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Stochastic Transport in PPCD Discharges

1. We begin by creating a class for the FLPSCS which will contain all of our I O functions Starting this class will start background inner classes that check for received messages and the PLC trigger this class will also have functions that call threads to perform solicited actions set programming run mode def init self status settings Connect to the FLPSCS DGH module and PLC trigger serial specs 32 bytes 19200 bps 8 data bits 1 stop bit no parity Everything except bps and portname are defaults in pyserial Automatically opens pser if port is not none self status status self settings settings threading Thread init self print FLPSCS backend initialized reader start sorter start jJ OUTPUT QUEUE csq will be the list of commands to be sent either for queries or programming INPUT QUEUES brq is the bulk serial received queue which takes in all serial data from the flpscs srq is the status received queue sorted from brq this data will update the UI crq is the command received queue sorted from brq _sendbaton will be a lock that prevents run mode toggle while sending commands mainbaton prevents queueing up of commands when in run mode def run self sender start diplea start plcb self status start set unused delays to 100us for i in range 9 20 cs comsyn SetPulseWidth 0 i 100 csq put cs cs comsy
2. 0 26 0 0 0 1 02 03 p m Figure 3 16 The q profile from a representative data point with calculated island widths for n 8 16 fluctuations Appendix F using the q profile from a representative data point 3 16 confirms that the islands are small while there is still an overlap between neighboring islands The calculated Chirikov stochasticity parameter Appendix F for this region is almost exactly unity DRAFT December 19 2010 57 Chapter 4 Summary and suggestions for future work 4 1 Summary While many approximations were made in the previous calculations to arrive at these results the clear x b B scaling characteristic of stochastic transport clearly showed through And although stochastic energy transport is the dominant energy transport mechanism in standard discharges in MST due to the large fluctuation am plitudes it is evident from this work that stochastic energy transport is still capable of describing the behavior seen in the mid radius of the plasma in PPCD discharges a regime of very low fluctuation amplitudes The agreement between experimental and calculated stochastic transport is seen in all of the data points in the data set and for discharges where high core transport results in a flat T profile to the mid radius the stochastic transport at the mid radius becomes a strong determinant in the evolution of the core electron temperature It is very interesting to note that stoc
3. Te eV 1000 500 2 0 uM 0 2 4 6 8 D an G Figure 3 14 Core temperature of various PPCD discharges vs n 8 16 RMS fluctuation amplitude for all data points This demonstrates a strong argument for the presence of stochastic thermal transport in the mid radius region of PPCD discharges even in the case of very low relative fluctuation amplitudes Because the choice of Ar allowed the calculation to fit the measured data so well it is important examine the implications of this value Generally the autocorrelation length is of the order of a field line s transit length At the core where the field is completely toroidal the autocorrelation length is on the order of the major radius Lae TR 5m At the reversal surface where the field is compeltely poloidal we have Lae Ta 1 5m The small Dm that results from our choice of Ar and the resulting small Lac suggests that our calculation for stochastic transport was applied in in a regime of very low field stochasticity A calculation of the island overlap DRAFT December 19 2010 Xe versus b n 8 16 modes BG MMC ME x 771 40 x TEE Kx l a 30 x EG E x x 920b are LX X 10 E Calculated Xe x E Measured Xe x C RS M ris 0 10 30 40 20 b G Figure 3 15 A plot of measured ye red vs calculated stochastic Xe for all data points DRAFT December 19 2010 56 0 2 0 1 q vs Radius 0 0 0 1
4. 4 2 Suggestions for refinements and future work 4 2 1 Measured transport calculation refinements Two approximations were made when calculating the measured transport set ting the change in electron thermal energy W 0 and omission of the convective term in the power balance equation One of the reasons setting W 0 in the power balance equation was necessary because large errors result from taking the time derivative of T data points which already have inherently large error bars due to the high plasma temperatures As previously mentioned for the PPCD data set this results in an over estimate of Xe as the stored thermal energy is either constant or increasing throughout the enhanced confinement period However short of hardware improvements to the TS system itself there still exist a few approximations that can still be made to either minimize the effect of omitting W or offer a more reasonable lower bound to W One method is to take advantage of the characteristic that is prominent in many PPCD discharges the plateauing of core T This can occur at any point in the discharge and generally the temperature profile remains mostly unchanged for the remaining duration of the enhanced confinement period Barring significant changes in the density profile between time points selecting data points from only this plateau period would lend much more accuracy in omitting the change in thermal energy term Unfortunately this method automa
5. C cos 8c Z cos 0 C 2sin c Z2sin 0 2C Z sin dc sin dz cos dc cos dz C Z 2C Z cos dz c C 9 C sin c Z sin 1 a E Z4cos 0 10 The above result is identical to the formulation Darren Craig used in the mode analysis write up Regardless of the flavor of expressing the stationary error fields it is clear to see the impact of the error fields on not only the mode amplitude but also the phase of the mode DRAFT December 19 2010 86 Appendix D MSTFit Misc Note MSTFit solves the Grad Shafranov equation by solving for the parallel current density mp WoR 2rRP It does so by iteratively creating a set of flux surfaces solving for F and taking T and n data to create a P profile D 1 Transport Calculations Po feaa D 1 ea D 2 fora D 3 D 4 DRAFT December 19 2010 8T Ti vs radius from 500kA PPCD discharges 03 04 05 00 04 02 pim p m Figure D 1 T profile used for fitting and transport calculations where 7 is the neoclassical Spitzer restivity As 7 depends on the Z effective profile an assumed profile for Z effective that is flat in the core and decreasing at the edge has been applied to all the equilibrium reconstructions Z and T profile D 1 Loss terms include electron to ion collision losses Me Wa LN T Mi Tcoll Pi 3 where To is the classical electron ion coulomb col
6. 2 8 and the new ca pability to record optimum alignment positions and automatically adjust to them on subsequent run days DRAFT December 19 2010 Figure 2 9 Automated beam line alignment program with beam spot im ages for both lasers for stages w m and d DRAFT December 19 2010 19 20 2 2 Laser Power Supply Upgrade As previously mentioned a major upgrade to the Spectron laser Thomson scat tering system has been the replacement of the old commercial power supplies with one made by UW Madison s Physical Sciences Lab allowing for improved operational versatility of the Spectron lasers over the previous commercial incarnation The new power supply features fast IGBT switched capacitor banks that can be fired at various pulse width and duration This system is composed of a capacitor chassis pulse drive chassis capacitor control box and master control box The heart of the control system is C Assembly code programmed onto Rabbit microprocessor boards These boards perform various functions such as controlling IGBT pulse duration and timing setting capacitor charge voltage and simmer toggle as well as performing various status and error checks This power supply was used in conjunction with two Jorway 221 A tim ing modules which are capable of generating a 16 second long sequence of pulses with lus resolution The fruits of the labor that went into the Spectron TS system is the capability to operate at a variety
7. The result is an overall lower fluctuation level that is less influenced by the presence of stationary error fields 3 4 DRAFT December 19 2010 Database and Reprocessed mode amplitudes 20 Database Bx ReprocessedB 15 1 B J T I n T i o Il Mi PII Ml VUA nil nl 2 10 IN AM AY WI TT Ill WV WII MI 4 2 f QU VH MIHI TI TOI i5 L Ju Hi 4 2 l i 5 l I n haan M a AH NU he hwy ral wl ih 15 16 17 18 Time window ms Figure 3 4 RMS mode amplitudes of n 8 16 tearing modes comparing database fluctuation amplitudes vs fluctuation amplitudes corrected for stationary fields DRAFT December 19 2010 39 3 2 2 Core Temperature and mode activity The T profiles of PPCD discharges exhibit a flat region from the core to mid radius suggesting high transport that ties core T behavior to the mid radius This connection between behavior in the core and mid radius offers a strong motivation to examine the impact of transport in the stochastic mid radius region directly on the behavior of the core electron temperature It should be noted here that the proper calculation of stochastic transport such as performed in Biewer et al is an involved process not within the scope of this work However because stochastic transport Us B2 relies on the the diffusion of magnetic field lines as given byt Dm zz mR much information could still be gained by a proper ex
8. S Sarf N E Lanier S C Prager and M R Stoneking Increased confinement and 8 by inductive poloidal current drive in the reversed field pinch Phys Rev Lett 79 1 62 5 Jan 1997 J Lei P M Schoch D R Demers U Shah and K A Connor Core electrostatic fluctuations and particle transport in a reversed field pinch Phys Rev Lett 89 27 275001 Dec 2002 A F Almagri S Assadi S C Prager J S Sarff and D W Kerst Equilibrium studies of a poloidal divertor pinch with a reversed toroidal field Phys Fluids B 4 12 4080 Aug 1992 D Craig Magnetic mode analysis in MST 2005 T M Biewer ELECTRON THERMAL TRANSPORT IN THE MADISON SYM METRIC TORUS PhD thesis University of Wisconsin at Madison 2002 D Craig D J Den Hartog G Fiksel V I Davydenko and A A Ivanov First charge exchange recombination spectroscopy and motional stark effect results from DRAFT December 19 2010 23 24 25 26 27 28 65 the madison symmetric torus reversed field pinch volume 72 pages 1008 1011 AIP 2001 J K Anderson Measurement of the Electrical Resistivity Profile in the Madison Symmetric Torus PhD thesis University of Wisconsin at Madison 2001 J A Krommes C Oberman and R G Kleva C R Sovinec Magnetohydronynamic Simulations of Noninductive Helicity Injec tion in the Reversed Field Pinch and Tokamak PhD thesis University of Wiscon sin at Madison 1995 G M Za
