Description of the friction force and degree of uncertainty with and without wigglers Work supported by US DOE, Office of NP, SBIR program, contract #

Description of the friction force and degree of uncertainty with

and without wigglers

Work supported by US DOE, Office of NP, SBIR program,

contract # DE-FG03-01ER83313

Tech-X Corporation5621 Arapahoe Ave., Suite ABoulder, Colorado 80303http://www.txcorp.com

David Bruhwiler

G. Bell, R. Busby, P. Messmer & the VORPAL teamBrookhaven National Lab:A. Fedotov, V. Litvinenko & I. Ben-ZviJINR, Dubna:A. Sidorin

RHIC Electron Cooling Collaboration Workshop

May 24-26, 2006

Simulating dynamical friction… p. 2BNL Workshop – May 24, 2006

Motivation & Background

• There’s a need for high-fidelity simulations of dynamical friction, making a minimum of physical assumptions– We are using the VORPAL code

• C. Nieter and J.R. Cary, Journal of Computational Physics (2004)

• http://www-beams.colorado.edu/vorpal/

• Goals of the simulations– Resolve differences in theory, asymptotics, parametric models– Quantify the effect of magnetic field errors

• Original numerical approach– use algorithm from astrophysical dynamics community– success w/ idealized case of magnetized cooling

• Bruhwiler et al., AIP Conf. Proc. 773 (Bensheim, 2004)

• Fedotov et al., AIP Conf. Proc. 821 (2005); PRST-AB (2006)

• Recent efforts– Effect of wiggler fields for “unmagnetized” approach


Outline

• Description of modified numerical approach– Required for wiggler simulations– Semi-analytic binary collision model w/ operator

splitting• G. Bell et al., AIP Conf. Proc. 821 (2005)

• Statistical uncertainty due to diffusive dynamics– use of averaging, central limit theorem, numerical

tricks

• Question of maximum impact parameter– finite time effects– incomplete and asymmetric collisions

• Reduction of friction force by wiggler dynamics


New simulation approach: Operator Splitting

• Numerical technique used for ODE’s & PDE’s• Consider Lorentz force equations

• Robust 2nd-order ‘Boris’ uses operator splitting– J. Boris, Proc. Conf. Num. Sim. Plasmas, (1970), p. 3.

• Add external E, B fields via operator splitting– Hermite algorithm: drift + coulomb fields– Boris ‘kick’: all external E, B fields

• Benchmark w/ pure Hermite alg. for constant B||

vx .

2 ;2 ; ;2 ;2 ttttt xvvvx EBE

extextCoulomb m

q

m

qBvEE

.v


Operator splitting is implemented for ‘reduced’ model

• Use analytical two-body theory for ion/e- pairs– handle each pair separately in center-of-mass frame– calculate initial orbit parameters in relevant plane– advance dynamics for a fixed time step

• electron’s new position and velocity are known• changes to ion position/velocity are small perturbations

– total ion shift is sum of individual changes

• Major improvement in simulation capability– Benchmarked with “Hermite algorithm”– Parallelizes well to 64 processors & beyond– Roughly an order of magnitude faster– Enables treatment of arbitrary external fields– e-/e- interactions can be included via electrostatic PIC


0

0

0

0

x

xR

xRT

),,(0 zyxx

z

x

y

x

y

z

00 vx

)0,,( yx

Semi-analytic ‘Reduced’ Model for Binary Collisions

• Must find the plane in which partial orbit occurs– necessary rotations (yaw, pitch, roll) are complete– transformations are messy, but straightforward– “initial” positions & velocities obtained in this plane

• Then standard orbital parameters are calculated

Simulating dynamical friction… p. 7BNL Workshop – May 24, 2006Hermite Algorithm Binary Collision Model

Hermite & Binary Model agree well for constant B


Diffusive dynamics can obscure friction

• For a single pass through the cooler– Diffusive velocity kicks are larger than velocity drag– Consistent with theory

• For sufficiently warm electrons ( “linear plasma” regime)– numerical trick of e-/e+ pairs can suppress diffusion

• Only remaining tactic is to generate 100’s of trajectories– Central Limit Theorem states that mean velocity drag is drawn

from a Gaussian distribution, with rms reduced by Ntraj1/2 as

compared to the rms spread of the original distribution

– Hence, error bars are +/- 1 rms / Ntraj1/2

• How to generate 100’s of trajectories for each run?– in past, we used “task farming” approach to automate many runs– now, we use “micro-particles” and parallel processing– run 8 trajectories simultaneously


Noise reduction 1: “micro-particle” electrons1 macroparticle per electron 266 macroparticles per electron


