vlasov methods for single-bunch longitudinal beam dynamics m. venturini lbnl ilc-dr workshop,...

Vlasov Methods forSingle-Bunch Longitudinal Beam Dynamics

M. Venturini

LBNL

ILC-DR Workshop, Ithaca, Sept-26-06

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Outline

• Direct methods for the numerical solution of the (nonlinear) Vlasov equation

• Instability thresholds from linearized Vlasov equation– Critique of Oide-Yukoya’s discretization method

Illustration of critique in case of coasting beams Bunched beams. Two case studies (SLC-DR, NLC-MDR)

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Anatoly Vlasov(1908-1975)

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Anatoly Vlasov(1908-1975)

Reminder of form of Vlasov equation

E

z

Ep

zq

/

,/

0)],,([

p

ffqFq

q

fp

fc

RF focusing Collective Force Damping

Fokker-Planck extension (radiation effects)

Fokker-Planck extension (radiation effects)

nn

Rinqcc nZeIdqzzzwI z ˆ)()'()'( /

0

Quantum Excitations

p

fpf

ptp

ffqFq

q

fp

f

dsc

2)],,([

Vlasov equation expresses beam density conservation along particle orbits

w(q - q’)(q’)dq’

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Direct Vlasov methodsvs. macroparticle simulations

• Pros:– Avoids random fluctuations caused by finite number of macroparticles– Can resolve fine structures in low density regions of phase space– “Cleaner” detection of instability

• Cons:– Computationally more intensive– Density representation on a grid introduces spurious smoothing.

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Numerical method to solve Vlasov Eq.

p

q

Beam density

at present time t defined on grid f =fij

Beam density

at present time t defined on grid f =fij

At later time t + twe want value of

density on this grid point

At later time t + twe want value of

density on this grid point

find imageaccording to backward

mapping

find imageaccording to backward

mapping

In general backward imagedoes not fall on grid point:

Interpolation neededto determine f

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Example of a simple drift

Beam densityat later time

Beam densityat present time

Mapping for a drift, M->: p’ = p, q’ = q + p

f(q’,p’,) = f(q,p,)

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f

qi2i 1i

),( qf

),( qf

Value of f is determined byinterpolation using e.g. valuesof f on adjacent grid points

p

Beam densityAt later time

Beam densityAt present time

Example of a simple drift (cont’d)

f(q’,p’,) = f(q’-p’,p’,)

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Detect instability by looking at evolution of moments of distribution

• Start from equilibrium (Haissinski solution)• Instability develops from small mismatch of computed Haiss. solution

• SLC DR wake potential model (K. Bane)• N= 1.86 1010

• Growth rate of instability: 11.1 synch. prds

2nd moment of energy spread 3rd moment of energy spread

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Consistent with macroparticle simulationsfor Broad-Band resonator model

• Contributing to the effort of benchmarking existing tools for single-bunch longitudinal dynamics

• Comparison against macroparticle simulations (Heifets)

Current Threshold

Vlasov calculationVlasov calculation

Macroparticlesimulation

Macroparticlesimulation

No

rmal

ized

cu

rren

t

Macroparticle simulation includes radiation effects

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- 1.5 0 1.5q

- 1.5

0

1.5

p

q=1.2

- 3 - 2 - 1 0 1 2 3 4q

0.1

0.2

0.3

0.4

egrahcytisned

q=1.2

- 1.5 0 1.5q

- 1.5

0

1.5

p

q=3.2

- 3 - 2 - 1 0 1 2 3 4q

0.1

0.2

0.3

0.4 q=3.2

- 1.5 0 1.5q

- 1.5

0

1.5

p

q=9.6

- 3 - 2 - 1 0 1 2 3 4q

0.1

0.2

0.3

0.4 q=9.6

Charge Density

2 cm

prdsynch.2.0 prdsynch.5.0 prdsynch.5.1

Direct methods allow for fine resolutionin phase space

tail

z/z

E/

E

head

Microbunching from CSR-driven instability

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Tackling the linear problem

• Techniques to solve analytically the linearized Vlasov equation for coasting beams have been known since Landau (O’Neil, Sessler)

• Numerical methods must be applied, i.e., – truncated mode-expansion (Sacherer).– Oide-Yukoya discretization (represent action on grid)

• Theory for coasting beams can be stretched to cover bunches in some (important) cases (Boussard criterion) …

• … but in general no analytical solutions are known for bunched beams

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It boils down to solving an integral equation…

• Assume time dependence ~ expior think Laplace transform)

• Express linearized Vlasov Eq. using action-angle variables (relative to

motion at equilibrium). Do FT with respect to angle variable

0)]([)()( ' JfJfJmΩ mm L

Synchrotron tune includingincoherent tuneshift

Integral operatorAzimuthal mode no.