9. both lasers is calculated for each stage and their overall deviation from their respective reference points is calculated If this deviation is beyond 10 pixels the XY deviations are sent to the calibrated motor controller subVI which corrects for alignment 2 7 DRAFT December 19 2010 18 beam_alignment visual_11 28 05 vi Fle Edt vew project Gperate Tools Whdow Hep veo elm T trigger waitims Camera camera Camera 2 camera uxo ae new picture Ble Edit wew Project Operate Tode Window 2 8 c m ict Application Fon 7 S ia Figure 2 8 Previous beam line alignment interface consisting of two sep arate VI s One VI images the beam spot right the other offers manual controls for the user to correct for beam mis alignment left In this subVI the previously calculated movement to pixel calibrations have been hardcoded The X Y pixel deviation inputs to this VI are multiplied by their appropriate calibration factor and sent to the motor controller for automated align ment A copy of the back panel along with calibration values can be found in the Appendix A The end result is the automated 2 9 optimization of a manual laser align ment correction method that eliminates the need for an experienced TS operator to constantly observe and manually correct the beam alignment
10. cation was broken down into four main parallel threads 2 11 a serial send thread and a serial read thread a sorting thread and a status checking thread The sending thread would monitor an internal command queue and send any commands from the command queue to the FLPSCS with a built in delay that allowed the FLPSCS to re spond with its expected solicited response The receiver thread would monitor inputs on serial port and send all responses to a response queue that is then processed by a sorting thread which identifies the response and updates the appropriate status indi cator The status checking thread periodically fills the command queue with queries for charge voltage simmer status and other various updates When the status thread is operating there is a thread lock on the status thread that ensures it has priority for populating the command queue When a user programs the system for operation the program thread requests the lock waiting for the status check to finish first if necessary and then proceeds to fill the command queue with programming commands A similar lock is used for the serial send thread itself to disable the sending of commands when the system is in Run mode DRAFT December 19 2010 amp lnitializations Import relevant packages Initialize program status threadlock and initialize program mode run mode threadlock Initialize voltage calibrations and FLPSCS syntax Initialize FIFO command an
11. d 3 1 m a 0 0 0 1 0 2 0 3 0 4 0 5 Figure 3 9 A plot of Xe with and without particle transport contributions and can be seen to be very small 3 9 Therefore convective transport contributions have been omitted from the following transport calculations and the the task becomes to properly calculate the local heating and loss terms in order to solve for Ne VT Xe Where Q is the total electron heat flux Po W Paci Prad Qe 4r rp Where the terms in the numerator are the volume integrated values of ohmic heating power change in electron thermal energy electron to ion collisional losses and radiative losses and the denominator is the surface area of the torus enclosing the volume With the current profile generated by MSTFit it is possible to calculate the local ohmic heating in the plasma by Po f gJ dV The change in thermal energy W was set to zero for this calculation as all of the data points for the PPCD data set were either from periods of constant or increasing T Setting W 0 in these situations at most results in an overestimation of the value of Xe Alternatives DRAFT December 19 2010 AT to the omission of W are discussed in the summary section Electron to ion loss Pero is based on a simple collisional model and the radiated power P aq is empirically set to be proportional to the ohmic heating power More detail is offered in the appendix C In order to properly identify t
12. go back to the outer while loop Class _plcb threading Thread Declare DGH run mode thread Continuously loop Wait for run mode request Acquire program mode and serial send threadlocks Enter run mode e If in run mode check for exit run mode request Exit run mode Release all threadlocks Class reader threading Thread Declare serial read thread Constantly read on serial port Put all received serial port messages into brq bulk response queue Class sorter threading Thread Declare response sorting thread Wait on brq for any messages Search for appropriate keyword and put message into corresponding queue Figure 2 12 Algorithm for FLPSCS Control page 2 DRAFT December 19 2010 26 2 2 2 Timing Algorithm and Implementation The design of the upgraded power supply system already allows for repeated triggering at 1ms intervals allowing for much more potential than the commercial power supplies However many different components of the TS system had to be coordinated together for the Spectron lasers to fully take advantage of the new power supplies While the timing could have been implemented in a variety of ways and is indeed in the process of being upgraded to a more compact system a very functionally elegant solution was implemented with the hardware available at the time The main focus of proper timing implementation is the capability of reliably and consistently generating 2J laser pulses while a
13. imaged by the CCD cameras During day to day operation a Thomson Scattering operator would monitor the alignment and beam position When the beam deviated from the center of of the turning mirror a command could be sent to a turning mirror earlier in the beamline to adjust the position of the beam spot Hardware limitations of the camera imaging equipment allow only one beam spot on one turning mirror can be imaged per laser fire Additionally the variations in orientation of mirror mounts motor control hysteresis and camera positioning setups between turning mirror stages meant that the same commands sent to different turning mirror stages could have very different impacts on DRAFT December 19 2010 Figure 2 3 Mirror mount and Pulnix Camera in a beamline how they changed the beam alignment Due to these difficulties ensuring proper alignment of the beamline required constant supervision by an experienced Thomson scattering operator At the start of the day the operator would find the ideal beam position for optimum collection of scattered photons As the day progressed the operator would need to correlate any deviation in beam alignment with a reduction in scattered photons suggesting a misalignment Then the user would input the appropriate commands to the turning mirrors to readjust the beam positions to what was optimal Building on this system an automation upgrade to the alignment system was implemented DRAF
14. of pulse burst modes where multiple laser pulses can be achieved with one flashlamp pulse enabling laser pulse repetition rates from 1kHz 12 5kHz per laser The two Spectron lasers were also operated in an interleaved format on a regular basis resulting in 2kHz 25kHz data acquisition rates Achieving these modes of operation involves direct control over both the flashlamp as well as Pockels cell timing DRAFT December 19 2010 21 2 2 1 Python Interface The control and timing system of the power supply is referred to as the Flash Lamp Power Supply Control System FLPSCS and interfaces with a PC via simple serial communication for setting parameters such as pulse width pulse delay flashlamp voltage etc for each channel as well as querying various statuses like capacitor charge and simmer status While robust the serial communication scheme requires the user to manually enter numerous commands and status checks per each channel For the Spectron lasers four channels were used for each laser 8 channels total and for the Fast Thomson 6 all 19 channels of the power supply would ultimately be used The FLPSCS was programmed to take properly formatted serial commands and send a response immediately after Voltage programming commands are received with a confirmation a query for previous settings returns the set value and status or error checks give the corresponding state of the system Aside from the solicited responses there ar
15. stochastic region where the n 8 16 modes reside This process relied on the use of a 2D equilibrium reconstruction code called MSTFit which is outlined below DRAFT December 19 2010 42 3 3 Equilibrium Reconstruction using MSTFit MSTFit is a 2D equilibrium reconstruction code which solves the Grad Shafranov equation in a torus dp 1dF QUSE ee ce Sy eum do Where F v RB E fl Oe OU pei 55 022 The plasma current pressure and reversal and pinch parameters are taken as inputs and a current profile is created from this input If there are additional di agnostics such as MSE or probe measurements the data is compared to expected values for the MSTFit generated equilibrium and used in a y minimization routine to additionally constrain the fit The pressure input is the straightforward product of electron from Thomson scattering and ion temperature and density A full write up of how MSTFit works is presented in Anderson s thesis DRAFT December 19 2010 43 3 3 1 Fitting Due to the high core electron temperatures associated with high current PPCD discharges and current limitations with TS hardware s ability to properly resolve those temperatures the Te profile in the core of a high current PPCD discharge often has quite a bit of structure and large error bars It was beyond the scope of this work to determine whether this structure was real and occurring in the plasma or systematic and an artif
16. the b measured at the wall The thermal velocity is a simple conversion from the measured electron temperature T The one parameter that is not well constrained by data in the calculation for stochastic transport is the mode radial width Ar The mode radial width can be in terpreted as a variety of values from the full width half maximum of the fluctuation s radial eigenfunction to a tearing mode s island width The radial component of fluc tuations on MST while peaked at their specific rational surfaces also span the minor radius of the plasma A half width of the fluctuation for the n 8 16 fluctuations would have Ar 0 2m The Ar term can also be interpreted as the island width associated with the fluctuation As Y stoch scales directly with what we choose for this value it is important to note its effects For an initial estimate a calculation with Ar taken to be 0 1m on the order of an n 10 island results in a reasonable autocorrelation length of Lac 1m The calculated stochastic transport for this value of Ar gives a Xe stoch approximately twice that which is measured 3 12 DRAFT December 19 2010 51 Measured Xe vs Calculated Xe 1000 0 100 0 N e o 10 0 E ab x L 1 0 Measured Xe Calculated Xe Oo a 3 a 0 15 0 20 0 25 0 30 p m Figure 3 12 A plot of measured x red vs Xe obtained from the previous calculation Instead the mode radial width was adjuste
