Chaotic Analysis of Numerical Plasma Simulations

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Chatic Analysis f Numerical Plasma Simulatins Pster 2T19 1991 Meeting f the American Physical Sciety Divisin f

f Physics University f WiscnsinMadisn Madisn, WI 5376 PLPI87 These PLP Reprts are infrmal and preliminary and as

1 Chatic Analysis f Numerical Plasma Simulatins Pster 2T Meeting f the American Physical Sciety Divisin f Plasma Physics Tampa Bay, Flrida Christpher Watts D. Newman, J.C. Sprtt, E.J. Zita Dept. f Physics University f WiscnsinMadisn Madisn, WI 5376 PLPI87 These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated. They are fr private circulatin nly and are nt t be further transmitted withut cnsent f the authr r majr prfessr.

2 Abstract: Data generated by nume rical simulatins f plasma prcesses were eami ned fr evidence f lw dimensinal chas. Analysi s included the genera tin f phase plts and Pinc are sectins, and the determi natin f the crrelatin dimensin an d the largest Lyapunv epnent. Data frm a simulatin f in cnvective cell turbulence shwed evidence f a lw di mensin ranging frm 4.3 t 5.9, depending n the mde, fr the case with n driving ter m. In the driven case the dimensin was >8. Data frm the DEBS cde simulatin f an RFP indic ates a pssible dimensin <3, hwever mre data is required t cnfi rm this result.

Mtivatin: I. Numerical simulatins have sme advantages: a) There are fewer mdes, and pssibly simpler dynamics. b) The simulatins are (in principle) reprducible.

3 Mtivatin: I. Numerical simulatins have sme advantages: a) There are fewer mdes, and pssibly simpler dynamics. b) The simulatins are (in principle) reprducible. c) One can generate an "infinite" amunt f steady state data II. Evidence fr chatic behavir in numerical simulatins wuld encurage a mre thrugh search fr evidence in data frm plasma eperiments. Recall last year's results:

.. N Evidence f Lw Dimensinal Chas in MST Eperimental Data Crrelatin Dimensin f Edge Tridal Magnetic Fluctuatins 6 4 2 2 4 Edge Magnetic Fluctuatins 6.16 18.

. (OJ CIi 4 v @ varius embeddings, 22 jul 9, #123 dim =21 c dim =25 A dim =23 dim =27 I 35k pints; 15 5ms 't = 5J..1s W=5J..1s.2.2.4.6 lg radius. 1 1.2 1.4.4.3.2. 1.1.2 '.

4 .. N Evidence f Lw Dimensinal Chas in MST Eperimental Data Crrelatin Dimensin f Edge Tridal Magnetic Fluctuatins Edge Magnetic Fluctuatins times 22 a ( (I.) CIi a " (OJ CIi 4 varius embeddings, 22 jul 9, #123 dim =21 c dim =25 A dim =23 dim =27 I 35k pints; 15 5ms 't = 5J..1s W=5J..1s lg radius '.3 Tridal Gap Vlts. 1)L1..L2. O...I.....l 6 ' ffi times (I.) 5 1 a.... " CIi 8 Crrelatin Dimensin f Tridal Gap Vlts 21 ct 9, #12 dim = 5 2k dim = 7 c dim=9 dim =11 M dim =13 14k pints t = 56ms 't = 28J..1s W=2J..1s lg radius.5 1

$a The fractal dimensin f the system indicates the$

5 Mtivatin fr Chas Studies: Identifying lw dimensin chas in signal means that determinant prcesses gvern the system. a The fractal dimensin f the system indicates the number f PDE's need t describe the system.

6 Time Lag Embeddings a) The time series f a single quantity is sufficient t determine the chatic nature f the system b) Recnstruct the attractr in an M dimensin embedding space by creating time lag vectrs: =[(t),(t+'t),... X(t+(Ml)'t). Analgy: A 2D cylindrical shell in a 3D embedding space Time Series f the Lrenz Attractr 2 15 (t+12) 1 " :11Vll'VU 1\, \ I time (s) (t) Lrenz Attractr in 3D timelag Phase Space

7 Lyapunv Epnent a) The largest Lyapunv epnent measures the epnential divergence f nearby trajectries. nl A= lim l L I n!(xk) n n k=o d X k b) A psitive Lyapunv epnent is indicative f chas. c) Using embedding crdinates, measure the divergence f tw nearby pints. Average this ver all pints in the attractr C 1""'4 + >< (t)

$Crrelatin Dimensin a) The crrelatin dimensin is a measure f the nnunifrmity f a chatic attractr. It is clsely related t the fractinal dimensin ccupied by the chatic attractr. 2 15 1 5... +.$ C(r) = lim _1 N) N 2.. l:;i:] 1 J c) The slpe f the lg r lg C(r) plt is the crrelatin dimensin D2. D = lim lgc(r) 2 r O " ' lgr 1 15 2 1 2 3.

