Collisionless Relaxation in Beam-Plasma Systems [thesis]

179 Pages·2001·16.791 MB·English

Collisionless Relaxation in Beam-Plasma Systems
by
Ekaterina Yu. Backhaus
M.A. (University of Texas in Austin) 1995
A dissertation submitted in partial satisfaction of the
requirements for the degree of
Doctor of Philosophy
in
Physics
in the
GRADUATE DIVISION
of the
UNIVERSITY of CALIFORNIA at BERKELEY
Committee in charge:
Professor Jonathan S. Wurtele, Chair
Professor Roger W. Falcone
Professor Charles K. Birdsall
Spring 2001
The dissertation of Ekaterina Yu. Backhaus is approved:
Chair Date
Date
Date
University of California at Berkeley
Spring 2001
Collisionless Relaxation in Beam-Plasma Systems
Copyright Spring 2001
by
Ekaterina Yu. Backhaus
1
Abstract
Collisionless Relaxation in Beam-Plasma Systems
by
Ekaterina Yu. Backhaus
Doctor of Philosophy in Physics
University of California at Berkeley
Professor Jonathan S. Wurtele, Chair
This thesis reports the results from the theoretical investigations, both numerical and an-
alytical, of collisionless relaxation phenomena in beam-plasma systems. Many results of
this work can also be appliedto other lossless systems of plasma physics, beam physics and
astrophysics.
Di(cid:11)erent aspects of the physics of collisionless relaxation and its modeling are
addressed. A new theoretical framework, named Coupled Moment Equations (CME), is
derived and used in numerical and analytical studies of the relaxation of second order mo-
ments such as beam size and emittance oscillations. Thistechnique extends the well-known
envelopeequationformalism,anditcanbeappliedtogeneralsystemswithnonlinearforces.
Itisbasedona systematicmoment expansionofthe Vlasov equation. Incontrast to theen-
velope equation, which is derived assuming constant rms beam emittance, the CME model
allows the emittance to vary through coupling to higher order moments. The CME model
is implemented in slab geometry in the absence of return currents. The CME simulation
yields rms beam sizes, velocity spreads and emittances that are in good agreement with
particle-in-cell (PIC) simulations for a wide range of system parameters.
The mechanism of relaxation is also considered within the framework of the CME
system. It isdiscovered thattherapidrelaxationor beamsizeoscillationscanbe attributed
to a resonant coupling between di(cid:11)erent modes of the system. A simple analytical estimate
of the relaxation time is developed.
The(cid:12)nalstateofthesystemreachedaftertherelaxationiscompleteisinvestigated.
New and accurate analytical results for the second order moments in the phase-mixed
2
state are obtained. Unlike previous results, these connect the (cid:12)nal values of the second
order moments with the initial beam mismatch. These analytical estimates are in good
agreement with the CME model and PIC simulations. Predictions for the (cid:12)nal density and
temperature are developed that show main important features of the spatial dependence
of the pro(cid:12)les. Di(cid:11)erent aspect of the (cid:12)nal coarse-grained state such as its non-thermal
nature, the appearance of ’hot’ regions on the periphery and the core-halo character of the
density are investigated.
Professor Jonathan S. Wurtele
Dissertation Committee Chair
iii
To my grandmothers, whose di(cid:11)erent lives were always an inspiration...
