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

2. 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)

3. 3 Anatoly Vlasov (1908-1975)

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

5. 5 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.

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

7. 7 Example of a simple drift Beam density at later time Beam density at present time Mapping for a drift, M  ->  : p’ = p, q’ = q + p  f(q’,p’,  ) = f(q,p,  )

8. 8 Value of f is determined by interpolation using e.g. values of f on adjacent grid points Beam density At later time Beam density At present time Example of a simple drift (cont’d) f(q’,p’,  ) = f(q’-p’ ,p’,  )

9. 9 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 10 10 Growth rate of instability: 11.1 synch. prds 2 nd moment of energy spread 3 rd moment of energy spread

10. 10 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) Current Threshold Vlasov calculation Macroparticle simulation Macroparticle simulation Normalized current Macroparticle simulation includes radiation effects

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

12. 12 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

13. 13 It boils down to solving an integral equation… Assume time dependence ~ exp  i   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)

14. 14 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) matrix e-value problem Term can vanish making the equation ‘singular’ (Integral equation of the ‘third kind’) Discretize

15. 15 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  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!

16. 16 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. In both cases: look for Im  > 0 as signature for instability

17. 17 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 really unstable mode

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

19. 19 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 of determinant D(  ) contour of integration on complex plane no. of roots of D(  ) Contribution from arc vanishes No. of windings of D(u) = around 0 as u (on real axis) goes from – to + infinity i with Im  > 0

20. 20 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 of determinant D(  ) contour of integration on complex plane no. of roots with Im  > 0 Contribution from arc vanishes Change of phase of D(u) as u (on real axis) goes from – to + infinity Winding of D(  ) on complex plane as  varies along the real axis i

21. 21 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

22. 22 Oide-Yukoya analysis consistent with Vlasov calculations in time domain Spectrum of unstable modes Threshold Numerical solution of Vlasov Eq. in time domain Linear theory

23. 23 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 Energy deviation Ic =0.048 pC/V

24. 24 Improved method is in good agreement with Oide-Yokoya, simulations One root of D(  ) found with 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 jump to initiate a Newton search: Find:  = 1.86 + 0.0023*i Energy spread

25. 25 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

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

27. 27 O-Y detects some spurious unstable modes Im  Spectrum of unstable modes Time domain simulations show no instability Simulations Linear theory

28. 28 Modified linear analysis correctly detects absence of unstable mode Im  Re  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

29. 29 Convergence of results against mesh refinement may 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

30. 30 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.

31. 31

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

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

34. 34 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 linearities cause saturation, turn-off burst Radiation damping relax beam back closer to equilibrium Bursting cycle

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

36. 36 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 longitudinal bunch density equilibrium with potential well-distortion