17. the turning DRAFT December 19 2010 11 mirror located at the East Wall Xm CA T Crum Ci Ym Cymw as Cain Cons 2 1 The beam dump can be affected by two turning mirror stages Xa xS Aswd w T Cray T rmd Am gy Corn Ya AywaAw En Cywd w ED dA T CymdCm Here it is also useful to note that for the beam dump X and Y are modified by changes in both the wall turning mirror stage as well as the stage above MST However changes from the wall stage are generally unwanted and instead corrected for by the above MST stage Calibration was performed by taking repeated measurements of changes in X and Y on the stages above MST and in the beam dump by adjusting A and C at the wall and above MST then the coefficients were computed and hardcoded into the auto alignment software It is also important to note that there is often a fair amount of hysteresis in the turning mirror motors such that for example moving 1000 units in A and then 1000 units in A will not result in returning to the original position As such the above a s and c s must be solved for twice once in positive moves in A and C and again for the opposite direction DRAFT December 19 2010 12 2 1 2 Alignment Algorithm Much of the calibration and implementation of the automated alignment pro cedure relies on the use of the National Instruments Image Acquisition NI IMAQ development package of which the following functions were highly util
18. toroidal magnetic field is strongest at the core falling off radially and eventually reversing while the poloidal field increases radially fig 1 1 The reversal of the toroidal field gives the RFP Reversed Field Pinch its name The field configuration of the RFP arises from the solution to Taylor s equilib rium in a cylinder which can be obtained by minimizing energy with global magnetic helicity held constant this is approximately the case in conditions where the timescale for energy dissipation is much shorter than the dissipation of global magnetic helicity Meeting this criteria results in the requirement that V x B AB One may set A to a constant and take the curl of this equation in a cylindrical basis to arrive at solutions to the magnetic field to be Bessel functions These solutions are similar to the fields DRAFT December 19 2010 Br Reversed Figure 1 1 Illustration of RFP field configuration seen in an RFP It is important to note that the plasma current is proportional to B with V x B uoJ and as plasma pressure is related to the current and B by J x B VP we see that this requires plasmas in a Taylor equilibrium to have a flat pressure profile VxB A BxB 0 Ho Ho VP JxBe As real RFP s do not have perfect cylindrical symmetry have higher plasma pressure in the core than the edge and are heated ohmically they do not operate in the Taylor equilibrium configuration instead constantly
19. DB chan s ss 15 ss stat s ss 18 ss4 17 19 chan int chan 3 4 clist append chan slist append stat else print oops except DRAFT December 19 2010 colors 75 pass create an array of all Yellow status colors the length of our channel list temp for i in range 24 temp append Y dlist append temp temp for i in range 24 temp append i dlist append temp temp for i in range 16 temp append Y dlist append temp temp for i in range 16 temp append i dlist append temp is if the status Simmering else the indicators are red for i in range len clist2 if sligt2 i se B colors2 append G elif slist2 i 2 N colors2 append R elif slist2 1 2 D colors2 append G elif slist2 ij C colors2 append R else print oops for i in range len clist if slist 1 5 5 colors append G elif alise r eUN if clist pipesees colors append G elif elist i 3 1 colors append R elit slist i 2 D colors append G elit slhist i ee 6 colors append R elit eliat ij F colors append R else print oops self status dlist No fault if the response self status clist colors clist2 colorg2 time sleep 4 Manual toggles manual toggle of fake PLC trigger class fakerun def init se
20. ST at poloidal angle 241 allowing it only to resolve the n number of the mode The finite number of physical coils used to measure the field also limit the resolution of the n number By spatially Fourier decomposing the signal 64 coefficients from the summation can be determined allowing modes up to n 31 to be resolved Modes with n number greater than 31 will have their fluctuation amplitudes and energy aliased into the resolved n modes however the small amplitudes of the higher n modes are on the order of digitizer bit noise and contribute little to the overall mode amplitude In PPCD discharges the fluctuation amplitudes can be reduced to a point where variations in the coil to coil calibrations require a more careful and specific treatment of the mode amplitudes as stationary error fields can obscure the amplitude of a measured fluctuation While real effects of stationary fields on fluctuations such as causing mode locking can occur and have a real physical impact on the plasma examining the impact of these modes on PPCD plasma discharges is beyond the scope of this work DRAFT December 19 2010 36 When stationary and slowly varying error fields n Yn or Zn 4 z2 y2 are present in the array See appendix C 0 3 the calculated mode amplitude becomes Be o2 t 62 0 x2 yn 2 as 2s On t ynld or B C2 Z 2C Z cos dz c It can be seen that an error field can change the amplit
21. STOCHASTIC TRANSPORT IN PPCD DISCHARGES You MING YANG A dissertation submitted in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE PHYSICS at the UNIVERSITY OF WISCONSIN MADISON 2010 Contents 1 Introduction 1 1 1 The Reversed Field Pinch o deux ok 6 SR xou Te oO 1 2 Thomson Scattering 6 2 1 Beam gio ouod ni E PTS arie ed pe cS vae Le POLES 6 2 2 Laser Power Supply Upgrade n 404 Re URL eh Se Be 20 3 MST Plasma Characteristics 31 3 1 Improved Confinement Discharges ll llsn 3l 3 2 Mode Activity and Electron Temperature Evolution 3l 3 3 Equilibrium Reconstruction using MSTFit 42 4 Summary and suggestions for future work 57 ZI Summary aiuta the SN ees Ne AS A ae dE ev Oe o desk a eure OR 57 4 2 Suggestions for refinements and future work 59 A Alignment 66 B FLPSCS Interface 72 C Mode Analysis D MSTFit Misc D 1 Transport Calculations E Calculating Xs F Island width calculation il 81 86 86 89 91 Chapter 1 Introduction 1 1 The Reversed Field Pinch The Reversed Field Pinch RFP is a type of plasma confinement device used in the pursuit of understanding the physics that govern magnetically confined plasmas RFP s are characterized by a toroidal shape with an often circular cross section and a low externally applied toroidal field In an RFP the equilibrium field configuration is one where the
22. T December 19 2010 10 2 1 1 Calibration In principle the positions of the beam spots in the stage above MST stage m in fig 1 1 and in the beam dump d determines the path that the beam takes through the machine To adjust the position of the beam spot above the the machine requires adjusting the tilt of the turning mirror on the East Wall w adjusting turning mirror at stage w will also affect the beam s subsequent path m and final position in the beam dump d imaged by the beam dump camera Similarly adjusting the tilt of turning mirror m just changes the final position of the spot d The tilt of the turning mirrors are adjusted via orthogonal directions subsequently referred to as directions Ay and Cw for the wall and Am and Cm for above the machine Due to their varying orientation adjusting just one stage in one direction for example A will have an impact on both the X and Y positions of the beam center in subsequent stages We can write Xm and Ym X4 and Y as the coordinates in pixels of the beam center as recorded by the Pulnix cameras Using coefficients a c to convert from units of translation to change in pixel position and using the representation Gz ym to represent the change in X as a result of adjusting the turning mirror on the wall on the stage above the machine etc we can write the relationship between adjustments in A s and C s with X s and Y s For the changes in X Y in stage above MST due to a change in
23. act of the diagnostic Therefore rather than using a spline fit one that offers more freedom in fitting for the electron temperature profile a polynomial fit which makes a relatively good assumption that the Te profile is flat from the core to the mid radius and still captures the characteristics of the temperature gradient in the mid radius was used for the T profile allowing for a consistent and reliable fit between different sets of data 3 7 Te v CP ease UU disc EN y T edge The model generates the best fit possible by adjusting the domain wv and parameters a and b Varying v adjusts the position of the gradient while a 4 and b 1 2 adjust the steepness of the falloff While robust the polynomial model can only fit monotonically decreasing func tions with one gradient region As a result many PPCD discharges that appear to have two flat top regions a large flat core and a smaller flat region located near the reversal surface 3 8 were omitted from the dataset due to the inaccuracy of the tem DRAFT December 19 2010 44 Fit to Te data using free splines Fit to Te data using polynomial model Figure 3 7 Example of a free spline fit on the left versus the polynomial temperature fit on the right perature fit in this region It is worth mentioning that there is strong motivation for a capability to better resolve and fit this region as cursory observations suggest that possibly all PPCD discharges sho
24. amination of the measured fluctuations in the T gradient region To examine the potential impact of stochastic transport on PPCD performance core temperatures from high current PPCD shots from June 30 2009 were plotted against the RMS sum of their n 8 16 mode amplitudes All of the discharges from the set had the same voltage and timing programming although the start and end times for good PPCD would vary Specific shots of moderate to good quality PPCD during times starting 12 14ms and ending 17 20ms were chosen Core electron temperature was plotted with the RMS n 8 16 mode amplitudes and showed a clear positive correlation between a high core temperature of a specific discharge and a low RMS mode amplitude in the stochastic region 3 5 Plotting the core Te vs the core n 5 7 modes 3 6 shows very little correlation between core Te and fluctuation amplitude The next step in the investigation involved a local transport analysis in the region DRAFT December 19 2010 Te versus b n 8 16 modes 2500 2000 x S 1500 E di v OX i 1000 500 0 0 2 4 6 8 Diwan G Figure 3 5 Core temperature of various PPCD discharges vs n 8 16 RMS fluctuation amplitude Te versus b n 5 7 modes 2500 2000 S 1500 x X x 2 1000 500 0 0 5 10 15 20 Dewan G Figure 3 6 Core temperature of various PPCD discharges vs n 5 7 RMS fluctuation amplitude DRAFT December 19 2010 A of interest the
