )L R R tρ ϑ ρ ϑ

All fields are now evaluated at the guiding centre position REuler-Lagrange equations for ρ and θ give

2 2 2

/

1 02

Zeb m

d L d m ZeB constdt dt

ϑ

ρ ϑ ρ ρϑ

= − ≡ Ω

⎛ ⎞∂ ⎛ ⎞= + = ⇒ Ω =⎜ ⎟⎜ ⎟∂ ⎝ ⎠⎝ ⎠

Gyro-averaged LagrangianSo far we have shown that there are two constants of motion: Total energy and magnetic moment:

222

2 2mvmvE Ze

Bμ ρ ⊥= + Φ = Ω =

Axisymmetry makes Lagrangian independent ζ and toroidal momentum pζ is an exact constant of motion.

( ) ( )2

2

Assume that:

2

A A A B B BmL v B v B Ze v A v A B ZeB

mv BLp ZeAB

θ ζ θ ζ

θ ζ θ ζθ ζ θ ζ

ζζ ζ

θ ζ θ ζ

μ

ζ

= ∇ + ∇ = ∇ + ∇

= + + + − − Φ

∂≡ = +∂

Particle Orbits - bananaAssume particle is moving on a flux surface with magnetic field magnitude between Bmin and Bmax. Define:

20 0

2 2

2 B v Bmv v Bμλ ⊥≡ =

A particle can never enter the region where B E Zeμ > − Φ

All particles must then satisfy 0 ≤ λ ≤ (B0 /Bmin).

0 0

max min

B BB B

λ≤ ≤Particles trapped by the mirror force, move on the outboard side of the flux surface.

This trapped particle orbits are called banana orbits.

Particle Orbits - bananaLet us estimate the width of a banana orbit, using tokamakordering, where the aspect ratio ε=a/R is a small parameter

( )( )20 0 0/ 1 ( ) ,B B R R T B Bζ ζ ϑ ζε ε+

where index 0 indicates quantity measured at magnetic axis0

00

cos , so | | 11B BR R r B B v v v

Bζ

ζζ

θ λ εε

⎛ ⎞= + ≈ = ⇒ = ± − ≈⎜ ⎟⎜ ⎟± ⎝ ⎠

Now, using invariant property of pζ we compute…1, but so thatp

pp

mR vr r B

Ze Rv

r

ψ ψ ψψ

ρ ε

ΔΔ = ∇ Δ ⇒ Δ = ≈ ∇

∇

ΔΔ

Ω∼

Particle Orbits - bananaThe width of this orbit depends inversely on Ωp, which is smaller than gyro-frequency since Bp<<B, it also much smaller for ions than electrons.

Let us estimate the diffusive transport coefficient that results from random walk with the characteristic step size of the banana orbit.

Which is bigger than ion diffusivity of the passing particles by about a factor of 50

2

2

( ) , 2 and

2

bi t eff t eff ii

bi pi ii

f x f vχ ν ε ε ν

χ ε ρ ν

= Δ =

Interesting questionsAny realistic configuration is not going to be axisymmetric.For example, MAST Resonant Magnetic Perturbation coils are designed to perturb the magnetic field, create magnetic islands

Interesting connection to chaos

Is magnetic field still hamiltonian? Vacuum models perform simple line tracing.

What are the implications for particle transport in such configurations?