C(r) = lim _1 N) N 2.. l:;i:] 1 J c) The slpe f the lg r lg C(r) plt is the crrelatin dimensin D2. D = lim lgc(r) 2 r O " ' lgr 1 15 2 1 2 3.

8 Crrelatin Dimensin a) The crrelatin dimensin is a measure f the nnunifrmity f a chatic attractr. It is clsely related t the fractinal dimensin ccupied by the chatic attractr >< 5 Cunting Pairs f Pints at Varius Radii b) T estimate the crrelatin dimensin, use embedding crdinates and cmpute the crrelatin integral by cunting nearest neighbr pairs within a given radius. C(r) = lim _1 N) N 2.. l:;i:] 1 J c) The slpe f the lg r lg C(r) plt is the crrelatin dimensin D2. D = lim lgc(r) 2 r O " ' lgr {j bo (t) Crrelatin f Lrenz Strange Attractr in varius embeddings D2 = ± <> A)( <> cc c,<>'<> '<> <> cc )( <)'<> D CO X <> '<> D O X X c <> C )( <><> CC Ll.AO C O A c O )( e'i!)()( 1k pints <>,," a t=.3 't=.3 W=.IS <> embedding: 5 A embedding: 7 c embedding: 9 embedding: 11 )( embedding: r 1 I I, I I lg radius 2

$D a 2ni _ ay2 4DL(nani )_n ve v ay z vnni is the in trapped particle fractin E is the aspect rati y is the inin cllisin frequency v* is the diamagnetic drift velcity Fr a mre thrugh discussin f the$

9 Trapped In Cnvective Cell Turbulence The mdel simulates trapped in mdes which may be imprtant in the cre f tkamaks. The mdel is 2D, and energy is cnserved. ani at + VD ay + ')'Ill + * ani. D a 2ni _ ay2 4DL(nani )_n ve v ay z vnni is the in trapped particle fractin E is the aspect rati y is the inin cllisin frequency v* is the diamagnetic drift velcity Fr a mre thrugh discussin f the mdel and underlying physics, see David Newman, 3T22.

10 Fr the cases belw, a single nnlinearity is included: the dminant EB nnlinearity. The 2 cases presented, driven and undriven, are results frm a 1313 mde calculatin. Fr a given mde, bth the real part f the amplitude A and the energy I A I 2 were eamined fr evidence f chas.

11 Tiln.e Series f the (,3) :mde Energy undriven, unda:mped case r_, 1 LL _L L L ti:rne (arbitrary u.nits) Ti:me Series fr the (,3) :mde A:m p litu e 4 d 3 1 r r I_ v p e_d c_a s_e, L L L L L ti:rne (artirary u.nits).15 Ti:me Se r ies f Plidal M: a gnetic Field.1.5 I time

Case #1: Undriven, Undamped In this case, the mdel had n

12 Case #1: Undriven, Undamped In this case, the mdel had n driving r damping terms. Shwn are the energies in the (,1) and (,3) mdes. Estimates f the crrelatin dimensins fr the tw mdes are 4.3 and 5.9, respectively. An estimate f the largest Lyapunv epnent fr the (,1) mde is.28 ±.5 bits/time unit. Significant is nly the fact that this number is psitive with a small errr.