25. ata Contains the x pixel of the calculated beam center ditto for y Boolean Eul offset of the ROI mask d offset in y of above Figure A 2 align create back DRAFT December 19 2010 69 autosteer fpga2 vi Block Diagram Ele Edit View Project Operate Tools Window Help amp il kallot 13pt Application Font Fv star gt Last Shot Number FL Enter Alignment Date mm dd yy pa E Write Images Figure A 3 align runtime backtop DRAFT December 19 2010 70 autosteer fpgaz vi Block Diagram ETE Ele Edit View Project Operate Tools Window Help amp amp m 25 balat 13t Application Font 1 image backup1a Moe P wala walc Above MSRove MSTC aka Fi
26. ay to day operation of the sys tem Additionally a pulse burst upgrade to the laser system has greatly increased the amount of temperature data that can be collected during each MST discharge or shot allowing for resolution of fast plasma dynamics for fluctuation measurements Stephens and the time resolved evolution of the plasma electron temperature profile The latter capability has allowed for unprecedented single shot analysis of discharges that have a high shot to shot variability such as Pulsed Parallel Current Drive PPCD discharges Single shot analysis of PPCD performance suggests a possi ble correlation between performance and m 1 n 8 15 magnetic fluctuations near the reversal surface An investigation of the effect of stochastic transport in this improved confinement regime is presented DRAFT December 19 2010 Chapter 2 Thomson Scattering 2 1 Beamline The Thomson scattering diagnostic on MST currently uses two commercial Spec tron SL858 lasers located in a laser room across a public hallway from the MST experiment After exiting the laser head the laser beam is directed underneath and across the hallway which separates Thomson scattering from MST over and above the machine and finally down vertically through the machine vessel as can be seen in figure 2 1 Reusch et al A seven element lens is used to focus scattered laser light onto 21 optical fiber bundles This scattered light travels to filter polyc
27. d for one data point one time point in one discharge such that the experimental and calculated Xe agreed this resulted in Ar 14m and an autocorrelation length Lace 0 5m This value of Ar was used for all of the data points Plotting the calculated x 5 5 and measured Xe meas against a data 3 13 clearly shows strong agreement between measured and calculated stochas tic transport As this analysis uses local values for all of the parameters in the transport calcu lations it was possible to expand the calculation beyond the data set where the core T tracked with n 8 16 modes The same calculations for the measured and expected stochastic transport was applied to all of the properly fit data points available The inclusion of these additional data points show more scatter between the core T rela tive to RMS fluctuation amplitude 3 14 however the agreement between measured DRAFT December 19 2010 52 Xe versus b n 8 16 modes 90 ee Pee CALLES 7 40 j TE 30 EX E E xK E xr kK X X 10 gt x x Calculated Xe x E Measured Xe x 0 ER cm TENTE NT T l tajila 0 10 30 40 20 b G Figure 3 13 A plot of measured x red vs calculated stochastic Xe for select data points DRAFT December 19 2010 transport and expected stochastic transport remains clear 3 15 DRAFT December 19 2010 53 54 Te versus b n 8 16 modes 2500 2000 1500
28. d status queues Main Thread Class flpscs threading Thread Declare main thread Upon initialization of main thread Start thread to read serial Start thread to sort received serial messages Upon execution of main thread Start thread to write to serial Set pulse widths of all unused channels 9 20 to 100ps Default 1ms Set FLPSCS via DGH to program mode continuous operation Start status checking thread Start simmer and CCS toggle threads interfaces with mouseclick on GUI radio buttons Programming thread Class command threading Thread Declare program thread Upon initialization of program thread Wait to acquire programming threadlock Readin voltages pulse delays and pulse widths Send desired programming values to command syntax processing thread Put properly formatted commands into serial send queue Status thread e Class status threading Thread Decare status checking thread Upon initialization of status thread Continuously looped with While 1condition allowing thread to run in background Acquire programming threadlock Send status requests to command syntax processing thread Put properly formatted commands into serial send queue For 8 seconds Get status values from response queue Isolate channel number and corresponding state Update color array with new status any response that takes over 8 seconds has an indeterminate status Write status to GUI fr
29. data at rates ranging from 2kHz to 25kHz in pulse burst mode fig 2 13 The Jorway J221 A modules function by receiving two input arrays one for time us after an input trigger and one to turn any number of channels referenced via bit number 1 high low corresponding to that time For example two arrays consisting of 0 5 10 and 6 2 0 would have the Jorway turn on channels 3 and 2 on at t 0 and then leave only Ch 2 on after t 5ys then turning everything off at t 10us The Jorway modules are interfaced with GPIB via low level Labview drivers DRAFT December 19 2010 28 2 2 3 Timing Schemes By using the basic pulse burst hardware and timing scheme above the simple modification of certain parameters such as flashlamp pulse widths and delays and fast J221 triggering allows for a variety of TS operation modes Single Pulse Operation fig 2 14 The most basic and commonly used mode of operation is the interleaved mode of two lasers firing in Single Pulse mode In this mode after the MST trigger the macro J221 sends master triggers spaced 1ms apart to the FLPSCS For each master trigger received by the FLPSCS after a 10s jitter at t 0 the four channels driving the first laser s flashlamps fire for a 150js duration pulse At t 140us5 to allow for an optimum population inversion in the rods the fast J221 triggers the Q switch for the first laser and digitizers After a 500us delay the four channels driving the second las
30. e additional unsolicited responses the FLPSCS is capable of sending Between discharges the system performs a self diagnostic error check for a potential Gate Driver Board fault and will send an unsolicited error message if such a fault exists The same occurs if the emergency stop is triggered Additionally if the system is in E stop mode or Run mode it will not send any responses to the user This motivated the creation of a graphical user interface that could automate many of the aforementioned tasks develop the best implementation to interface with the control system as well as reduce the workload required of a TS operator when programming the power supply for the desired mode of operation DRAFT December 19 2010 22 Reader Thread Sender Thread t Command Queue Response Queue FLPSCS 9 AS U User Input Status Thread Sorting Thread Status Indicator Update GUI Front End Figure 2 10 Flow chart for FLPSCS Control Interface DRAFT December 19 2010 23 For ease of use testing and modification a GUI interface in Python was created The back end was programmed in Python and the front end utilized PyQT a Python wrapper around the QT graphics package Formatting of the serial commands and graphical implementation aside the main efforts of the power supply interface went into the method of handling the serial I O Taking advantage of Python s multi threading capabilities the I O communi
31. e request if self flag 0 DRAFT December 19 2010 self flag l mainbaton acquire sendbaton acquire print run mode modi dghser write 1SB09 r n elif flag 1 pass 79 this portion acquires the _csq lock so that status and command button cannot access it the brq join will make sure the bulk response queue for now we assume this starts filling up while DO send serial command for run mode elif trigger off elif a 0 if self flag 1 print off runmode self flag 0 modi dghser write 1CB09 r n time sleep 2 _mainbaton release sendbaton release elif flag 0 pass except pass class _reader threading Thread Class which will read all serial communications into the sort into various solicited and unsolicited lists def init self threading Thread init self self started 0 def run self while 1 modi pass a pser readline print a if a find Power On Init Complete 1 if self started 0 self started l stetus start elegit modi _brq put a class sorter threading Thread def run self while 1 b _brq get af b find Crq put elif b find crq put elif b find srq put elif b find Srq put else zerqsput QU e Ug ve Jg x g gie AES GA Go ARA D n PulseWidth 1 MonitorVoltage 1 Status 1 c s DRAFT December 19 2010 i
32. ed for each mirror stage The output of this routine is stored as a proprietary IMAQ data file format which simply consists of the image mask stored as a png and X Y pixel coordinate DRAFT December 19 2010 17 Wal walc Above MSR amp ove MSTC Bi123 23 1237 HI HI HI HI a Figure 2 7 Reference call to motor controller subVI with X Y pixel devi ations in stages m and d taken as inputs of the laser beam embedded as ASCII data within the png file 2 6 As each file contains separate X Y pixel information one per each laser there are two files stored per laser stage containing the same image mask and different center coordinates Very often the relative alignment of the components in the beamline such as the orientation of the cameras relative to the turning mirrors and the fiber mount and lens setup are undisturbed for long periods of time and these calibration files may be used for extended periods of time without a need for recalibration Once the calibration files have been created they may be used by the runtime VI for day to day operation The runtime VI begins by loading the six calibration files and proceeds to wait on six consecutive camera triggers in a specific laser firing sequence After each camera trigger the imaged laser pulse is cropped by its respective image mask and the new beam center is calculated The beam center for
