Vlasov Methods forSingle-Bunch Longitudinal Beam Dynamics

M. Venturini

LBNL

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

2

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)

4

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’

5

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(

21

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

28

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