self consistents a coupled-bunch instability … · coupled bunch instability (cbi) for arbitrary...

Self-consistent Simulations for Collective EffectsCoupled Bunch Instability (CBI) for Arbitrary Fillings

SELF-CONSISTENT SIMULATIONS ANDANALYSIS OF THE COUPLED-BUNCH

INSTABILITY FOR ARBITRARY MULTI-BUNCHCONFIGURATIONS

Gabriele Bassi

Brookhaven National Laboratory, NSLS-II

Gabriele Bassi Brookhaven National Laboratory, NSLS-II NOCE 2017 1/24


OUTLINE

NSLS-II Beam Intensity IncreasingSelf-consistent Parallel Tracking Code SPACECoupled Bunch Instability (CBI) for ArbitraryFilling Patterns

- Theoretical Analysis- Benchmarking Theory and Simulations- Measurements of the CBI Driven by the Resistive Wall

Instability in NSLS-II



NSLS-II BEAM INTENSITY INCREASING



SPACE: General Features and CapabilitiesSPACE: Parallelization

SELF-CONSISTENT PARALLEL TRACKING CODE SPACE

SPACE(Self-consistent Parallel Algorithm for Collective Effects)

Efficient study of short and long-range wakefield effects in 6Dphase-space via parallel processing communications.Study of slow head-tail effect + coupled-bunch instabilities.Passive higher harmonic cavity effects with arbitrary fillings.Landau damping from betatron tune spread.Microwave instability.Efficient methods for density estimation from particles.Localized wakefield effects.

G. Bassi et al., Phys. Rev. Acc. Beams 19 024401, 2016.



SPACE: General Features and CapabilitiesSPACE: Parallelization

GENERAL PARALLEL STRUCTURE

M bunches each with N simulations particles distributed to Mprocessors.Short-range (single bunch) wakefield interaction calculated inserial (locally).Long-range wakefield calculation done in parallel (globally) viamaster-to-slave processor communications by storing the“history” of moments of the bunches.For efficient study of microwave instability, the calculation isdone in parallel by distributing N/M simulation particles to Mprocessors.



Theoretical AnalysisMeasurements

ANALYTICAL TREATMENT OF THE CBI FOR ARBITRARY FILLINGS

Motivation:- Many storage ring modes of operation require multi-bunch

configurations that differ from uniform fillings: gap in the fillingsfor ion clearing (NSLS-II), special multi-bunch configurationswith a single high charge bunch (camshaft) .

- Desire to operate with the most stable multi-bunch configuration.The treatment is based on the formulation of an eigenvalueproblem defined by the complex frequency shifts of the uniformfilling pattern case.The numerical solution of the eigenvalue problem allows thestudy of instability thresholds via the determination of theeigenvalue with the largest imaginary part.As a complementary tool to the computation of the eigenvaluespectrum we discuss the Gerschgorin Circle Theorem.The application of the Gerschgorin Circle Theorem is useful for

A) a rapid localization of the eigenvalues in the complex plane.B) very efficient perturbative studies of uniform filling patterns.




TRANSVERSE EIGENANALYSIS

The transverse phase space densities m

(, , x, p

x

, t), associated tothe mth bunch with N

m

particles, satisfy the following system ofM-coupled Vlasov equations for 0 m M 1

@ m

@t

@

m

@+

!2s

@

m

@+ p

x

@ m

@x

!2x

@ m

@p

x

A

x

+1X

k=0

M1X

m

0=0

N

m

0

" Z

1

d 0W1 0 + a

k

m

0m

T0

+1Z

1

dx

0x

0m

0 0, x

0, t a

k

m

0m

T0#@

m

@p

x

= 0,

where a

k

m

0m

= k + m

0m

M

and m

(, x, t) =R

ddp

x

m

(, , x, p

x

, t).




EQUATIONS FOR THE EVOLUTION OF DIPOLE MOMENTS

The time evolution of the dipole moments hxm

i =R

d d dx dp

x

x m

and hpx

m

i =R

d d dx dp

x

p

x

m

is found by integrating by parts theVlasov equations using the boundary conditions for

m

d

2

dt

2 hxm

i+ !2hxm

i = A

x

+1X

k=0

M1X

m

0=0

f

a

k

m

0m

T0

N

m

0hxm

0i

t a

k

m

0m

T0

,

where f (x) +1R

1d

+1R

1d 0W1( 0 + x)()( 0).

Rearranging and omitting the brackets (xm

understood as hxm

i)TIME EVOLUTION BUNCH CENTROID

x

m

(t) + !2x

m

(t) = A

x

+1X

k=0

f

k

T0

M

N[m+k]x[m+k]

t k

T0

M

,

where [m + k] = m + k Mb(m + k)/Mc, with bxc the floor function(gives the largest integer less or equal to x).