33. eld line undergoes allowing for the calculation Xe MAL vr a as a direct comparison to Xe stoch given by the Rechester Rosenbluth model However the above equation for ye is only assumed valid for regions of high stochasticity A numerical calculation of the of Xe Ar compared with Yemeas and Xe stoch in this case would give insight into the the applicability of stochastic transport in this regime as well as allow for a better estimate of the actual stochasticity of the field in the mid radius DRAFT December 19 2010 61 Te versus normalized b nz8 16 modes 2500 L 500kA PPCD x 2000 500kA standard PT Point x x ra 400kA Standard x S 1500 AK J 9 PE o x p 1000 x 500 a 0 000 0 005 0 010 0 015 Diwan B arb Figure 4 1 tbpp Probing the limits of stochastic energy transport The previous refinements would improve the accuracy the results seen for the 500kA PPCD discharges and also be useful in the many other operating regimes MST is capable of An examination of peak core T with respect RMS fluctuation amplitudes for four different operating modes 500kA PPCD 500kA standard 400kA PPCD and 400kA standard discharges shows a clear continuation of relationship seen previously 4 1 The same analysis done in this thesis could easily be applied to those different operating regimes and would greatly increase our understanding of the scope of applicability of the stochastic energy transport mechani
34. er s flashlamps fire for a 150s duration pulse 140s into the second pulse at t 640ys the fast J221 triggers the digitizers and Q switch for the second laser This results in two pulses spaced 500us apart repeating every lms for 2kHz operation Triple Pulse Operation fig 2 15 An excellent demonstration of the pulse burst mode of operation is the interleaved mode of two lasers firing in Triple Pulse mode In this mode after the MST trigger the macro J221 sends master triggers spaced lms apart to the FLPSCS For each master trigger received by the FLPSCS after a 10s jitter at t 0 the four channels driving the first laser s flashlamps fire for a 310us duration pulse At t 140ys to allow for an optimum population inversion in the rods the fast J221 triggers the Q switch for the first laser and digitizers Each laser DRAFT December 19 2010 29 Sync and MST Macro 10us Fast ETC EAC3E Laser 1 IR 1 100 200 300 400 500 600 700 800 900 1000 Single Pulse Interleaved Mode 2kHz Laser 2 a 100 200 300 400 500 600 700 800 900 1000 Figure 2 14 Two lasers interleaved in Single pulse mode capable of 2kHz operation for 15ms pulse only partially depopulates the rod and after a short 80us repopulation period another laser pulse of 2J energy can be fired again The Q switch and digitizers are triggered again at t 220us and t 300us For the second laser a pulse delay of 40s is used The second laser s
35. era3 aM trigger wait ms 7 jj 360000 9 Laser 1 49 Laser 2 Di Force Align Fere f A a Beam center 2 r 2 XCentroid DI bo 2l Y Centroid e po o l el Date of Alignment mm dd yy 3 test amp Save reference 640x480 1X Signed 16 bit image 0 115 1 FLP Lamp channelbit Camera Cameral Camera2 Cameras ao trigger wait ms 1 360000 Laser 1 O Laser 2 Force Align 1 Laser 1 so Calculate Beam Center i e Beam center XCentoid 535 19 1 YCentrod gesee Date of Alignment mm dd yy 13 test 4 Figure 2 5 Create reference VI before and after manual selection of ROI to crop DRAFT December 19 2010 16 Figure 2 6 Reference image mask file used for auto alignment aforementioned intensity averaging giving the pixel locations of the intensity centroid which calculates the center of the intensity in the X and Y directions as Ma I x estu 2 2 2 3 My I y Sg en Where x and y represent a pixel s coordinates and I x y is the numerical value of the intensity for a pixel at that given x or y position and the sum is taken over all pixels For the eight bit pixels used in the alignment code the intensity can range from 0 to 255 where 0 is a black pixel with no energy and 255 is a white pixel with the most energy As the orientation of the cameras and mirrors vary between stages a different image mask is created and us
36. gure A 4 align runtime backbot DRAFT December 19 2010 71 steerer vi Block Diagram Ele Edit View Project Operate Tools Window Help ola a n P 25 eal os 13t Application Font viori 85 15 2b i Xand Y FES mod C move 227 3201402 6 379950628 Navigation Figure A 5 align calib back DRAFT December 19 2010 Appendix B FLPSCS Interface Initializations This is the code used to manage the FLPSCS power supply it will be able to address all of the FLPSCS functions however it is intended to be imported into another script or GUI and fed inputs from there Note that all timing is somewhat order of magnitude as time clock goes by the cpu clock which varies between computers and also time sleep has lims accuracy import threading serial Queue time ctypes mainbaton threading Lock sendbaton threading Lock dwps ctypes c_ulong 0 dwpc ctypes c_ulong 1 pb ctypes c_byte qb ctypes pointer pb modi load the device driver for the USB D i o board a ctypes windll LoadLibrary adsapi32 pshort ctypes c_ushort i ctypes c_ulong 0 dhandle ctypes c_long dhandle2 ctypes
37. hastic transport in MST is generally as sociated with poor energy confinement due to the large fluctuation amplitudes in standard discharges However in this situation the stochastic transport mechanism DRAFT December 19 2010 58 plays a significant role even in regimes of very low magnetic fluctuation amplitudes where very small transport Xe 10 still scales following what is predicted by the Rechester Rosenbluth model for stochastic transport The result is that the regions of stochastic transport in these PPCD discharges exhibit some of the lowest transport in the plasma saw in his work the stochastic transport mechanism In line with what Biewer is applicable in regimes of very low fluctuation amplitudes as long as the field is still stochastic What is interesting in this situation is that estimates of the island widths for the n 8 16 regions in our 500kA PPCD data set and the spacing between rational surfaces suggest that the island overlap and resulting stochasticity is actually quite small This is reflected in the value which we set our mode radial width as well The strong agreement between our data and the model for stochastic transport that should be at its limit of applicability greatly encourages additional investigation Some further refinements are suggested in the following pages for a more detailed investigation into probing the limits of the stochastic transport mechanism DRAFT December 19 2010 59
38. he region of interest in the plasma to examine the local transport with respect to fluctuation induced stochasticity we examine the q profile generated by MSTFit The rational surfaces radial locations at which the q profile becomes rational q 7 give the positions at which the fluctuations of interest reside The m 1 n 8 16 tearing modes would be centered on radial locations where q 1 8 1 9 etc As any stochastic transport would be the result of an overlap of the islands centered on these locations Xe is evaluated at the geometric mean of the n 8 15 rational surface locations A plot of the local Xe vs the n 8 15 mode amplitudes shows that x scales linearly with the squared sum of the fluctuation amplitudes 3 10 DRAFT December 19 2010 Xe versus b n 8 16 modes Brown ee ae eae T 40 yee iz 30 x E E 10 KK 0 s oss PETRI 4 0 10 20 30 40 bZ G Figure 3 10 A plot of local ye vs rms fluctuation amplitudes of n 8 16 modes DRAFT December 19 2010 48 49 2p pt Bt Bp Br 1E m 1 n 11 ee N LLLLLELLLLLLLLLT LELLLLLL Br Bth Bz vd ERIT ES I Libis Litiit io va Figure 3 11 A plot of the m 1 n 11 eigenmode for Br Bt and Bp 3 3 3 Xe and stochasticity limits This is a good moment to compare this result with an estimate of the expected transport due to a stochastic field The stochastic energy transport in a collisionless plasma can be expre
39. hromators P containing filters that select out specific wavelength regions to pass onto avalanche photodiode detectors The incident laser beam undergoes 5 turns before scattering off of the plasma and the overall length of the beam path is over 15m With this setup the beam alignment can be observed to drift throughout the course of the day This drift is more pronounced during summers and has been attributed to thermal expansion and contraction the building undergoes in various parts of the 15m beamline If left DRAFT December 19 2010 Elevation Laser Head Camera N Camera 7 Vacuum Vessel CO Pumping Duct im B D Laser ISIS Housing eam Dump Head Port Hole Limiter Chamfered MST Wall core laser beamline 5 cm Thick Al Wall Figure 2 1 TS Beam PHtitAfofieebarroon2049 MST above and TS collection optics below Fiber FOV Scattered Signal Fiber FOV Scattered Signal Good Alignment Bad Alignment Figure 2 2 Illustrations of good alignment and poorly aligned scattered laser signal relative to optical fiber bundle uncorrected the beam can drift out of the field of view of the collecting optics 2 2 resulting in reduced scattered light and poorer quality of data To counteract this the beamline is equipped with remotely actuated turning mirror mounts and CCD cameras mounted as shown in figure 2 3 The beam spots of the lasers incident on the turning mirrors are
40. igure 3 2 A comparison of transport in Standard vs PPCD discharges taken from 2009 CMPD and CMSO Winter School Lecture by Dr Sarff Green line is calculated stochastic transport vs measured transport 0 20 m 1 n 6 15 0 15 island widths 0 10 0 05 0 00 0 05 0 0 0 2 0 4 0 6 0 8 1 0 r a Figure 3 3 A q profile of a standard discharge showing the calculated widths of magnetic islands centered in their rational surface The island widths are greatly reduced in PPCD discharges and core stochasticity is small however stochasticity near the reversal surface can still dominate transport 8 of 2 23 DRAFT December 19 2010 33 34 amplitudes The smallness of these mode amplitudes mean that the mode data requires a more careful treatment of the measured magnetic signals to arrive at the proper fluctuation amplitudes In this regime small error fields that arise during a plasma discharge are often on the order of the measured fluctuations and correcting for this contribution is crucial to the understanding of transport in this regime DRAFT December 19 2010 35 3 2 1 The toroidal Array on MST The magnetic field in MST as measured by the toroidal array can be written as an infinite sum of its Fourier components see Appendix C B 3 c cos n den n 0 The field is measured unintrusively by an array of 64 passive magnetic coils that are spaced evenly along the toroidal direction of M