iv
Contents
List of Figures vi
1 Introduction 1
2 Illustration of Beam Propagation Through Long Plasma Systems 10
2.1 Numerical Example. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 Predictions for the Rms Dynamics Using the Traditional Envelope Equation 16
2.3 Estimates of Equilibrium Bulk Properties . . . . . . . . . . . . . . . . . . . 18
3 Coupled Moment Equation Model 27
3.1 Derivation of the Coupled Moment Equations . . . . . . . . . . . . . . . . . 27
3.2 Numerical Integration of the Coupled Moment Equations . . . . . . . . . . 33
3.2.1 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.3 Accuracy Aspects of the Numerical Model . . . . . . . . . . . . . . . . . . . 54
3.3.1 CME and Conservation Laws . . . . . . . . . . . . . . . . . . . . . . 54
3.3.2 Truncation of the In(cid:12)nite CME System . . . . . . . . . . . . . . . . 56
3.3.3 Normalization of the CME Equations . . . . . . . . . . . . . . . . . 63
3.4 Description of Beam Dynamics in a Symmetrically Weighted Hermite Rep-
resentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 Description of Collisionless Relaxation 77
4.1 Analysis of Collisionless Relaxation Using the CME Model. . . . . . . . . . 79
4.2 Relaxation Time Estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
4.3 Resonant Coupling Strength and Force Nonlinearity . . . . . . . . . . . . . 96
4.4 Resonant Coupling Strength and Self-Consistency of the Force . . . . . . . 100
5 Coarse-grained Equilibrium State 102
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5.2 Coarse-grained Equilibrium State - Numerical Examples . . . . . . . . . . . 105
5.3 Comparison of Numerical Stationary Results with the Previous Theory . . 114
5.4 Non-thermal Character of the Final Coarse-grained Equilibrium. . . . . . . 123
5.5 Simple Analytical Model for the Asymptotic State . . . . . . . . . . . . . . 125
5.6 Density Pro(cid:12)le in the Asymptotic State of a System with an External Force 135
v
6 Conclusions 152
A CME derivation through the evolution of beam moments. 156
Bibliography 158
vi
List of Figures
2.1 The setup of the system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 PIC propagation curves for the beam with the initial gaussian distribution
and the mismatch of (cid:11) = 5 for (a) rms beam size, (b) rms velocity and (c)
the emittance. All quantities are normalized to their initial values. . . . . . 15
2.3 Comparison between propagation curves for the rms beam size generated
by the envelope equation (|{) and PIC simulation ( ) for the initial
(cid:1) (cid:1) (cid:1)
mismatch of (cid:11) =5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.4 Comparisonbetweenthenewanalyticalpredictions( )givenbyEqs. (2.33),(2.38)
(cid:15)
and (2.40), predictions of Ref. [1] (2) given by Eqs. (2.13)-(2.15) and PIC
results ( ) for (a) the rms beam properties and (b) the emittance vs the
4
initial mismatch (cid:11) of the initially gaussian beam. . . . . . . . . . . . . . . 24
2.5 Comparisonbetweenthenewanalyticalpredictions( )givenbyEqs. (2.33),(2.39)
(cid:15)
and (2.41), predictions of Ref. [1] (2) given by Eqs. (2.16)-(2.18) and PIC
results ( ) for (a) the rms beam properties and (b) the emittance vs the
4
initial mismatch (cid:11) of the initially uniform beam. . . . . . . . . . . . . . . . 26
3.1 Comparison of results for the evolution of the rms beam size obtained by a
PIC simulation ( ) and by a CME integration with di(cid:11)erent truncation
(cid:1) (cid:1) (cid:1)
orders jmax =2 (|{) and jmax =4 ({ {). . . . . . . . . . . . . . . . . . . 34
(cid:1)
3.2 The unphysical growth of the amplitude of the rms oscillations obtained in
the course of CME integration with jmax = 20 (|{) together with the PIC
simulation prediction ( ). . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
(cid:1) (cid:1) (cid:1)
3.3 Comparisons of propagation curves for (a) rms beam size, (b) rms velocity
and (c) the emittance obtained using the CME model (|{) with jmax =20,
(cid:1) = 10 and a PIC simulation ( ) for the initially gaussian beam with a
(cid:1) (cid:1) (cid:1)
