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