41. ized for this purpose a function to calculate the geometric center of intensity of a black and white image in a selection region of interest the ability to store and retrieve custom regions of interest and data The implementation for the automated alignment setup was then broken down into two components Once ideal alignment had been achieved via manually adjusting the turning mirrors a create reference program would be executed to record the beam positions on each turning mirror for each laser A region of interest ROI for each turning mirror is selected to calculate the beam center used to crop out bright scattered artifacts from mirror mounts reflections etc then the ROI and center of the beam spot is saved to a reference file An autosteering program is then used for day to day operation After a data taking discharge the camera is triggered with individual laser pulses recording the location of the beam on each of the turning mirrors The beam center is then calculated using the same ROIs as those stored in the reference files and the new beam center compared to the stored reference position If the pixel deviation X s and Y s is higher than 10 pixels total on any stage the turning mirrors are adjusted via the calibration calculated above The end result is an automated beam alignment program that requires much less user supervision than the previous iteration DRAFT December 19 2010 13 Alignment Example 1 De
42. l every few minutes Once optimum alignment has been achieved the next step is to create calibration files used in the automated alignment program A Labview VI was created for such a purpose and has a front panel which allows the user to select a specific turning mirror stage and laser to create references for The full FOV of the camera captures the scattered light of not only the beam spot imaged on the turning mirror but also additional artifacts due to scattered light such as the mirror mount and sometimes reflections of the camera itself To most effectively utilize NI IMAQ s built in intensity averaging subVI the camera image is displayed on the front panel and the user may select an outline within the image to be cropped to a specific region of interest ROI for processing 2 5 In the back panel upon execution the program waits on a camera acquire trigger which acquires what a specific camera stage sees within a 16ms time window This trigger is sent to the camera right before a laser of interest is fired allowing for the selection of one laser beam on one camera stage This acquired image is displayed on the front panel where the user selects an outline to use as an ROI This ROI serves as an image mask cropping out any of the image beyond the selected region The remaining region is then input into the NI IMAQ Centroid subVI which does the DRAFT December 19 2010 15 FLP Lamp channelbit Camera Cameral Camera2 Cam
43. lamps are fired for the same 310us duration pulse The fast J221 triggers the digitizers and Q switch for the second laser at t 180ys 260us 340us This results in two pulses spaced 40s apart repeating every 1ms allowing 25kHz burst operation for 6 pulses every lms DRAFT December 19 2010 Sync and MST Macro 10us Fast Trigger on PL sitter iic 9 5 Laser 1 1B Tl 100 200 300 400 500 600 700 800 900 1000 Triple Pulse Burst Interleaved Mode 25kHz Laser 2 n 100 200 300 400 500 600 700 800 900 1000 Figure 2 15 Two lasers interleaved in Triple pulse pulse burst mode capa ble 25kHz operation for 6 pulses every 1ms DRAFT December 19 2010 3l Chapter 3 MST Plasma Characteristics 3 1 Improved Confinement Discharges As previously mentioned the gradient in parallel current can be a drive for the instabilities that degrade energy confinement One means of improving confinement is therefore to reduce the gradient which drives the instabilities by techniques such as Pulsed Poloidal Current Drive PPCD By externally driving parallel current along the edge of the plasma 3 1 PPCD attempts to flatten the parallel current profile While extremely effective PPCD discharges have high shot to shot variability The same plasma current density and PPCD programming can often result in disparate temperature evolution 3 2 Mode Activity and Electron Temperature Evolution The reduced fluc
44. lf self indie 0 or Done charging is timed out 1 dlist 0 dlist 3 dlist 2 DRAFT December 19 2010 indicators are green it will be yellow def go self if self indie 0 self indie 1 rt putil else rt put 0 self indie 0 toggle of CCS part 1 class ccstog def iuit self self ccs off off off off off off off off def go self ccsnum if self ccos ccsnum off self ccs ccsnum on ccst put ccsnum elif self ccs ccsnum on self ccs ccsnum off ccst put ccsnum 1 else print holy crap CCS exception toggle of CCS part 2 class ccsrun threading Thread def run self while 1 ccsnum ccst get mainbaton acquire if ccsnum 50 a comsyn EnableCCS 0 ccsnum csq put a else a comsyn DisableCCS 0 abs ccsnum csq put a time sleep 3 mainbaton release Toggle of simmer part 1 class simmertog def init self self sim off off off off off off off off def go self simmernum if self sim simmernum off self sim simmernum on st put simmernum elif self sim simmernum on self sim simmernum off st put simmernum 1 else print holy crap simmer exception Toggle of simmer part 2 class simmerrun threading Thread def run self while 1 si
45. lision time as well as radiative loss which is empirically scaled to Po r Prad PoCraa EY where is a coefficient C44 is based off data from the bolometer DRAFT December 19 2010 88 By combining all of these terms we can arrive at the local electron heat flux Q Po We Paoi Prad Qe 4r rp Dividing this value by the density and temperature gradient allows us to arrive at a thermal transport coefficient Xe Qe Ne VTe Xe MSTFit also generates a q profile from the reconstructed equilibrium using the rBz RB definition q DRAFT December 19 2010 89 Appendix E Calculating Xs Xe VTeDm E 1 Bi Tha p m E 2 T La S E 3 k B ky JBT E 4 1 ym d B3 B Ts D d o IP n DB r B R B gt TL m RBg Bg R n r B B where B B B DRAFT December 19 2010 and ki Aky Ar l ae which gives TL o m RBg Bg Ak A ki R r E DRAFT December 19 2010 rs 90 91 Appendix F Island width calculation The island width was calculated using the equation F 1 Wmn n Bs a The modes were assumed to be dominated by m 1 fluctuations The values for ra Bg and q were taken from the MSTFit generated q profile its derivative and equilibrium field profile and the local Don was obtained from reprocessed edge coil data scaled using the same eigenfunctions as
46. mnum st get mainbaton acquire if simnum gt 0 a comsyn EnableSS 0 simnum _csq put a print enabling simnum else DRAFT December 19 2010 a comsyn DisableSS 0 abs simnum csq put a print disabling simnum time sleep 3 mainbaton release Background threads class sender threading Thread def run self should send serial of qt get for now print while 1 cs _csq get _mainbaton acquire sendbaton acquire while 1 modi pser write cs print cs if cs find SetPulseDelay 05 1 sendbaton release mainbaton release break elif cs find SetPulseDelay 09 1 sendbaton release mainbaton release break elif cs find GetGDBFault 09 1 sendbaton release mainbaton release break elif cs find SetPulseWidth 19 1 sendbaton release mainbaton release break elses cs _csq get DELEZ try cs _csq get 0 except sendbaton release _mainbaton release break time sleep 3 Thread which will continuously check for a PLC trigger and upon receiving one wait for other processes to finish before locking them out class plcb threading Thread def init self status threading Thread init self self flag 0 self status status Zchecks interface for PLc trigger def run self while 1 time sleep 20 try a rt get 0 if e Zprint Received runmod
47. n SetPulseWidth 0 1 100 DRAFT December 19 2010 74 tsq put cs Set continuous mode modi dghser write 1D00400 r n time sleep 1 dghser write 1SBOB r n time sleep 4 modi status self status start simmerrun start cesrun start Programming thread class command threading Thread def init self laser input threading Thread init self self input input self laser laser def run self mainbaton acquire for i in range 4 creating input O0 setoutput voltage nn vvvv cs comsyn SetOutputVoltage 0 chanlist self laser i abs self input 0 i 10 caliblist self laser 2 i slopes self laser i _csq put cs create input 0 set width delay nn vvvv for ii in range 2 if self input 1 i 2 ii lt 450 cs comsyn proglist ii 0 chanlist self laser i self input 1 i 2 ii else cs comsyn proglist ii 0 chanlist self laser i 150 esq put es time sleep 4 cs comsyn GetPulseWidth 0 chanlist self laser i self input 1 ii _csq put cs time sleep 2 ticl time clock while time clock ticl lt 5 try em erg get 0 ind c find PulseWidth if ind 1 print c ind 411 ind 4 13 print c ind 414 ind 4 20 else pass except pass mainbaton release Status thread class status threading Th
48. on MST is to reduce the stochas ticity of the plasma thereby reducing the stochastically driven transport This is done by reducing the plasma s own drive to reduce the gradient in the parallel current profile To do this we apply an external current profile control method called Pulsed Poloidal Current Drive PPCD PPCD inductively drives poloidal current in the edge where it is the parallel current of the plasma This modification of the current profile offers a dramatic improvement in plasma properties including the reduction of island widths and a correspondingly lower level of stochasticity the elimination of sawtooth crashes healed flux surfaces and much higher core electron temperatures as a result of a dramatic improvement in electron energy confinement While PPCD dis charges show much better performance on the metrics of energy confinement and peak DRAFT December 19 2010 4 core plasma temperature they also exhibit high shot to shot variability between dis charges that have very similar starting conditions This shot to shot variability offers an opportunity to examine the impact of stochasticity on varying confinement quality Understanding and predicting the causes of this disparate behavior will also benefit not only the optimization of PPCD but more advanced current drive techniques in the future A diagnostic that has been crucial in helping illuminate the behavior of RFP plasmas such as those in MST is the Thom
49. ont panel Release threadlock Figure 2 11 Algorithm for FLPSCS Control page 1 DRAFT December 19 2010 24 Manual Toggles Class fakerun Define run mode toggle Upon toggle either puts 1 or 0 into the run mode queue Class ccstog Toggle of CCS part 1 Define Capacitor Charging Supply CCS toggle Upon Initialization with channel number If CCS is off put command to toggle on in CCS queue If CCS is on put command to toggle off in CCS queue Class ccsrun Toggle of CCS part 2 Upon initialization continuously looped Wait for command in CCS queue Acquire serial send threadlock Enable or disable the respective CCS channel Release serial send threadlock Note Due to the waits and threadlocks the PyOT GUI will lock up if the interface toggle is directly send to the serial thread hence the need for splitting up the CCS interface Class simmertog summerrun Toggle of simmers parts 1 and 2 Simmer toggle implementation is identical to the CCS thread Background Threads Class sender threading Thread Declare serial send thread Upon initialization of serial send thread begin outer while loop Wait on serial send queue for formatted command Acquire program mode threadlock once command is to be sent and enter inner while loop Send command Send any remaining commands in queue with 0 3 seconds delay processing limitation of FLPSCS Release threadlock and