Mode frequency(unknown)

Mode amplitude(unknown)

iimm eeJff )(

iimm eeJff )(

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The integral equation is `pathological’:

• Convergence of finite-dimension approximation is not guaranteed for singular integral equations

• For general convergence the operator M is approximating should be “compact” (Warnock)

• Convergence of finite-dimension approximation is not guaranteed for singular integral equations

• For general convergence the operator M is approximating should be “compact” (Warnock)

0)]([)()( ' JfJfJmΩ mm L

ff

M matrix e-value problem

Term can vanish making the equation ‘singular’

(Integral equation of the ‘third kind’)

Discretize

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Nature of problem is best illustrated in case of coasting beams

• Linearized V. equation can be solved analytically (e.g. gauss beam in energy spread)

0)()(2

)2/exp( 2

dppfiIpfp pp

0)()(

2

)2/exp( 2

dppfiIpfp pp

Current parameter I includes Z/nmomentum compaction, etc; can be a complex no.

• Low current: spectrum of eigenvalues is continuous = real axis.– Corresponding “eigenfunctions” (Van Kampen modes)

are not actual functions but Dirac-like distributions

• High current: Isolated complex eigenvalues emerge with Im >0

• Using finite-dimensional approximations = trying to approximate a delta-function. Numerically, it may not be a good thing!

• Using finite-dimensional approximations = trying to approximate a delta-function. Numerically, it may not be a good thing!

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Two ways of solving the linear equation for coasting beams

1. Analytical solution (Landau’s prescription) -- this is also the computationally `safe’ way:

1. Divide both terms of Eq. by - p. 2. Integrate. Remove p-integral of f(p) from both terms. 3. Integral expression valid for Im >0; extend to entire plane by analytic

continuation

2. Oide-Yokoya style discretization:1. Represent f(p) on a grid. 2. Solve the eigenvalue problem of finite-dim approximation.

0)()(2

)2/exp( 2

dppfiIpfp pp

0)()(

2

)2/exp( 2

dppfiIpfp pp

dpp

ppiI

)2/exp(

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2

• In both cases: look for Im > 0 as signature for instability

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Coasting beam: Oide-Yukoya discretization indicates instability when there is none

• Choose I = real number; theory threshold for instability is I = 1.43

Eigenvalue spectrum

below (theory) threshold

• Theory says all eigenvalues should be on real axis…

• … yet most calculated e-values have a significant Im >0

Eigenvalue spectrum

above (theory) threshold

only this eigenvalue

corresponds to a reallyunstable mode

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How do we cure the singularity ?

• Regularize integral equation by simple replacement of the unknown

function: )()()( JfJmJg mm

0)('

)()( '

Jm

JgJg m

m L

0)(L1det)( D

is compact; discretization is OK

Regularized equation

Equation to solve is more complicated than simple

eigenvalue problem

‘Old’ unknown‘New’ unknown

L

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A way to determine if there are unstable modes without actually computing the zeros of determinant D()

• Use properties of analytic functions to determine

no. of zero’s of D()=0 (Stupakov)

roots ofdeterminant D()

contour of integrationon complex plane

no. of roots of D()

d

D

Dn

)(

)('

2

1

Contributionfrom arc vanishes

No. of windings of D(u) = around 0 as u (on real axis) goes from – to + infinity i

with Im > 0

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Fix the current. For instability, look for no. of roots of D()=0 with Im >0

• Use properties of analytic functions to determine

no. of zero’s of D()=0 (Stupakov)

roots ofdeterminant D()

contour of integrationon complex plane

no. of roots with Im > 0

d

D

Dn

)(

)('