EQUATIONS FOR THE EVOLUTION OF MODE xµ

Defining the mode xµ by

xµ(t) =M1X

m=0

x

m

(t)ei2mµ/M, x

m

(t) =1M

M1X

µ=0

xµ(t)ei2mµ/M,

it follows that xµ are coupled and satisfy the equations of motionTIME EVOLUTION MODE xµ

¨xµ(t) + !2

xµ(t) = A

x

M

1X

k=0

f

k

T0

M

e

i2µk/M

M1X

µ0=0

xµ0

t k

T0

M

M1X

m=0

N

m

e

i2m(µ0µ)/M.

Remark: for N

m

= N (uniform filling) the modes xµ are uncoupled.




MULTI-BUNCH MODE x

(µ)m

The general solution of the non-collective equations of motion¨xµ(t) + !2

xµ(t) = 0 reads

xµ(t) = A1e

i! t + A2e

i! t =12

hxµ(0)

i

!

˙xµ(0)

e

i! t

+

xµ(0) +i

!

˙xµ(0)

e

i! t

i.

x

m

2 R =) xµ = x

?Mµ, thus define the multi-bunch mode as

x

(µ)m

(t) =1M

xµ(t)e

i2mµ/M + x

Mµ(t)ei2mµ/M

=1M

xµ(t)e

i2mµ/M + x

?µ(t)e

i2mµ/M

=2M

Rexµ(t) cos

2µm

M

Imxµ(t) sin2µm

M

.




PERTURBATIVE SOLUTION

Look for a perturbative solution identifying the perturbation by theparameter .Without loss of generality, assume for the perturbative solution theformPERTURBATIVE SOLUTION MODE xµ

xµ(t) = A1e

i(!+)t + A2e

i(!+)t = aµe

i(!+)t, 2 C,

where aµ = xµ(0) and we chose A1 = 0.Defining 1 Im, !

r

Re and assuming Imxµ(0) = 0, themulti-bunch mode x

(µ)m

takes the form

PERTURBATIVE SOLUTION MULTI-BUNCH MODE x

(µ)m

x

(µ)m

(t) = ae

t

cos

2µm

M

cos(! + !r

)t + sin2µm

M

sin(! + !r

)t,

where a = 2Rexµ(0)/M and we set the perturbation parameter = 1.Gabriele Bassi Brookhaven National Laboratory, NSLS-II NOCE 2017 11/24



EIGENVALUE EQUATION

The perturbative solution for xµ leads to the eigenvalue equationh

A

x

N

2!

1X

k=0

f

k

T0

M

e

i2µk/M

e

i!kT0/M iaµ

+M1X

µ0=0µ0 6=µ

hA

x

2!M

1X

k=0

f

k

T0

M

e

i2µk/M

e

i!kT0/M

M1X

m=0

N

m

e

i2m(µ0µ)/M

iaµ0 = 0.

or, equivalentlyEIGENVALUE EQUATION FOR ARBITRARY FILLING PATTERNS

(BI)a = 0, Bµµ0 =U

µ

NM

M1X

m=0

N

m

e

i2m(µ0µ)/M, a = [a0, .., a

M1]T

where U

µ are the eigenvalues of the uniform filling pattern case(Bµµ0 = U

µ if µ0 = µ, 0 otherwise)

U

µ = i

I

b

Mc

4(E0/e)

+1X

p=1

(pM+µ)!0+!

2Z

?1(pM+µ)!0+!

.




Solving for the characteristic polynomial p() = |B I| = 0 andassuming M distinct eigenvalues

m

, the general solution x

g

µ(t) isgiven by

x

g

µ(t) =M1X

m=0

c

m

aµm

e

i(!+m

)t,

where am

= [a0 m

, .., a

M1 m

]T are the eigenvectors associated to theeigenvalues

m

.Since the sum of the eigenvalues of B is equal to the trace of B, itfollows that the sum of the complex frequency shifts µ for arbitraryfilling patterns is equal to the sum of the complex frequency shifts U

µ

for uniform filling patterns

M1X

µ=0

µ = Tr B =)M1X

µ=0

µ =M1X

µ=0

U

µ .




GERSCHGORIN CIRCLE THEOREM

Estimation of eigenvalue spectrum very accurate for strictlydiagonally dominant matrices (off-diagonal terms small with respectto diagonal terms), i.e. for a n n complex matrix A = (a

ij

) forwhich

|aii

| >nX

j=1j 6=i

|aij

|, 0 i n.

GERSCHGORIN’S THEOREM

If is an eigenvalue of an n n matrix A = (aij

), then, for some i,

| a

ii

| nX

j=1j 6=i

|aij

| r

i

, 0 i n.

r

i

defines the radius of a Gerschgorin row disc. Defining similarly theradius of a Gerschgorin column disc, we have that all of theeigenvalues of the matrix A are contained in the intersection of theunion of all the row discs and column discs.S. Gerschgorin, Izv. Akad. Nauk. USSR Otd. Fiz-Mat. Nauk 7, 749-754, (1931).




