UPGRADE OF THE GSP GYROKINETIC CODE MID-YEAR PROGRESS REPORT

Transcription

1 12/6/211 1 UPGRADE OF THE GSP GYROKINETIC CODE MID-YEAR PROGRESS REPORT George Wlke gwlke@umd.edu December 6, 211 Supersor: Wllam Dorland, Dept. of Physcs bdorland@umd.edu Abstract: Smulatons of turbulent plasma n a strong magnetc feld can take adantage of the gyroknetc approxmaton, the result of whch s a closed set of equatons that can be soled numercally. An exstng code, GSP, uses a noel and hghly effcent soluton method to sole the nonlnear 5D gyroknetc equaton. In ths project, we wll seek to change the elocty-space representaton n GSP. Ths transformaton wll smplfy the ncluson of a new collson operator and make the algorthm more sutable for smulatons of turbulence n Tokamak plasmas, whle retanng the effcency and accuracy of the orgnal code.

2 12/6/211 Outlne 2 Outlne Oerew of Gyroknetcs Descrpton of GSP Changes to Algorthm

3 12/6/211 Background 3 Oerew GSP s a Partcle-In-Cell (PIC) code to sole the Gyroknetc equaton to study the eoluton of turbulence n hghly magnetzed plasma Gyroknetc theory treats the many rapdly crculatng charges as charged rngs These rngs drft parallel and perpendcular to the background magnetc feld

4 12/6/211 Background 4 Oerew The partcles n the code are not ntended to smulate the physcal partcles, nor the gyro-aeraged rngs of charge. The eoluton of many physcal charges are descrbed by a statstcal dstrbuton functon n phase space: f r,, f F f Performng an asymptotc expanson and gyroaeragng the Fokker-Planck equaton, we can get a dynamcal equaton for f : the gyroknetc equaton,

5 12/6/211 Background 5 The Gyroknetc Equaton f t z f D F Ez f D f C F Where: f s the perturbed dstrbuton functon to be soled for The gen background equlbrum dstrbuton s: The (characterstc) drft elocty s: C f f R,, The partcle weght s w s the collson operator f F D c E zˆ B m m 2kT F n e 2kT 2 (angle brackets sgnfy the gyro-aeragng operaton at constant R)

6 12/6/211 Outlne 6 Outlne Oerew of Gyroknetcs Descrpton of GSP Changes to Algorthm

7 12/6/211 Algorthm 7 The Gyroknetc Equaton f t z f D F Ez f D f C F To sole ths equaton, we dentfy characterstc trajectores n phase space so that: The left hand sde becomes d f dt zˆ D s the characterstc elocty Characterstc cures are constant n elocty space only under specal condtons: Unform equlbrum magnetc feld B Specal choce of elocty coordnates: 2 m 2B E 1 2 m 2 2

8 12/6/211 Algorthm 8 GSP Code Step 1: Intalze partcles n phase space Step 2: Predctor Step: Calculate felds at step n Calculate marker weghts along characterstcs for step n+1/2 Adance marker postons for half a tmestep Update weghts wth collson operator for step n+1/2 Step 3: Corrector Step: Calculate felds at step n+1/2 Calculate marker weghts along characterstcs for step n+1 Adance marker postons for a full tmestep Update weghts wth collson operator for step n+1 Step 4: Output results Repeat Steps 2-4 N T tmes

9 12/6/211 Algorthm 9 Flowchart Calculate Felds Update Weghts Adance Markers Apply Collson Operator

10 12/6/211 Algorthm 1 Flowchart Calculate Felds Update Weghts Adance Markers Apply Collson Operator

11 12/6/211 Algorthm 11 Calculate Felds D B c E zˆ Once we know the electrostatc potental, we know the electrc feld: E The gyro-aeragng operaton s smplfed n Fourer space: ~ E k ~ ~ k E k J J ( s the -th order Bessel functon) J eb mc

12 12/6/211 Algorthm 12 Calculate Felds Frst, depost charges onto grd n R-space Fourer transform grd Apply Posson s equaton to calculate the potental: ~ d d f J R k Issue: J s expense to calculate explctly eery tme Soluton: Dscretze on a grd n elocty space Store J n a table at the releant dscrete alues of and k

13 12/6/211 Algorthm 13 Calculate Felds Once potental s calculated, fnd the gyro-aeraged felds n k-space ~ E R k ~ J k Fourer transform back nto grd n R-space Interpolate to fnd felds at marker postons

