Ray tracing for ballooning modes in quasi-symmetric devices · 2005-09-08 · September 8, 2005...

September 8, 2005 National Stellarator Theory Teleconference: Ware 1

Ray tracing for ballooning modes inquasi-symmetric devices

A. S. WareUniversity of Montana

9/8/2005

Collaborators: E. Mondloch, University of MontanaR. Sanchez, D. A. Spong, D. del-Castillo-Negrete, ORNL

September 8, 2005 National Stellarator Theory Teleconference: Ware 2

OverviewOverview

Using COBRAVMEC to solve theballooning equationRange of parameters in ballooning

space

Ray equationsIntegrating the ray equations

Results for two configurationsQPS, NCSX, (still working on HSX)

September 8, 2005 National Stellarator Theory Teleconference: Ware 3

We use COBRAVMEC to solve the idealMHD ballooning equationGiven a VMEC equilibrium (wout file),

COBRAVMEC obtains the MHD eigenvalue as afunction of the normalized flux, S, the field linelabel, α = qθ - ζ, and the ballooning parameter, θk= kq/kα.

To obtain:

Solving the ballooningeigenvalue equation

!

" = " S,#,$k( )

!

"

"#c1 # | S,#

k,$( )

" ˆ %

"#

&

' (

)

* + + c2 # | S,#

k,$( ) ˆ % + ,c3 # | S,#

k,$( ) ˆ % = 0

September 8, 2005 National Stellarator Theory Teleconference: Ware 4

Issue concerning using the safetyfactor as the radial coordinate

For ray tracing, it is usefulto use the safety factor, q,as the radial coordinateFor monotonic q-profiles

this is not an issue (QPS)For profiles with shear

reversals (NCSX and HSX)this proves difficult forinterpolation

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Ro

tati

on

al

tran

sfo

rm, !

S

HSX

NCSX

QPS

September 8, 2005 National Stellarator Theory Teleconference: Ware 5

Issue concerning using the safetyfactor as the radial coordinate For now, we are using ray tracing only over

the portion of flux surfaces with a monotonicq-profile

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

Ro

tati

on

al

tran

sfo

rm, !

S

HSX

NCSX

QPS

0

0.2

0.4

0.6

0.8

1

1.2

0 0.2 0.4 0.6 0.8 1

(dP/d!)/(dP/d!) max

S

HSX

NCSX

QPS

Region of max pressuregradient is included

September 8, 2005 National Stellarator Theory Teleconference: Ware 6

The Ray Equations

Dewar and Glassers ray tracing theory can be usedto obtain information about the global eigenmodes

Using the values of λ on a 49x49x91 grid in (q,θk,α),we integrate the ray equations Range for θk is 0 to π and for α is 0 to 2π

!

" = " q,#,$k( )

!

˙ q ="#

"kq

, ˙ $ ="#

"k$, ˙ k q = %

"#

"q, ˙ k $ = %

"#

"$

September 8, 2005 National Stellarator Theory Teleconference: Ware 7

All of the unstable ballooning modesthat we have examined for QPS,NCSX, and HSX, are localized inballooning space

We have also examined some rays forconfigurations for which the3-dimensional shaping has beenreduced

Results for three quasi-symmetricstellarators

September 8, 2005 National Stellarator Theory Teleconference: Ware 8

Transition from 2D to 3D:The “equivalent” tokamak

Let ν represent an artificial parameter used to transform afixed boundary from axisymmetric to nonaxisymmetric

ν = 0 is the “equivalent” tokamak case ν = 1 is the QPS (or NCSX or HSX) case

Use VMEC to calculate equilibria for 0 ≤ ν ≤ 1, keeping ιand <|B|> fixed!

R ",#( ) = Rbc m,0( )cos m"( )m

$ +% Rbc m,n( )cos m" & n#( )m,n'0

$

Z ",#( ) = Zbs m,0( )sin m"( )m

$ +% Zbs m,n( )sin m" & n#( )m,n'0

$

September 8, 2005 National Stellarator Theory Teleconference: Ware 9

QPS at β = 2.5%

The equivalent tokamakcase (i.e., ν=0)All surfaces are stable Surfaces of constant λ are

independent of αA constant λ surface and

a ray on that surface areshown

September 8, 2005 National Stellarator Theory Teleconference: Ware 10

QPS at β = 2.5%

Moderate shaping (ν=0.25)All surfaces are still stable Surfaces of constant λ are

extended along (but notindependent of) α

A constant λ surface anda ray on that surface areshown

September 8, 2005 National Stellarator Theory Teleconference: Ware 11

QPS at β = 2.5%

Moderate shaping (ν=0.25)All surfaces are still stable Surfaces of constant λ are

extended along (but notindependent of) α

A constant λ surface anda ray on that surface areshown

September 8, 2005 National Stellarator Theory Teleconference: Ware 12

QPS at β = 2.04%

Standard QPS (ν=1) Localized region of

weakly unstable surfaces Surfaces of constant λ are

localized to a small rangein α

The marginal stable λsurface is shown

September 8, 2005 National Stellarator Theory Teleconference: Ware 13

QPS at β = 4.0%

Standard QPS (ν=1) Larger region of

unstable surfaces The unstable λ =-0.5673

surface and a ray on thatsurface are shown

September 8, 2005 National Stellarator Theory Teleconference: Ware 14

NCSX at β = 4.88%

Weakly shaped (ν=0.05)All surfaces are stable Surfaces of constant λ are

still bumpy and extendedin α

Range in θk is now 0 to 2πA stable λ shown along

with a ray on that surface

September 8, 2005 National Stellarator Theory Teleconference: Ware 15

NCSX at β = 4.88%

Standard NCSX (ν=1) Localized regions of

instability Surfaces of constant λ are

localized in α Range in θk is now 0 to 2π The marginal stability

surface is shown Region of instability

localized to the edge(lowest q values)

September 8, 2005 National Stellarator Theory Teleconference: Ware 16

NCSX at β = 5%with a p = p0(1 - S)2 profile

“Tokamak” case (ν=0)No unstable regions Standard cylindrical

type rays

September 8, 2005 National Stellarator Theory Teleconference: Ware 17

NCSX at β = 5%with a p = p0(1 - S)2 profile

Standard NCSX (ν=1) Larger unstable regions Regions of instability

localized in α