2. A. Z. Ghalam, T. Katsouleas (USC) C. Huang, V. Decyk, W.Mori(UCLA) G. Rumolo and F.Zimmermann(CERN) U C L A 3-D Parallel Simulation Model of Continuous Beam-Cloud Interactions.

3. Outline Introduction –Existing models for beam-cloud interaction (Single kick approximation) What is QuickPIC and why use it for e-cloud problem? –Challenges of using a continuous model for beam-cloud interaction Results: –Benchmark Results: Beam dynamics in LHC and CERN-SPS QuickPIC and HEAD-TAIL (Single kick code) comparison –Physics Results: Role of cloud image charges Dipole magnet effects on beam stability

4. Only valid when cloud wakefield is linear. (i.e., W y =  y) Existing Codes for Beam-Electron Cloud interaction modeling Cloud Slab Single Kick Approximation: Assumes cloud is concentrated at one thin slab in the ring. Reduces computational costs. Beam coordinates are updated after kick back to kick point by transformation matrix: Beam receives a kick from The cloud at the interaction point Electron Cloud free region

5. QuickPIC Model: QuickPIC is a 3-D parallel PIC code originally developed for plasma wakefield accelerator research Is extensively benchmarked for beam plasma interactions Capable of running over long range of beam propagation owing to its parallel processing capability. Why use PIC-Plasma Codes for Electron Cloud Modeling? Motivation: Want continuously model the beam-cloud interaction. Electron cloud is a (non-neutral) plasma. Explicit PIC-Plasma has highest fidelity (lowest level approximation)

6. Maxwell’s equations for field solver Lorentz force updates particle’s position and momentum Interpolate to particles Particle positions Lorentz Force push particles weight to grid tt Computational cycle (at each step in time) What Is a Fully Explicit Particle-in-cell Code?

7. Description of the Simulation Model (QuickPIC) QuickPIC is a 3D PIC (particle In Cell) code with parallel processing capability It Uses Quasi- Static or frozen field approximation (  >>  z ) 3D Maxwell equations are reduced to 2D equations The need for solving 2D equations results in larger time steps of the 3D push,enabling a time savings of 2-3 orders of magnitude compared to traditional PIC models

8. Description of the Simulation Model,(Continued) QuickPIC Cycle: Explicit 3-D PIC with 3 time saving innovations: 1.Moving window follows the beam. 2.Separate time steps for beam and cloud dynamics 3. Use local nature of wake equations to form 3-D wakefields from a sequence of 2-D Poisson solves.

9. Changes and modifications to QuickPIC Adding Synchrotron and Betatron motions By adding external focusing forces in transverse and longitudinal directions Adding Chromaticity effect By adding external magnetic field and modifying our 3-D magnetic pusher. Transverse Equations of Motion Longitudinal Equations of Motion Q x : Horizontal tune Q y : Vertical tune ∆Q x : Chromatic shift in x ∆Q y : Chromatic shift in y  p/p 0 : momentum spread  : relativisitc factor  : slippage factor F cl : Total force by cloud

10. Computational Challenges of applying a plasma PIC code to circular accelerator modeling to circular accelerator modeling Length of beam-plasma interaction to be modeled  e- cloud is a 2000 turns * 27Km = 54000Km long plasma!  10 7 steps, 100m long * 10 6 particles = 10 13 particle-steps! Solution: Parallel computing First order leap frog pusher introduces small numerical frequency shift but tune desired to 4 decimal places! Solution : Advanced Pusher

11. z y x Node 3Node 2Node 1Node 0 x y Node 3 Node 2 Node 1 Node 0 beam plasma Parallelization QuickPIC uses 2 different domain decompositions: Beam decomposition along Z Cloud decomposition along Y MPI is used for communication between nodes

12. Overcoming Numerical Tune Shift Overcoming Numerical Tune Shift Solving harmonic Oscillator with Leap Frog pusher results in a tune shift (Quadratic in time step): Modified Pusher:

13. Benchmarking the Code Against HEAD-TAIL for LHC ring Red : QuickPIC Blue : HEAD-TAIL Horizontal Spot Size of the beam predicted by the two codes QuickPIC simulations are performed On USC’s Linux cluster on 32 processors for 28 days Beam Density in Horizontal Plane Y Z

14. Benchmarking: Single Kick QuickPIC vs. HEAD-TAIL Red : QuickPIC(Single Kick Mode) Blue : HEAD-TAIL For accurate benchmarking, QuickPIC is modified to be in the single kick regime Good agreement between the two code. LHC parameters have been used for benchmarking purpose.