Noise Reduction 2: using correlated positrons266 macroparticles per electron

133 macroparticles per electron133 macroparticles per positron


Present simulation parameters – questions regarding max, finite interaction time, asymmetric collisions

• e- Beam parameters– density: 9.50E13 e-/m3

– RMS e- velocities [x,y,z]: 2.8x105, 2.8x105, 9.0x104 m/s–

• Wiggler parameters– Length: 80 m– Wavelength: 8 cm– Sections: 10– Field on axis: – Interaction time: 2.47 ns (beam frame)

• Problem setup in VORPAL– Domain size: 0.8 mm x 0.8 mm x 0.8 mm– Gold ions per domain: 8– Electrons per domain (actual): 4.86x104

– Periodic domain

108

TGBw 001. 10


Finite interaction time modifies Coulomb log

• The unmagnetized longitudinal friction can be written

– above form has the usual problem with logarithmic divergence

– assumes sufficient time for large max collisions to complete

• Explicit treatment of finite interaction time leads to modified Coulomb logarithm:

• Assuming the log can be pulled outside of the integral– and that– we can use the following Coulomb log:

• v is characteristic relative ion/e- velocity

2

max

2

min

min

max

21

21

ln

v

v 2/12222

|||| sincos vvVvVvVv

vmin

2

max

min

max 21ln

v


What is max, really?

• One should not be worrying about detailed calculations of max & min

– the Coulomb log should be “large”

– uncertainties in calculation are ~1/ln(max/min)

• Nevertheless, max is a concern– how large must the simulation domain be?

– run-time scales as Lx3

• very painful, but large Lx possible with supercomputers

• What happens in simulations?– maximum possible is ~0.5 Lx

– we choose Lx~Vion* and then check with 2*Lx

• typically, differences are within the noise

• if noise is sufficiently low, one sees weak growth of friction


For pe<<2, max is ~rbeam

• However, one must include finite effects:

• Strictly speaking, one requires Lx ~ 2*rbeam

– for RHIC parameters, there is no time for “shielding”– finite time effects yield a modified Coulomb log that is

essentially equal to the original case

• Choosing Lx ~ 2*rbeam would be a waste of resources

– in practice, results change very little for Vion*Lxrbeam

– inherent uncertainty of Coulomb log formulation is >10%

– uncertainties due to size of Lx are of the same order

• Note: correct treatment of detailed wiggler fields and misalignments will also require Lx ~ 2*rbeam – see talk by George Bell tomorrow

2

max

min

max 21ln

v

min

2ln

v vmaxfor


Wiggler approach to RHIC cooler

• Advantages of a wiggler– provides focussing for e- beam– suppresses recombination

• Modest fields (~10 Gauss) effectively reduce recombination via ‘wiggle’ motion of electrons:

• Negative effects of ‘wiggle’ motion on cooling?– intuition suggests an increase of the minimum impact

parameter in the Coulomb logarithm:

– this has now been confirmed via VORPAL simulations

][][104.1~ 232

GBmxvk ww

beamw

gyrow

w min


Friction force reduction for 10 G wiggler

• VORPAL simulations confirm the expected reduction in dynamical friction due to wiggler

Longitudinal Friction

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

80.0

0.00 0.26 0.59 0.84 1.23 1.57

angle

Fri

ctio

n i

n e

V/m

Fz no undulator

Fz w ith 10G undulator

No undulator/1.2

Transverse Friction

0.0

10.0

20.0

30.0

40.0

50.0

60.0

70.0

0.00 0.26 0.59 0.84 1.23 1.57

angle

Fri

ctio

n in

eV

/m

Fx no undulator

Fx w ith 10G undulator

No undulator/1.2


Conclusions

• Reliable first-principles simulation of dynamical friction is now routine with VORPAL– unmagnetized, wiggler, solenoids, errors– requires (can take advantage of) parallel computers

• scales well up to 64 processors and probably higher

– typically ignores e-/e- interactions• these are included via electrostatic PIC, when necessary

• VORPAL simulations confirmed that the effect of a wiggler is to increase min


Acknowledgements

We thank A. Burov, P. Schoessow, P. Stoltz, V. Yakimenko & the Accelerator Physics Group of the RHIC Electron Cooling Project for many helpful discussions. Discussions with A.K. Jain regarding solenoid field errors were especially helpful.

Work at Tech-X Corp. was supported by the U.S. Dept. of Energy under grants DE-FG03-01ER83313 and DE-FG02-04ER84094. We used resources of NERSC.

Documents

Description of the friction force and degree of uncertainty with and without wigglers Work supported by US DOE, Office of NP, SBIR program, contract #