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