BENCHMARKING THEORY AND SIMULATIONS

Study of the coupled-bunch instability driven by the HOMs of the7-cell PETRA-III RF Cavity (used during commissioning phase 1 ofthe NSLS-II storage ring).

TABLE : Transverse HOMs of the 7-cell PETRA-III RF Cavity

f

r

, MHz R

sh,?, M/m Q? R

sh,?/Q?, /m860.25 14.7 55700 263.91867.12 17.5 56800 308.1869.55 56.1 58200 963.92870.96 19.7 59400 331.651043.53 83.6 40400 2069.311047.44 26.2 40900 640.591089.13 17.0 49400 344.131465.13 15.5 54600 283.881545.34 26.8 44300 604.97




UNIFORM BUNCH TRAIN WITH A GAP

-1000

-500

0

500

1000

1500

0 200 400 600 800 1000 1200

gro

wth

rat

e (1

/s)

bunch mode

numericanalytic

0 200 400 600 800 1000 1200

bu

nc

h t

rain

bunch number m

Mg=1320Mg=1200Mg=1000Mg=800Mg=600

a)

Growth rates uniform filling Bunch trains with different gaps

0

200

400

600

800

1000

1200

1400

1 10 100 1000

Gro

wth

ra

te (

1/s

)

Mg

numericanalytic

-1000

-500

0

500

1000

1500

-1500 -1000 -500 0 500 1000

ImΩ

(1/s

)

ReΩ (1/s)

µ=1170

µ = 944

µ = 316

a) Mg = 1200Mg = 1000

Mg = 800Mg = 600

Fastest instability vs. bunch train with a gap Fast Eigenvalue Estimation: Gerschgorin Circle Theorem




NSLS-II MEASUREMENTS WITH GAP IN THE FILL PATTERN(Operational lattice with 3DW gaps closed at zero linear chromaticity)

Calculated CBI threshold from resistive wall impedance I

th

= 22mA.Fill Pattern Monitor I

b

(single bunch current)

Uniform I

b

M = 688 (gap = 632)I

th

= 20.5mA

Uniform I

b

M = 352 (gap = 968)I

th

= 20.5mA

Non-uniform I

b

M = 600 (gap = 620)I

th

= 24.3mA




NSLS-II MEASUREMENTS WITH GAP IN THE FILL PATTERNFill Pattern from Oscilloscope Tune Measurement from TBT Data

Uniform I

b

M = 688I

th

= 20.5mA

Uniform I

b

M = 1284I

th

= 20.5mA




NSLS-II MEASUREMENTS WITH GAP IN THE FILL PATTERNFill Pattern from Oscilloscope Tune Measurement from TBT Data

Non-uniform I

b

M = 600I

th

= 24.3mA

I

th

= 24.3mA (higher threshold possibly explained by betatron tune spread across the bunch train induced by single bunch effects)

Uniform I

b

M = 352I

th

= 20.5mA




THANK YOU!




BACK-UPSLIDES




ELEMENTARY CASE: TWO BUNCHES

For M = 2, and with 2N = N

T

, the eigenvalue problem readsU

0 NU

0NU

1 U

1

= 0, N =N0 N1

N0 + N1, N0 + N1 = 2N,

and is easily solved with eigenvalues

0,1 =U

0 + U

12

± 12

q(U

0 U

1 )2 + 4N

2

U

0 U

1 , 0 + 1 = U

0 + U

1 ,

and corresponding eigenvectors

a0 =

2

41,2U

1 N

U

1 U

0 q

(U

0 U

1 )2 + 4N

2

U

0 U

1

3

5T

a1 =

2

4 2U

0 N

U

1 U

0 q

(U

0 U

1 )2 + 4N

2

U

0 U

1

, 1

3

5T

.




ELEMENTARY CASE: TWO BUNCHES

-100

0

100

200

300

400

500

600

700

0 0.5 1 1.5 2

ImΩ

(1/s

)

N0/N

a)ImΩ0ImΩ1

-300

-250

-200

-150

-100

-50

0

50

100

0 0.5 1 1.5 2

Re

Ω (

1/s

)

N0/N

b)ReΩ0ReΩ1

0

100

200

300

400

500

600

700

800

0 0.5 1 1.5 2

gro

wth

rate

(1/s

)

N0/N

c)numericanalytic




ELEMENTARY CASE 2: THREE BUNCHES

In the case M = 3 we discuss the configurationN0 = N1 = 3N/2,N2 = 0, with 3N = N

T

, which describes aconfiguration with a missing bunch. The corresponding eigenvalueproblem reads

|B I| =

U

0 aU

0 a

U

0a

U

1 U

1 aU

1aU

2 a

U

2 U

2

= 0, a =

1 +p

3i

2,

where denotes complex conjugate. It follows that

B I| = h2 Tr B+

34

U

0 U

1 + U

0 U

2 + U

1 U

2

i= 0,

where Tr B = U

0 + U

1 + U

2 .


self consistents a coupled-bunch instability … · coupled bunch instability (cbi) for arbitrary...

Documents