Ergodic heat transport analysis in non-aligned coordinate systems S. Gnter, K. Lackner, Q. Yu IPP...
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Transcript of Ergodic heat transport analysis in non-aligned coordinate systems S. Gnter, K. Lackner, Q. Yu IPP...
Ergodic heat transport analysis in non-aligned coordinate systems
S. Günter, K. Lackner, Q. YuIPP Garching
• Problems with non-aligned coordinates?• Description of new scheme• First results
Problems with non-aligned coordinates
Radial heat transport in multiple helicity magnetic fields enhanced by stochasticity
For large ||/ (realistic values for hot plasmas ~ 1010) careful treatment of parallel heat flux required
Problems with non-aligned coordinates
TbbTTt
//2
Usually coordinate system aligned with magnetic field lines used,see e.g. Runov et al. (for static magnetic field)
But for non-linear MHD calculations dynamically evolving magnetic fields need to be considered!
For large ||/ (realistic values for hot plasmas ~ 1010):
small gradient along magnetic field lines cause large errors in temperature profile
BBb
An example: Interaction of NTMs with different helicities
No simultaneous large NTMs of different helicities observed in experiments
jBS p
Analytic theory: • for NTMs stabilising effect of additional helical field can be proven for small values of ||/ (incomplete temperature flattening)• effect vanishes for ||/
Is there an effect remaining for realistic values of ||/ ?
If so: new stabilisation method for NTMs can be propsed:stabilisation by external helical perturbation fields
An example: Interaction of NTMs with perturbation fields
Many other problems, but: so far no non-linear MHD code can dealwith realistic ||/
Proposal for a solution in non-aligned coordinate system
1//11//102
0 21
qbqbTTt
2//10//11//012
1 qbqbqbTTt
3//11//12//022
2 qbqbqbTTt
1111//0// 21
TbTbq
211001//1// TbTbTbq
312011//2// TbTbTbq
In the following, for simplicity (not in the code): Cartesian coordinates in radial direction, Fourier decomposition within the unperturbed flux surface, only one perturbation field component
Heat conduction equation for different Fourier components of temperature:
BBb ii
… …
//2 qbTT
t
Tbq ////
To close the equations one should not truncate the Fourier series in T, but in q heat flux along perturbed magnetic field line remains finite (nearly vanishing temperature gradients)
Fourier decomposition for perturbation
2//10//11//012
1 qbqbqbTTt
3//11//12//022
2 qbqbqbTTt
1111//0// 21
TbTbq
211001//1// TbTbTbq
312011//2// TbTbTbq
BBb ii
//2 qbTT
t
Tbq ////
To lowest order (for explanation): include only terms up to first order in q
T2 adjusts itself such that q||1 becomes small
1//11//102
0 21
qbqbTTt
In the following, for simplicity (not in the code): Cartesian coordinates in radial direction, Fourier decomposition within the unperturbed flux surface, only one perturbation field component
Heat conduction equation for different Fourier components of temperature:
Fourier decomposition for perturbation
If one truncates in T (just as example, holds for any order):
Cut after lowest order in temperature
1101022
//02
0 1TkbibTbTT
t rr
Enhancement of radial transport (T1 would adjust to cancel the first term)
To close the equations one should not truncate the Fourier series in T, but in q heat flux along perturbed magnetic field line remains finite (nearly vanishing temperature gradients)
What about the radial derivatives?
rrebb 11 bkib
perturbation field:
1102
0 qr
bTTt r
10112
1 qbkiTTt
rqq
bii
r
)12/()12/(
111
iqbki1
01
1122
2 qr
bTTt r
rqq
bii
r
)12/()12/(
111
Introduces an additional error or order (r)2 , but equations for each grid point ensure vanishing temperature gradients along perturbed field lines
simplest discretisationat i’s grid point
2
)2/1()2/1(
0111
ii qqbk
new scheme
Convergence properties: single magnetic island
||/= 108
Still convergence only (r)-2
But: relative error reduced by factor of 10Improvement increases for larger ||/
(~ (||/)1/2)
-2
Example: Magnetic islands of two helicities
||/= 1010
Magnetic islands seen in temperature contours, but still strong gradient in ergodic region
Example: Magnetic islands of two helicities
||/= 1012
Temperature gradient vanishes in ergodic region due to increased radial transport along magnetic field lines
Example: Magnetic islands of two helicities
Coming back to: Interaction of NTMs with different helicities
Is there an effect remaining for realistic values of ||/ ?
If so: new stabilisation method for NTMs can be propsed:stabilisation by external helical perturbation fields (next talk)
YES!
Summary and conclusion
• New scheme for solving heat conduction equation in non-aligned coordinates developed
• Successful test for realistic (and even higher) values of ||/
• Method can be used in general (toroidal) non-linear MHD codes
• Generalisation to 3d Cartesian grid straightforward