Vlasov Methods for SingleBunch Longitudinal Beam Dynamics M

Vlasov Methods for Single-Bunch Longitudinal Beam Dynamics M. Venturini LBNL 1 ILC-DR Workshop, Ithaca, Sept-26 -06

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 ü Illustration of critique in case of coasting beams ü Bunched beams. Two case studies (SLC-DR, NLC-MDR) 2

Anatoly Vlasov (1908 -1975) 3

Reminder of form of Vlasov equation expresses beam density conservation along particle orbits Anatoly Vlasov (1908 -1975) Fokker-Planck extension (radiation effects) RF focusing Collective Force w(q - q’)r(q’)dq’ Damping Quantum Excitations 4

Direct Vlasov methods vs. 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. 5

Numerical method to solve Vlasov Eq. Beam density at present time t defined on grid f =fij At later time t + Dt we want value of density on this grid point find image according In general backward image to backward does not fall on grid point: mapping Interpolation needed to determine f 6

Example of a simple drift Mapping for a drift, Mt->t. Dt : p’ = p, q’ = q + p. Dt f(q’, p’, t+Dt) = f(q, p, t) Beam density at later time Beam density at present time 7

Example of a simple drift (cont’d) f(q’, p’, t+Dt) = f(q’-p’Dt, p’, t) Beam density At later time Beam density At present time Value of f is determined by interpolation using e. g. values of f on adjacent grid points 8

Detect instability by looking at evolution of moments of distribution • Start from equilibrium (Haissinski solution) • Instability develops from small mismatch of computed Haiss. solution 2 nd moment of energy spread • SLC DR wake potential model (K. Bane) • N= 1. 86 1010 • Growth rate of instability: 11. 1 synch. prds 3 rd moment of energy spread 9

Consistent with macroparticle simulations for Broad-Band resonator model • Contributing to the effort of benchmarking existing tools for single-bunch longitudinal dynamics • Comparison against macroparticle simulations (Heifets) Normalized current Current Threshold Macroparticle simulation Vlasov calculation 10 Macroparticle simulation includes radiation effects

Direct methods allow for fine resolution in phase space DE/E Microbunching from CSR-driven instability 2 cm z/sz Charge Density head tail 11

Tackling the linear problem • Techniques to solve analytically the linearized Vlasov equation for coasting beams have been known since Landau (O’Neil, Sessler) • 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 • Numerical methods must be applied, i. e. , – truncated mode-expansion (Sacherer). – Oide-Yukoya discretization (represent action on grid) 12

It boils down to solving an integral equation… • Assume time dependence ~ exp(iwt) (or think Laplace transform) • Express linearized Vlasov Eq. using action-angle variables (relative to motion at equilibrium). Do FT with respect to angle variable Synchrotron tune including incoherent tuneshift Integral operator Azimuthal mode no. Mode frequency (unknown) Mode amplitude (unknown) 13

The integral equation is `pathological’: Term can vanish making the equation ‘singular’ (Integral equation of the ‘third kind’) Discretize matrix e-value problem • Convergence of finite-dimension approximation is not guaranteed for singular integral equations • For general convergence the operator M is approximating should be “compact” (Warnock) 14

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) Current parameter I includes Z/n momentum compaction, etc; can be a complex no. • Low current: spectrum of eigenvalues w 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 w>0 • Using finite-dimensional approximations = trying to approximate a delta-function. Numerically, it may not be a good thing! 15

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 w - p. 2. Integrate. Remove p-integral of f(p) from both terms. 3. Integral expression valid for Im w >0; extend to entire w-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. • In both cases: look for Im w> 0 as signature for instability 16

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 w >0 Eigenvalue spectrum above (theory) threshold only this eigenvalue corresponds to a really unstable mode 17

How do we cure the singularity ? • Regularize integral equation by simple replacement of the unknown function: ‘New’ unknown ‘Old’ unknown Regularized equation is compact; discretization is OK Equation to solve is more complicated than simple eigenvalue problem 18

A way to determine if there are unstable modes without actually computing the zeros of determinant D(w) • Use properties of analytic functions to determine no. of zero’s of D(w)=0 (Stupakov) contour of integration on complex plane Contribution from arc vanishes i roots of determinant D(w) No. of windings of D(u) = around 0 as u (on real axis) goes from – to + infinity no. of roots of D(w) with Im w > 0 19

Fix the current. For instability, look for no. of roots of D(w)=0 with Im w >0 • Winding of D(w) on complex Use properties of analytic functions toas determine plane w varies along the real axis no. of zero’s of D(w)=0 (Stupakov) contour of integration on complex plane Contribution from arc vanishes i no. of roots with Im w > 0 roots of determinant D(w) Change of phase of D(u) as u (on real axis) goes from – to + infinity 20

Case study 1: wake potential model for SLC DR Wake Potential • 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 21

Oide-Yukoya analysis consistent with Vlasov calculations in time domain Spectrum of unstable modes o t he r a Line Threshold ry Numerical solution of Vlasov Eq. in time domain 22

Unstable mode right above threshold has a dominant quadrupole (m=2) component Unstable mode for SLC DR: Energy deviation Density plot in action-angle coordinates Longitudinal coordinate Ic =0. 048 p. C/V 23

Improved method is in good agreement with Oide-Yokoya, simulations Plot of phase of D(w) in complex plane for a fixed current … One root of D(w) found with Im w > 0 Use location of phase jump to initiate a Newton search: Find: w = 1. 86 + 0. 0023*i Energy spread … compare to time-domain calculation done with Vlasov solver Extract growth rate by fitting, Find excellent agreement with theory (within fraction of 1 %) 24

Case study 2: wake potential model for NLC MDR (1996) Wake Potential • Numerical calculation of wake potential by K. Bane • Oide-Yukoya style analysis not completely consistent with numerical solution of Vlasov equation 25

Spectrum looks scattered Spectrum of unstable modes Im w Re w Are the scattered eigenvalues physical? 26

Si m ul at io th ar ne Li Spectrum of unstable modes Im w ns eo ry O-Y detects some spurious unstable modes Time domain simulations show no instability 27

Modified linear analysis correctly detects absence of unstable mode No unstable mode detected Current-scan: e-values with Im w>0 using O-Y discretization when using improved method Im w G One unstable mode detected when using improved method B Re w 28

Convergence of results against mesh refinement may help rule out spurious modes in O-Y Convergence is reached here • Black points -> 80 mesh pts in action J • Color points-> 136 mesh pts in action J Blow-up No convergence reached here 29

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. 30

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2 D Density function defined on cartesian grid • Propagation along coordinate lines done by symplectic integrator Drift Kick Drift 32

Coasting-beam model offers a good approx. to onset of instability, microbunching Particle density in phase space DE/E Particles with this energy deviation move in phase with traveling wave of unstable mode and are trapped in resonance p z/sz 33

Solution of VFP equation shows bursts and saw-tooth pattern for bunch length CSR signal from solution of VFP Eq. Bursting cycle Instability jump starts burst Saw-tooth in rms bunch length Non linearities cause saturation, turn-off burst Radiation damping relax beam back closer to equilibrium 34

NSLS VUV Storage Ring Charge Density Bunch Length (rms) z / sz Radiation Spectrum Radiation Power (single burst) 35

Current methods to solve linearized Vlasov Eq. are not generally satisfactory Example of longitudinal bunch density equilibrium with potential well-distortion • “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. 36