mismatch of (cid:11) =5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 Comparisons of predictions for the asymptotic (a) rms beam size, (b) rms
velocity obtained using PIC ( ) and CME ( ) simulations with di(cid:11)erent
(cid:15) 4
number of particles and equations used in the computation. An example of
an initially gaussian beam with (cid:11) = 5 is used. . . . . . . . . . . . . . . . . . 38
vii
th
3.5 Comparisons of propagation curves for normalized 4 order (a) spatial, (b)
velocity and (c) correlation moments obtained using the CME model (|{)
withjmax =20, (cid:1)=10 anda PICsimulation( ) forthe initiallygaussian
(cid:1) (cid:1) (cid:1)
beam with a mismatch of (cid:11) =5 . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Density pro(cid:12)les at (a) z = 0, (b) z = 1:5m, (c) z = 3m and (d) z = 4:5m
obtained by the CME integration (|{) and PIC simulation ( ) for the
(cid:1) (cid:1) (cid:1)
initial gaussian beam with a mismatch of (cid:11) =5. . . . . . . . . . . . . . . . 42
3.7 Temperaturepro(cid:12)lesat(a)z = 0, (b)z =1:5m, (c)z =3mand(d)z =4:5m
obtained by the CME integration (|{) and PIC simulation ( ) for the
(cid:1) (cid:1) (cid:1)
initial gaussian beam with a mismatch of (cid:11) =5. . . . . . . . . . . . . . . . 43
3.8 Comparisons of propagation curves for (a) rms beam size, (b) rms velocity
and (c) the emittance obtained using the CME model (|{) with jmax =20,
(cid:1) = 8 and a PIC simulation ( ) for the initially gaussian beam with a
(cid:1) (cid:1) (cid:1)
mismatch of (cid:11) =0:33. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
th
3.9 Comparisons of propagation curves for normalized 4 order (a) spatial, (b)
velocity and (c) correlation moments obtained using the CME model (|{)
with jmax =20, (cid:1) =8 and a PIC simulation ( ) for the initially gaussian
(cid:1) (cid:1) (cid:1)
beam with a mismatch of (cid:11) =0:33 . . . . . . . . . . . . . . . . . . . . . . . 45
3.10 Density pro(cid:12)les at (a) z = 0, (b) z = 1:5m, (c) z = 3m and (d) z = 4:5m
obtained by the CME integration (|{) and PIC simulation ( ) for the
(cid:1) (cid:1) (cid:1)
initial gaussian beam with a mismatch of (cid:11) =0:33. . . . . . . . . . . . . . 46
3.11 Temperaturepro(cid:12)lesat(a)z = 0, (b)z =1:5m, (c)z =3mand(d)z =4:5m
obtained by the CME integration (|{) and PIC simulation ( ) for the
(cid:1) (cid:1) (cid:1)
initial gaussian beam with a mismatch of (cid:11) =0:33. . . . . . . . . . . . . . 47
3.12 Comparisons of propagation curves for (a) rms beam size, (b) rms velocity
and (c) the emittance obtained using the CME model (|{) with jmax =20,
(cid:1) = 10 and a PIC simulation ( ) for the initially uniform beam with a
(cid:1) (cid:1) (cid:1)
mismatch of (cid:11) =5. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
th
3.13 Comparison of propagation curves for normalized 4 order (a) spatial, (b)
velocity and (c) correlation moments obtained using the CME model (|{)
with jmax =20, (cid:1)=10 and a PIC simulation ( ) for the initiallyuniform
(cid:1) (cid:1) (cid:1)
beam with a mismatch of (cid:11) =5 . . . . . . . . . . . . . . . . . . . . . . . . 50
3.14 Comparison of the dynamical CME results (3), analytical predictions ( )
(cid:15)
given by Eqs. (2.33), (2.38) and (2.40) and PIC results ( ) for (a) rms beam
4
properties and (b) the emittance of the initially gaussian beam vs the initial
mismatch (cid:11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.15 Comparison of the dynamical CME results (3), analytical predictions ( )
(cid:15)
given by Eqs. (2.33), (2.39) and (2.41) and PIC results ( ) for (a) rms beam
4
properties and (b) the emittance of the initially uniform beam vs the initial
mismatch (cid:11). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3.16 Comparison of the dynamical CME results ( ) and PIC predictions ( ) for
2 2 2 2 (cid:15) 4
thecorrelationratio x vx =((cid:27)x(cid:27)v)oftheinitiallygaussianbeamvstheinitial
h i
mismatch (cid:11). The solid line corresponds to a fully thermalized beam and is
plotted for a reference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53