14 12/6/211 Algorthm 14 Flowchart Calculate Felds Update Weghts Adance Markers Apply Collson Operator

15 12/6/211 Algorthm 15 Update Weghts and Adance Markers Ths Monte Carlo scheme [Aydemr,1994] doesn t model explctly, but rather the functon: f R w f F R The equaton for w along characterstcs s: w q T E z R F F D Update marker postons explctly usng zˆ D

16 12/6/211 Algorthm 16 Flowchart Calculate Felds Update Weghts Adance Markers Apply Collson Operator

17 12/6/211 Algorthm 17 Collson Operator Ptch-angle scatterng operator Dffuse n elocty space Energy s consered wth lke-partcle nteractons, or when there s a large dsparty n mass (such as ons and electrons) n s a constant parameter Collson operator defned n terms of h f F n h C h

18 Collson Operator 12/6/211 Algorthm 18 Goduno splttng: We hae already appled the non-collsonal part of the derate: Frst, conert f * to h* : Fnd the derates mplctly Inert the tr-dagonal matrx to obtan h n+1 1 * 1 n n C h t h h * * F f h * f f n F h f n n 1 1

19 12/6/211 Algorthm 19 Collson Operator Issues: As mplemented, the collson operator does not obey the proper conseratons laws The update wll use the operator from [Abel et al, 28], whch conseres partcles, momentum, energy, and obeys the Boltzmann H-theorem

20 12/6/211 Outlne 2 Outlne Oerew of Gyroknetcs Descrpton of GSP Changes to Algorthm

21 12/6/211 Status 21 Summary of Changes Calculate Felds Most dffcult task. Perform calculaton wth new elocty-space coordnates. Try to keep repeated calculaton of J effcent. Update Weghts Adance Markers Mnor changes f any Apply Collson Operator Rework conert ptch-angle operator to new coordnates. Apply Abel operator.

22 12/6/211 Status 22 Codng Status Calculate Felds In progress Update Weghts Adance Markers No changes needed Apply Collson Operator

23 12/6/211 Status 23 Integral for Electrostatc Potental ~ d d f J R k We re representng the dstrbuton as: f R F w( r, F, ) w ( r R ) (, ) ( ) so to calculate the ntegral, we just sum up the alues of for each partcle and normalze J w When s allowed to change, we need another way to lookng up J

24 12/6/211 Status 24 Current Velocty Grd Partcles nterpolated onto k grd Partcles already hae an assgned J J kx, ky,

25 New Velocty Grd 12/6/211 Status 25 B k k J B k k J J y x y x,,,,, Partcles scattered n 2D elocty- space Can moe around n elocty space dependng on the local magnetc feld ) B(z

26 12/6/211 Schedule 26 Tmelne Mlestone Phase I September October Transformed GK equaton dered and algorthm understood Phase II Noember January Changes coded and debugged Phase III February Aprl New code aldated, tested, and benchmarked Phase IV Aprl May Results organzed, presentaton prepared and gen

27 12/6/211 Summary 27 Summary Upgrade to GSP progressng slghtly behnd schedule Phase I (Gyroknetcs analytcs and algorthm understandng) took longer than ntally expected Currently on Phase II Codng stll on track to be completed before February

28 12/6/211 Summary 28 Questons?

29 12/6/211 Backup 29 Gyroknetc Deraton Summary Start wth Fokker-Planck equaton Gyroknetc orderng assumptons: ~ n ~ k k ~ D t f ~ F ~ Order : F ndependent of gyroangle 1 Order : F h T Maxwellan q f F h where 2 Order : The gyroknetc equaton

30 Gyroaeragng Fourer Transforms perp perp k perpperp k J e d e e d e e d e e cos ) ( 2 1 R k R k k R k R k R r k 12/6/211 Backup 3

31 Maxwell s Equatons Posson s Equaton: Ampere s Law: e e n q n q /6/211 Backup 31 e e T e n n d t f n 3 1,, r J E B c t c 4 1 J 3 f 1 d n en en e e

32 12/6/211 Algorthm 32 Update Weghts and Adance Markers Now adance markers along characterstc trajectores: x y z x y z t t t xˆ yˆ D D The markers elocty space coordnates do not need updatng As currently mplemented, markers hae and assgned, whch are constants of moton only n a unform magnetc feld Ths update wll change coordantes to E and whch reman constants of moton (up to the order requred) een when the magnetc feld s nonunform