Update of Microwave Instability Study in SuperKEKB Damping Ring Using Vlasov Fokker-Planck Solver

Update of Microwave Instability Update of Microwave Instability Study in SuperKEKB Damping Study in SuperKEKB Damping

Ring Using Vlasov Fokker-Planck Ring Using Vlasov Fokker-Planck SolverSolver

L. Wang, SLAC

In collaboration with H. Ikeda, K. Ohmi, K. Oide, D. Zhou

6.15.2012

KEKB Physics Meeting1

MotivationMotivation Systematic study the microwave instability using

Vlasov solver to validate the results: We checked the numerical parameters: mesh

size, domain, time step and the CSR impedance Our goal is to empirically find the appropriate

parameters/way for the simulation of MWI.

2

Wakes Wakes Geometry wake CSR Impedance(rectangular geometry 34mmX24mm) @Demin’s code (Japanese Journal of Applied Physics 51 (2012) 016401) The CSR Wake is given by the convolution of a 0.5 mm long Gaussian bunch with CSR impedance (frequency up to

477GHz)

-20 -15 -10 -5 0 5 10 15 20-150

-100

-50

0

50

100

150

S (z)

W||(V

/pC

)

Geometric Wake

Wakebunch shape

Geometry wake

-15 -10 -5 0 5 10 15

-1500

-1000

-500

0

500

1000

1500

S (z)

W||(V

/pC

)

16cell CSR Wake

Wakebunch shape

CSR wake

3

Microwave instability with different Microwave instability with different wakes (Yunhai’s code)wakes (Yunhai’s code)

With CSR wake only, the energy spread starts to increase near the nominal bunch current (N= 51010 ) The microwave instability starts at bunch population 61010 when both geometry wake and CSR wake are included

Geometry wake only

0 1 2 3 4 5 6

x 1010

0.99

1

1.01

1.02

1.03

1.04

1.05

Np

e/

e0,

z/

z0

Goemetry Wake only

z/

z0

e/

e0

0 2 4 6 8 10 12

x 1010

1

1.05

1.1

1.15

1.2

1.25

1.3

Np

e/

e0,

z/

z0

CSR Wake+Geometry Wake

z/

z0,csr and geometry wake

e/

e0,csr and geometry wake

z/

z0,csr wake

e/

e0,csr wake

CSR wake only & both g-wake and CSR wake 4

Numerical parameters: qmax=8, nn=300, ndt=1024

Effect of domain, mesh size and time stepEffect of domain, mesh size and time step The simulation is done in the normalized phase space,

which is rectangular region with maximum domain qmax (and minimum –qmax). (16X16)

The whole domain contains (2*nn+1) mesh points: (1001X1001)

The time step is given by ndt, the number of steps per synchrotron period.

0 2 4 6 8 10 12

x 1010

1

1.05

1.1

1.15

1.2

1.25

1.3

Np

e/

e0,

z/

z0

z/

z0,nn=300,qmax=8

e/

e0,nn=300,qmax=8

z/

z0,nn=500,qmax=8

e/

e0,nn=500,qmax=8

z/

z0,nn=500,qmax=11

e/

e0,nn=500,qmax=11

Requirement:nn: 500qmax: 8Time steps: 1024/syn. period

5

Effect of CSR impedance at high frequency

Previous study using impedance with frequency up to 500GHz;CSR impedance with frequency up to 1.4 THzThe wake is convoluted by a 0.2 mm Gaussian bunch

-15 -10 -5 0 5 10 15

-8000

-6000

-4000

-2000

0

2000

4000

S(z)

W (

V/p

C)More high frequency component

6

Although our bunch is long, but micro-wave instability occurs at micro-bunch level

With higher frequency CSR-1With higher frequency CSR-1

0 2 4 6 8 10 12

x 1010

1

1.05

1.1

1.15

1.2

1.25

1.3

Np

e/

e0,

z/

z0

With High frequency 1.5THz CSR impedance

There is a similar threshold, but a much stronger instability above the thresholdA clear Saw-tooth instability;

0 200 400 600 800 10000.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

t (syn. period)

z/

e

z

e

Near thresholdN=5.5E10

0 200 400 600 800 10000.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

t (syn. period)

z/

e

z

e

above threshold N=8.5E10

7

However, no saw-tooth instability found with the CSR impedance of f~477GHz

0 200 400 600 800 10000.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

t (syn. period)

z/

e

z, f~1.5THz

e, f~1.5THz

z, f~500GHz

e,f~500GHz

Higher frequency CSR effectHigher frequency CSR effect


High frequency CSR impedance is important:Saw-tooth type of instability is found with high frequency CSR impedance only

8

Higher frequency CSR effectHigher frequency CSR effect

above threshold N=8E10

High frequency CSR impedance is important:

9

f up to 500GHZ f up to 1500GHZ

With higher frequency CSR-2With higher frequency CSR-2instability near threshold N=5.5e10instability near threshold N=5.5e10

10

0 200 400 600 800 10000.98

1

1.02

1.04

1.06

1.08

1.1

1.12

1.14

t (syn. period)

z/

e

z

e

Phase plot

With higher frequency CSR-3With higher frequency CSR-3MWI above threshold N=8.5e10MWI above threshold N=8.5e10

110 200 400 600 800 1000

0.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

t (syn. period)

z/

e

z

e

Clear Saw-tooth instability occursPhase plot shows high order modes

Phase plot

Saw-tooth instability driven by CSR

Saw-tooth Period:110 syn. period517Hz

Time step checkTime step check


Ndt (steps per synchrotron period) =1024 is good enough

0 200 400 600 800 10000.9

1

1.1

1.2

1.3

1.4

1.5

1.6

1.7

1.8

t (syn. period)

z/

e

z, ndt=1024

e, ndt=1024

z, ndt=2048

e,ndt=2048

12

SummarySummary

13

The microwave instability in SuperKEKB Damping Ring is simulated using the Vlasov-Fokker-Planck code. It suggests a threshold slightly above the designed bunch current. PIC (Particle-In-Cell) code also confirms similar threshold

The simulation of microwave instability with CSR impedance is nontrivial. We carefully checked the numerical parameters used in the simulation: the good numerical parameters areqmax=8 (domain), nn=500(mesh), ndt=1024 (time step)

The most important finding: The high frequency part CSR impedance (f~1.4THz) plays an important role in the saw-tooth instability. High order modes observed

discussiondiscussion

14

How to include high frequency impedance:(1)Calculate high frequency CSR impedance It is straightforward and only a matter of CPU, using parallel computation.(2) convolution with a short Gaussian bunch, <=0.1mm

It would be very helpful (for both machine and beam dynamics) to observe the CSR & MWI in DR as proposed by Prof. Fukuma

(3) Instead of convolution with a Gaussian bunch, directly using impedance (Demin IPAC12, Bob) green function (direct FFT of impedance)? other approaches?

Thank You!Thank You!

15

Y. Cai and B. Warnock for fruitful discussions

Thanks to Demin for his great help on the simulations

Update of Microwave Instability Study in SuperKEKB Damping Ring Using Vlasov Fokker-Planck Solver

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