50. pointer dhandle c a DRV DeviceOpen i dhandle2 Create lists of commands for programming checking status and on off null at position zero shifts indices chanlist 1 and 2 to match lasers chanlist null chanlist append range 2 6 chanlist append range 6 10 proglist SetPulseWidth statuslist GetEOC GetSS setlist GetEOC GetSS list of channels for iterating through SetPulseDelay gt GetGDBFault caliblist 3836 3831 3835 3803 3845 3852 3822 3868 082 08 11 1 124 078 068 02 liege geal T for i in range 4 llcal append 10 caliblist 3 12cal append 10 caliblist 4 i caliblist 1 i i i aliblist 2 1i DRAFT December 19 2010 72 73 slopes llcal 12cal Create all of the internal variables we will need csq commands to be sent brq bulk responses srq status responses crq command responses erq error responses currently not implemented drq dgh responses sq simmer queue RT used to test the run mode toggle csq Queue Queue brq Queue Queue srq Queue Queue crq Queue Queue erq Queue Queue drq Queue Queue st Queue Queue ccst Queue Queue rt Queue Queue modi pser serial Serial COMl 19200 dghser serial Serial COM 19200 Main thread class flpscs threading Thread
51. r for absolute intensity measurements in the soft x ray region Rev Sci Instrum 12 1 1188 91 Jan 2001 H D Stephens Electron Temperature Structures Associated With Magnetic Tear ing Modes in the Madison Symmetric Torus PhD thesis University of Wisconsin at Madison 2010 SpectronLaser User manual for sl858g nd yag laser system J A Reusch M T Borchardt D J Den Hartog A F Falkowski D J Holly R O Connell and H D Stephens Multipoint thomson scattering diagnostic for the madison symmetric torus reversed field pinch Review of Scientific Instru ments 79 10 10E733 5 pp Oct 2008 H D Stephens M T Borchardt D J Den Hartog A F Falkowski D J Holly R O Connell and J A Reusch Calibration of a thomson scattering diagnostic for fluctuation measurements Review of Scientific Instruments 79 10 10E734 5 pp Oct 2008 TS alignment section of MST plasma wiki http hackserver physics wisc edu wiki index php TS_Alignment DRAFT December 19 2010 15 16 Ii 18 19 20 2 22 64 D J Den Hartog J R Ambuel M T Borchardt J A Reusch P E Robl and Y M Yang Pulse burst operation of standard nd yag lasers to be published in Journal of Physics Conference Series 2010 W S Harris D J Den Hartog and N C Hurst Initial operation of a pulse burst laser system for high repetition rate thomson scattering Rev Sci Instrum 81 10D505 2010 J
52. read thread which will send query for status commands into the def init self status self status status threading Thread init self cSq unless in run mode DRAFT December 19 2010 def run self while 1 mainbaton acqu ire self ticl time clock for i in range 4 for ii im statuslist cs comsyn ii 0 chanlist 1 1 _csq put cs cs comsyn ii 0 chanlist 2 i 6sq put cs send out serial commands but for now just print ticl time clock _mainbaton release nop dlist is default diist clist slist clist2 slist2 eolars colors2 list of channels and colors clist is list of channels that receive responses slist is list of statuses to be parsed as faults and converted into while time clock ticl lt 8 echan 0 stats voltage 0 try time sleep s _srq get 0 1 if s find EOC 1 ss s find EOC chan s ss stat s ss 10 ss 13 ss4 12 14 chan2 int chan 2 4 chan int chan 3 6 clist append chan slist append stat clist2 append chan2 slist2 append stat elif s find SS 1 ss s find S8 5 chan s ss 9 ss 11 stat s ss 12 ss 13 chan2 int chan 2 3 chan int chan 3 5 clist append chan slist append stat clist2 append chan2 slist2 append stat elif s find GDB 1 ss s find G
53. relaxing towards that state The Madison Symmetric Torus MST is one such RFP with a major radius of 1 5m and a minor radius of 0 52m MST plasma discharges are heated ohmically with plasma current primarily being driven toroidally in the core During MST op eration the plasma can be seen relaxing towards the Taylor equilibrium The field rBo RBg configuration of an RFP such as MST can be described by a parameter q r At the radial locations where the q becomes rational or q unstable modes reside and can grow the dominant modes being m 1 modes These modes are driven by the DRAFT December 19 2010 gradient in the parallel current and form islands centered on the rational surfaces In a standard discharge the overlap of these islands result in a stochastic equilib rium field throughout the entire plasma resulting in very high energy transport The Rechester Rosenbluth model of stochastic transport describes this stochastic energy diffusion for a collsionless plasma as a double diffusion process by which particles tied to the magnetic field lines diffusing radially proportional to the radial component of stochastic field Dm 5 collide and transfer energy at a rate proportional on the electrons parallel thermal velocity mfp Xe stoch v Dm Dare Tcollision Stochastic energy transport in MST has been successfully compared this model by Biewer et al One method of improving energy confinement
54. s empty is being filled brq and then 80 function that puts command inputs into proper syntax for FLPSCS communication def comsyn cmd id 0 nn vv ww takes in command device ID device number status byte etc and voltage for PulseWidthSet only ID defaults to zero nn and vv are optional note that device ID comes after cmd since default values are annoying in python rf yyl vv 96 4d vv if nml nn 9 6 2d nn if ww ww 96 6d 96 ww a str id str cmd str nn str vv str ww fill out whitespace a at 29 len a r n return a DRAFT December 19 2010 8l Appendix C Mode Analysis We can express the magnetic field measured by the toroidal array coils as X ansin nd b cos n n 0 where 1 fa B sin n d d o 0 if B Q cos n d o 0 are S DRAFT December 19 2010 or alternatively we can let B oo 82 ps Cncos nd den C 1a n 0 Cn cos n cos den sin nd sin den n 0 oo 3 cncos Sen cos no n 0 c sin 05 sin n 2 b cos n a sin no n 0 Giving CnC08 Sen Cn8in den tan den ande cos Sen sin Sen Cn Cn and C 1b An SP arb 2 P C 2 arctan gt C 3 TL DRAFT December 19 2010 83 C 0 3 Error Fields We can add Fourier decomposed s
55. slavsky and B V Chirikov Sov Phys Usp J S Sarff S A Hokin H Ji S C Prager and C R Sovinec Fluctuation and transport reduction in a reversed field pinch by inductive poloidal current drive Phys Rev Lett 72 23 3670 1994 B E et al Chapman Generation and confinement of hot ions and electrons in a reversed field pinch plasma Plas Phys Controll Fusion 52 12048 2010 DRAFT December 19 2010 Appendix A Alignment DRAFT December 19 2010 66 Figure A 1 align create front DRAFT December 19 2010 dH mopu sjool eje do jpeloig wei w3 eld se Wweag m x m a S w y m 67 68 DEK beam_alignment create_reference vi Block Diagram CN ola a n P 25 eal os iat Application rere vior sie 85 15 id allowed by dot camera number and then dot laser number i e 4 08 09 0 L1 for 4 08 09 camera zero laser 1 meo Date of Alignment mm dd yy E j Cd fi Drawing Mode Frame CE Boolean Ten 2 quz es jns End 2 BA B image backup IMRO e centz E R 8 jis rex centy mask E Ee e mask file describing the ROI that was used as well as appended d
56. sm DRAFT December 19 2010 62 Bibliography 1 J S Sarff J C Sprott and L Turner Equilibrium studies of a poloidal divertor pinch with a reversed toroidal field Phys Fluids 30 7 2155 62 March 1987 2 S Ortolani and D Schnack Magnetohydrodynamics of plasma relaxation 1993 3 R N Dexter D W Kerst T W Lovell Prager S C and J C Sprott The madison symmetric torus Fusion Technol 19 1 131 139 1991 4 A B Rechester and M N Rosenbluth Electron heat transport in a tokamak with destroyed magnetic surfaces Physical Review Letters 40 1 38 41 Jan 1978 5 T M Biewer C B Forest J K Anderson G Fiksel B Hudson S C Prager J S Sarff J C Wright D L Brower W X Ding and S D Terry Electron heat transport measured in a stochastic magnetic field Phys Rev Lett 91 4 045004 Jul 2003 6 J S Sarff S A Hokin H Ji S C Prager and C R Sovinec Fluctuation and transport reduction in a reversed field pinch by inductive poloidal current drive Phys Rev Lett 12 23 3670 73 June 1994 DRAFT December 19 2010 10 11 12 13 14 63 B E et al Chapman Reduced edge instability and improved confinement in the mst reversed field pinch Phys Rev Lett 87 20 205001 Nov 2001 D J Den Hartog and M Cekic Meas Sci Technol 5 1115 1123 1994 N E Lanier S P Gerhardt and D J Den Hartog Low cost robust filtered spectromete
57. son Scattering diagnostic Thomson scat tering TS is the process by which charged particles are accelerated by an incident electromagnetic field and re emit radiation due to that acceleration scattering the incident wave The frequency of the incident wave is shifted by the velocity of the charged particle and the direction of scatter By comparing the spectrum of scattered photons to what one would expect with a Maxwellian distribution at a given tempera ture the temperature of the scattering particles can be gleaned In low field magnetic plasma confinement devices such as those in the Madison Symmetric Torus MST passive diagnostics such as Electron Cyclotron Emission ECE do not work as the plasma is overdense and diagnostics such as Soft X Ray SXR experience contami nation from aluminum lines Therefore Thomson scattering is an invaluable tool for accurately determining the electron temperature of the plasma in these devices In MST the incident electromagnetic wave is a monochromatic Laser beam at 1064nm that travels 15m through an enclosed beamline ultimately passing through the plasma volume see Chapter 2 and scattering off of the electrons The scattered light from 21 radial positions is collected and from this data a temperature profile of the plasma DRAFT December 19 2010 can be created This thesis will describe the implementation of various upgrades to both the beamline and laser system that have streamlined the d
