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