15. Growth rate changes with number of kicks Growth rate changes with the number of kicks! QuickPIC Results for LHC HEAD-TAIL results for LHC Green : 4 Kicks/Turn Blue : 2 Kicks/Turn Red : 1Kick/Turn Aqua : 16Kicks/Turn

16. Mode Coupling-Evidance for Head-Tail Instability n cl =3e5/cm 3 n cl =6e5/cm 3 n cl =1e6/cm 3 CERN-SPS parameters for simulation Mode coupling is observed as the cloud density grows:

17. –Physics Results: Role of cloud image charges Dipole magnet effects on beam stability

18. Unperturbed Cloud Perturbed Cloud Cloud Effects on Tune Shift (CERN-SPS) Unperturbed Cloud Perturbed Cloud Measured Perturbed cloud Y Unperturbed cloud Beam Z Tune Shift due to a uniform cloud distribution: Actual tune shift is closer to the uniform cloud case. Reason lies on the role of cloud image forces

19. Interaction Between E-Cloud Forces on Beam Oppositely directed image and cloud forces create an equilibrium in which the beam is tilted. Tilt angle can be found by equating the two forces at the tail:

20. Study of the Long Term Beam Dynamics (a)(b)(c) -0.015 y(c/  p ) 0.015 -0.75 z(c/  p ) 0.75 D 0.2 Z(c/  p ) -0.2 -0.004 Y(c/  p ) 0.004 (a) (b) (c) These figures show that no matter how the beam is initially perturbed, the beam ends up having a similar long term dynamics 3 snapshots of beam over CERN-SPS ring. a) At t 0 =0.4ms (18 turns). b) At t 1 =t 0 +T B /4 c) At t2=t 0 + T B /2, where T B is the nominal betatron period (0.9  s),The beam is initially off centered 1mm from the axis of the pipe 3 snap shots of beam evolution over CERN-SPS a) At t=0 b) At t=136  s (6 turns) c) At t=0.8ms (35 turns). The beam is initially tilted Initially Off-Centered Beam Initially Tilted Beam

21. Effects of Dipole Magnets on Beam-cloud Interactions CERN_SPS Ring Specifications: 750 bending of length 6.26m. 70 percent bending sections. Straight sections 9m. Dipole Strength 0.117T. Modeling bending sections/Magnets on QuickPIC: Effect of magnetic field on cloud dynamics is significant Resolving the spatial profile of the magnets increases the run time by a factor of ten. Assume average B on the whole ring Cloud Density in Horizontal Plane B = 0B = 0.117T Shallow Cloud Compression Severe Cloud Compression 3D cloud density with magnetic field Vertical Plane Horizontal Plane

22. Dipole Magnets Reduce emittance growth and amplitude of centroid motion CentroidSpot Size Beam is initially displaced in vertical direction Less growth in both amplitude of centroid oscillation and spot size with dipole field

23. 2 Recent Results (Work in Collaboration with M. Pivi (SLAC) NLC damping ring. TESLA damping ring

24. Simulation box size is chosen as 10 times the transverse beam size with periodic boundaries. Instability threshold is at cloud density of 2e6/cm 3. No dipole magnet effects are considered in the simulation. Similar results have been obtained by HEAD-TAIL code (M. Pivi) Red : n cl = 1e6/cm 3 Blue : n cl = 2e6/cm 3 Green : n cl = 5e6/cm 3 Single Kick Modeling of the NLC Damping Ring

25. Single Kick modeling of TESLA Damping ring(2000 turns) n cl =1e4/cm 3 Vertical Spot Size(a.u.) No. of Macro-Particles Vertical Centroid(a.u.) n cl =1e4/cm 3 TESLA is modeled as 17Km ring with dipole magnet all around the ring. Simulation box size is chosen as 10 times that of the transverse beam size. Severe vertical spot size growh Severe particle loss. No severe growth in coherent centroid oscillation.

26. Summary Parallel QuickPIC enables continuous model of beam-cloud interaction over a long range. Single kick approximation overestimates the growth E-Clouds are a 54000 km plasma playgrounds! -- Rich physics in beam-cloud interaction

27. Future Research Couple the code to cloud formation codes for more realistic model  We assumed uniform cloud density(in saturation)  Electron density is non-uniform close to magnets Add elliptical boundaries to QuickPIC Study coupled bunch instability Courtesy: L. Wang Non-Uniform cloud density close to magnets