2

1

Contributionfrom arc vanishes

Change of phaseof D(u) as u (on real axis)goes from – to + infinity

Winding of D() on complexplane as varies along the real axis

i

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Case study 1: wake potential model for SLC DR

• Numerical calculation of wake potential by K. Bane

• This is a ‘good’ wake– Oide-Yukoya style analysis

seems to work well.– Detection of current

threshold consistent with numerical solution of Vlasov equation

– Consistent with modified linear analysis

Wake Potential Wake Potential

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Oide-Yukoya analysis consistent withVlasov calculations in time domain

Spectrum of unstable modes

ThresholdThresholdNumerical solution ofVlasov Eq. in time domain

Linear

theory

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Unstable mode right above threshold has a dominant quadrupole (m=2) component

Unstable mode for SLC DR: Density plot in action-angle coordinates

Longitudinal coordinate

En

erg

y d

evia

tion

Ic =0.048 pC/V

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Improved method is in good agreement with Oide-Yokoya, simulations

One root of D() foundwith Im > 0

Plot of phase of D() in complex plane for a fixed current …

Extract growth rate by fitting,Find excellent agreement with theory(within fraction of 1 %)

… compare to time-domain calculation done with Vlasov solver

Use location of phase jumpto initiate a Newton search:

Find: = 1.86 + 0.0023*i

En

ergy

sp

read

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Case study 2: wake potential model for NLC MDR (1996)

• Numerical calculation of wake potential by K. Bane

• Oide-Yukoya style analysis not completely consistent with numerical solution of Vlasov equation

Wake Potential Wake Potential

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Spectrum looks scattered

Im Im

Re Re

Spectrum of unstable modes

Are the scatteredeigenvalues physical?

Are the scatteredeigenvalues physical?

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O-Y detects some spurious unstable modes

Im Im Spectrum of unstable modes

Time domain simulations showno instability

Simul

atio

n

s

Line

ar

theo

ry

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Modified linear analysis correctly detects absence of unstable mode

Im Im

Re Re

One unstable mode detected when using improved method

One unstable mode detected when using improved method

Current-scan: e-values with Im >0 using O-Y discretization

No unstable mode detected when using improved method

G

B

No unstable mode detected when using improved method

G

B

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Convergence of results against mesh refinementmay help rule out spurious modes in O-Y

• Black points -> 80 mesh pts in action J

• Color points-> 136 mesh pts in action J

Convergence is reached here

Convergence is reached here

No convergence reached here

No convergence reached here

Blow-up

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Conclusions

• We have the numerical tools in place to study the longitudinal beam dynamics

• Study of the linearized Vlasov equation using discretization in action-angle space should be done with care.

– Possible ambiguity in detection of instability. – Certain cases may not be treatable by current methods (e.g. transformation to

action angle should be defined) – How generic are the results for the 2 shown examples of wake potential?– Agreement with simulations for BB wake model not very good (work in progress).

• For DR R&D, emphasis should be placed on good numerical model for impedance, wake-potential.

• Benchmark against measurements on existing machines.

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2D Density function defined on cartesian grid

• Propagation along coordinate lines done by symplectic integrator

p

q

KickKick

DriftDrift

DriftDrift

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Coasting-beam model offers a good approx. to onset of instability, microbunching

Particles with this energy deviationmove in phase with traveling waveof unstable mode and aretrapped in resonance

Particle density in phase space

z/z

pE

/E

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Solution of VFP equation shows bursts and saw-tooth pattern for bunch length

Saw-tooth in rms bunch length

CSR signal from solution of VFP Eq.

Instability jump starts

burst

Non linearitiescause saturation,

turn-off burst

Radiation dampingrelax beam back

closer to equilibrium

Bursting cycle

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Bunch Length (rms)

Radiation Power (single burst)

NSLS VUV Storage Ring

Radiation Spectrum

Charge Density

z / z

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Current methods to solve linearized Vlasov Eq. are not generally satisfactory

• “State of the art” method is by Oide-Yukoya.– includes effects of “potential well distortion” i.e. effect of collective effect

on incoherent tuneshift of synchrotron oscillations

• There is evidence that O-Y method sometimes fails to give the correct estimate of current threshold for instabilities (vs. particle simulations).

• Also, problems of convergence against mesh-size, etc.

Example of longitudinalbunch density equilibriumwith potential well-distortion