13 "k '" U b.o D X Crrelatin Functin f the Mde Energy due t dim=7 dim=9 dim =11 dim =13 dim =15 25 pints 't = 41 W=11 Trapped In Cnvective Cell Turbulence D D D D D D D D D D D 8 D = EB nnlinearity, undriven (,3) mde l radius

14 Crrelatin Functin f the Mde Energy due t Trapped In Cnvective Cell Turbulence dim=7 dim=9 1 I I dim = dim =13 dim =15 / 2 f I 25 pints 3 t 't = 29 W=ll '4 U 4 c Q.I 5 I 6 f 8 6 D2 = t O EB nnlinearity, un driven (,1) mde lq: radiu

15 Case #2: Driven, Damped In this case, the mdel included a nnzer driving term and viscus damping. Shwn 'is the amplitude f the (,3) mde. N Saturatin (cnstant slpe) regin is evident in the lg C(r) lg r plt, indicating that the crrelatin dimensin must be greater than 7.

16 1 2 3 'k U 4 CO Q 5 Crrelatin Functin f the Mde Amplitude due t dim=7 dim=9 dim =11 dim =13 dim =15 1 pints 't = 3 W =11 Trapped In Cnvective Cell Turbulence X 8 g 6 7 EB nnlinearity, driven (,3) mde lg radius

17 Simulatin f an RFP: A Reversed Field Pinch plasma was simulated using the DEBS cde. Current is kept cnstant while magnetic and electric fields are allwed t fluctuate. The simulatin was run with the parameters listed t the right. Fr mre detail cncerning the simulatin, see Elizabeth Zita,4Q26.

18 The time series used spans 45 Alfven times. The data recrd is far t shrt t draw any firm cnclusins frm. Hwever, the fllwing tentative statements can be made: a) The "burstiness" in the signal may be indicative f intermittency. b) A flattening f the slpe is seen spanning abut half a decade crrespnding t a crrelatin dimensin 2.2. Results are highly speculative. While the evidence suggests ptential lw dimensinal chas, a much lnger time recrd (n the rder 5) is needed t cnfirm this.

19 The Resistive MHD equatins reduce t: P d/dt = da/dt = S vb l1j Sp v.vv + S + UV2 B = VA, E = da/dt, J = VB, S = va'tr/a nrmalized with VA = B/" 41tl1, 'tr= 41ta2/ c211 Assume fields f frm B = B(r) ert i(m9+nkz) Parameters: Magnetic Reynlds number S=6l3 cnstant pinch parameter 8=1.59 als cnstant: tridal flu, resistivity, pressure

.. U bo,... 1 2 3 4 Crrelatin Functin f Plidal Magnetic Fluctuatins RFP Simulatin a dim=7 dim=9 <> dim =1 1 b.

20 .. U bo, Crrelatin Functin f Plidal Magnetic Fluctuatins RFP Simulatin a dim=7 dim=9 <> dim =1 1 b. dim =13 dim =15 24 pints t= 29 W=11 a a a a a D O b. b. a a D O a D O b. b. b. b. <> b. X 8 a b. a b. a <> D O b. b. D2 = lg radius

21 3 Jt U QQ I Crrelatin Functin f Tridal Magnetic Fluctuatins dim= 7 dim=9 dim =11 dim =13 dim =15 24 pints 't = 29 RFP Simulatin g D DO W= 11 OO O O D2 = lg radius 2 1

22 Cnel usins: a) There is strng evidence f lw dimensinal chas in the undriven case f trapped in mde turbulence. Varius mdes evince crrelatin dimensins ranging frm 4.3 t 5.9. b) The drivendamped case shws n evidence f chas belw a dimensin f 8. c) The RFP simulatin des shw tentative evidence indicating lw dimensinal chas, hwever further investigatin with mre data is necessary befre any firm cnclusins can be drawn.

23 Future Wrk: a) Cntinue studies f trapped in mde turbulence t determine whether r nt the size f the k space influences the crrelatin dimensin. b) Eamine the 3 different regimes f trapped in mde turbulence transient, steady state and slshing t determine any effect n dimensin. c) Cntinue generating the time series fr the RFP case t better establish whether r nt lw dimensinal chas eists. d) Using the equatins f the DEBS cde, analytically determine the spectrum f Lyapunv epnents.

GENERAL FORMULAS FOR FLAT-TOPPED WAVEFORMS. J.e. Sprott. Plasma Studies. University of Wisconsin

GENERAL FORMULAS FOR FLAT-TOPPED WAVEFORMS J.e. Sprtt PLP 924 September 1984 Plasma Studies University f Wiscnsin These PLP Reprts are infrmal and preliminary and as such may cntain errrs nt yet eliminated.