58. ssed as Xe stoch UTeDm where vr is the thermal velocity and the b magnetic diffusion Dm of a stochastic field is given by Dm Lac pr And Lae is the 0 auto correlation length of the stochastic field and can be approximated as Lac Bk where Aky is the spread in the mode spectrum and kj k B As the values for the equilibrium and perturbed field must be evaluated in the stochastic region of the plasma we make use of the equilibrium generated by MSTFit as well as calculated linear eigenmodes of the m 1 n 8 15 fluctuations allowing us to locate the region of interest for our calculations and convert from the measured b at the wall to the b local to the region of interest 3 11 DRAFT December 19 2010 50 m RBg Bg r n r B IB The spread in the mode spectrum appendix E Akj zAr can be numerically evaluated using the q profile and equilibrium Be B profiles gen erated by MSTFit Ar is defined to be a fluctuation s mode radial width The interpretation of this value is expanded upon later The MSTFit equilibrium allows us to directly evaluate the field in the gradient region 0 45T From the eigenmode plots previously calculated by Sovinec it is possible to convert the toroidal fluctuation amplitude measured at the wall to that of the radial amplitude in the stochastic region We see that the b of a specific eigenmode at its rational surface within the plasma is on the order of 2 3 times
59. t the same time minimizing the stress on the flashlamps As previously mentioned the timing scheme for the Fast Spectron is controlled by two Jorway J221 A modules and the FLPSCS Master Control chassis An MST trigger to begin TS data collection is sent to one of the two Jorway mod ules This module is used to control the macro ms timing and determines the frequency of the flashlamp burst pulses Once the macro Jorway triggers the master control chassis of the power supply there is a 10us jitter for the drive system to re spond before initiating the flashlamp pulses Once the flashlamp bursts begin a sync pulse with a lus jitter relative to the flashlamps is sent to the the second fast J221 module which then with triggers the Q switch and digitizers on order us tim ing generating a laser pulse and digitizing any data from the scattered signal This method allows the lasers to be Q switched within lus of their optimal charge time and digitize at the proper time with respect to the flashlamp bursts minimizing the DRAFT December 19 2010 2r Sync and Flashlamp s Burst Fast MST Macro 10us xd a4 J221 Trigger J221 Jitter 100 200 300 400 500 Time relative to Sync pulse us Figure 2 13 Flow chart for FLP timing of a pulse burst sequence time that the flashlamps are on while ensuring optimal pumping of the rods before lasing This implementation allowed for the capability of taking
60. tationary error fields n Yn to our coil mea surements as follows B o t an t sin nd b t cos n nsin no y cos nQ C 4 which can be expressed as supressing the t terms B X an tn sin nd bn yn cos ng C 5 or B p C cos n 6 Z cos nd 6 C 6 Here we have added a time dependence to the a s and bps to distinguish them from the stationary error fields We may repeat the same procedure outlined above to convert this signal into a total mode amplitude cosine and phase B cos n p If using equation A 5 B B cos nd dz B cos nd cos deltag sin n d sin dzg B cos dg cos nd B sin dg sin n Bid S as n sin nd b Yn cos n DRAFT December 19 2010 84 As with before this results in inserting time dependence back in B V an t tn balt Yn B a2 t b2 t z2 Yn 2 a t bn t yn C 7 and arc an Mn tn Tn g t CERA C 8 We may derive this using equation A 6 as well B Cncos no 6 Zncos no C cos nQ cos 0 sin nQ sin 6 Z cos n cos d sin n sin d C cos 0c Z cos d cos n C sin dc Z sin dz sin n DRAFT December 19 2010 85 Similar to before from B cos n p this results in B cos g B sin 6g B giving B OB C cos dc Z c0s dz C sin dc Z sin 0
61. tect Deviation in Stage m 2 Correct using calibration for w 3 Predict Deviation in d 4 Correct using calibration for m Figure 2 4 An example of using the calibrations to fix mis alignment in the stage above MST 2 1 8 Alignment Implementation This section refers to the implementation of the alignment algorithm within Lab view Small Labview icons and their features will be described and a more complete back panel can be seen in the appendix A The initial alignment of the TS system is an involved process discussed else where and will only be briefly outlined here The mirrors in the Spectron beamline are replaced with dichroic mirrors that allow propagation of visible HeNe diode lasers to which the 1064 beam is aligned down the beamline through MST and into the beam dump The bundle of collection fibers are aligned to a beam probe that is in serted into MST from the beam dump port In general with the incident laser beam DRAFT December 19 2010 14 exiting through the same port that the beam probe is inserted through a rough align ment can be achieved which results in at least some scattered laser light to be within the FOV of the fibers Additional manual adjustments of the turning mirrors allows the user to maximize the number of collected photons This last step is currently unavoidably time consuming as it requires a plasma for the incident laser beam to scatter off of allowing only one or two adjustments tota
62. tically excludes using the best performing PPCD discharges from the data set as the core temperature and stored electron energy for DRAFT December 19 2010 60 those discharges steadily increase until the end of PPCD For certain PPCD discharges there also exist periods of steady and smooth Te increase throughout the core and mid radius of the plasma For discharges where the density profile does not vary much during this period it is possible to arrive at an estimate of the W term by taking a linear time derivative between data points For the particle balance as previously noted there exists a particle transport reconstruction code ported from RFX that is capable of utilizing MSTFit s Da array data to more accurately correct for the convective term in the transport calculations Still this contribution was seen to be small in the handful of data points it was previously applied to 4 2 2 Stochastic transport calculation For a more rigorous calculation of the expected stochastic transport a direct numerical simulation of select PPCD discharges via field line tracing is required By using the linear eigenmodes calculated via DEBS which solves the nonlinear MHD equations in a cylindrical geometry The magnetic field configuration can be calculated by the field line tracing routine MAL using the DEBS results and measured fields as inputs Previously this has allowed for the direct calculation of the radial excursion Ar a fi
63. tuations levels from PPCD result in much smaller island widths and a correspondingly lower level of stochasticity throughout the majority of the plasma as can be seen in the results of a field line tracing simulation using inputs from PPCD discharges 3 2 Thermal transport in the core of the plasma is much DRAFT December 19 2010 32 m amg Controlled E a ad hoc auxiliary r a Figure 3 1 Cartoon of current profile control from 2009 MST proposal higher than what is predicted by the stochastic model and is assumed to be via another mechanism possibly electrostatic but its discussion is beyond the scope of this work There is however a region radially inward from the reversal surface where the calculated stochastic transport is of the order of the measured transport It can be seen that in this region the shape of the q profile is such that even when the fluctuations are greatly reduced many small amplitude fluctuations may still overlap creating a stochastic region 3 3 The focus of this thesis will be on comparing the thermal transport in this region with the stochastic model for high current PPCD discharges Because of the reduction of fluctuations in PPCD discharges and the high n number of the modes in the stochastic region the modes of interest have very small DRAFT December 19 2010 Standard 1000 100 Ke 0 02 04 06 08 10 02 04 06 08 1 ra ra F
64. ude of the calculated mode amplitude from the database value to what is actually present as well as altering the time dependence due to its non rotating nature In low fluctuation level situations a careful analysis of the fluctuating signal is then very beneficial A mode analysis script by John Sarff performs such a treatment We can write the field measured by the coils including the error field as B Cncos no c Zncos nd We choose a time window t to tz order 1ms to perform the mode analysis and average the total field measured during this period s X cncos nd z cos n 6 dt lt Bd gt ium Because the fluctuations of interest are rotating quickly past the coils in this time window it is noted if they are found not to be rotating later in the analysis and the specific discharge being analyzed is omitted from the data set their amplitude DRAFT December 19 2010 3T averages out to zero leaving just the stationary error field amplitude measured by that coil ua cos n 6z lt Bid gt um M Z cos n i By subtracting this component from the raw measured signal we are left with just the fluctuating component of the signal B 6 t lt B 0 gt S OD cos no 6 Z cos n 6 lt B d gt 2 C t cos n 6 Z cos n 6 Z cos n B t lt B gt M C t cos n
65. w this behavior to some extent Additionally the highest performing PPCD discharges seem to strongly correlate to a pronounced double flattening of the profile suggesting the possibility of two radially separated transport barriers resulting in better core energy confinement Once the magnetic equilibrium that best fits the data is generated a local trans port analysis can be performed DRAFT December 19 2010 45 Fit to Te data using polynomial model 2500 2000 1500 1000 Figure 3 8 An electron temperature profile displaying two pronounced gradient regions centered around r 0 3m and r 0 4m resulting in a poor temperature profile fit 3 3 2 Transport The transport of thermal energy within the plasma can be broken down into conductive and convective components 5 Qe Ls Xehie VT gl ede The energy transport resulting from particle flux is generally a small contribution to the overall transport additionally particle flux measurement data from MST s Da array was unreliable from the day the 500kA PPCD discharge data set was taken To verify the smallness of the convective contribution to energy transport an RFX particle transport reconstruction code was applied to data from one time point in one specific discharge to estimate the convective transport contribution to the overall xe DRAFT December 19 2010 46 shot 1090630049 at time 16 500000ms 1000 lt 7 5 100 E TRES Sonava 9 CER i J 105 MEI
66. with the Ye stocn calculations The DRAFT December 19 2010 92 Chirikov stochasticity parameter was calculated using the equation 1 Wn41 Wn 2 Tn41 Tn F 2 For the representative data point the stochasticity parameter between islands was n and m 1 E 8 and 9 1 22 9 and 10 1 19 10 and 11 1 22 11 and 12 1 05 12 and 13 0 99 13 and 14 1 09 14 and 15 1 06 15 and 16 2 27 DRAFT December 19 2010

Stochastic Transport in PPCD Discharges

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