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A moving boundary problem motivated by electric breakdown, I: Spectrum of linear perturbations

15 May 2009-Physica D: Nonlinear Phenomena (Elsevier)-Vol. 238, Iss: 9, pp 888-901

Abstract: An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be written as a Laplacian growth model regularized by a ‘kinetic undercooling’ boundary condition. We study the linear stability of uniformly translating circles that solve the problem in two dimensions. In a space of smooth perturbations of the circular shape, the stability operator is found to have a pure point spectrum. Except for the eigenvalue λ 0 = 0 for infinitesimal translations, all eigenvalues are shown to have negative real part. Therefore perturbations decay exponentially in time. We calculate the spectrum through a combination of asymptotic and series evaluation. In the limit of vanishing regularization parameter, all eigenvalues are found to approach zero in a singular fashion, and this asymptotic behavior is worked out in detail. A consideration of the eigenfunctions indicates that a strong intermediate growth may occur for generic initial perturbations. Both the linear and the nonlinear initial value problem are considered in a second paper.
Topics: Linear stability (59%), Boundary value problem (57%), Boundary problem (57%), Initial value problem (56%), Eigenfunction (55%)

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Physica D 238 (2009) 888–901
Contents lists available at ScienceDirect
Physica D
journal homepage: www.elsevier.com/locate/physd
A moving boundary problem motivated by electric breakdown, I: Spectrum of
linear perturbations
S. Tanveer
a
, L. Schäfer
b
, F. Brau
c
, U. Ebert
c,
a
Department of Mathematics, Ohio State University, 231 W 18th Ave, Columbus 43210, USA
b
Fachbereich Physik, Universität Duisburg-Essen, Lotharstr. 1, 47048 Duisburg, Germany
c
Centrum Wiskunde & Informatica (CWI), P.O.Box 94079, 1090GB Amsterdam, The Netherlands
a r t i c l e i n f o
Article history:
Received 1 September 2008
Received in revised form
17 February 2009
Accepted 20 February 2009
Available online 9 March 2009
Communicated by B. Sandstede
PACS:
47.54.-r
Keywords:
Moving boundary
Kinetic undercooling regularization
Linear stability analysis
Laplacian instability
Electric breakdown
a b s t r a c t
An interfacial approximation of the streamer stage in the evolution of sparks and lightning can be written
as a Laplacian growth model regularized by a ‘kinetic undercooling’ boundary condition. We study the
linear stability of uniformly translating circles that solve the problem in two dimensions. In a space of
smooth perturbations of the circular shape, the stability operator is found to have a pure point spectrum.
Except for the eigenvalue λ
0
= 0 for infinitesimal translations, all eigenvalues are shown to have negative
real part. Therefore perturbations decay exponentially in time. We calculate the spectrum through a
combination of asymptotic and series evaluation. In the limit of vanishing regularization parameter, all
eigenvalues are found to approach zero in a singular fashion, and this asymptotic behavior is worked out
in detail. A consideration of the eigenfunctions indicates that a strong intermediate growth may occur for
generic initial perturbations. Both the linear and the nonlinear initial value problem are considered in a
second paper.
© 2009 Elsevier B.V. All rights reserved.
1. Introduction
The motion of interfaces in a Laplacian field is of general interest
and has been a subject of intense study over many years (see for
instance the reviews [1,2]). Such problems arise in many physical
contexts, such as viscous fingering in multi-phase fluid flow [3–16],
dendritic crystal growth in the quasi-steady small Peclet number
limit [17–20], void electromigration [21–25] and a host of other
phenomena such as the growth of biological systems like bacterial
colonies or corals [26].
More recently, a similar mathematical problem has been
discovered in the study of ‘streamers’ [27–34] which occur during
the initial stage of electric breakdown and play an important role
both in the natural phenomena of sparks and lightning as well
as in numerous technical applications [33]. Streamers are weakly
ionized bodies growing into some nonionized medium due to an
externally applied electric field. This field is so strong that the
drifting electrons very efficiently create additional electron ion
pairs by impact ionization, and the nonlinear coupling between
ionized body and field further increases this effect.
Corresponding author.
E-mail address: ebert@cwi.nl (U. Ebert).
Models for negative streamers in simple gases like nitrogen or
argon are based on a set of partial differential equations for the
densities of electrons and of positive ions coupled to the electric
field [28–33]. Analysis and numerical solutions of these equations
reveal that in the front part of the streamer, a thin surface charge
layer develops where the electron density strongly exceeds the
ion density. Therefore the electric field E varies strongly when
crossing this layer. Right before the layer, it is enhanced, but in
the interior of the streamer, it is screened to such a low level
that impact ionization is suppressed and the electron current
transporting charge from the interior to the surface charge layer is
small. Consequently we may take the interior as being essentially
passive, and the growth of the streamer is governed by the surface
charge layer which is driven by the strong local field.
If the external field is very strong, the thickness ` of the surface
charge layer can become small compared to the typical diameter
2R of the streamer [34]. This suggests modeling this layer as
an interface separating the ionized interior from the nonionized
exterior region. In this model the variation of the potential ϕ
across the surface charge layer is replaced by a discontinuity on the
interface. Since the interior is considered passive, only the limiting
value ϕ
+
reached by approaching the interface from the outside
is relevant for the dynamical evolution, and analysis of results of
the PDE-model suggests [34–36] that with an appropriate gauge,
ϕ
+
is coupled to the limiting value E
+
of the electric field by the
0167-2789/$ see front matter © 2009 Elsevier B.V. All rights reserved.
doi:10.1016/j.physd.2009.02.012

S. Tanveer et al. / Physica D 238 (2009) 888–901 889
boundary condition
ϕ
+
= `n · E
+
. (1)
Here n is the outward normal on the interface, ` is the
regularization length corresponding to the interface thickness,
and E
+
= −∇ϕ
+
, where the
+
again indicates the limit of
approaching the interface from the outside. In the context of
dendritic crystal growth, this boundary condition is equivalent
to including the kinetic undercooling effect
1
while excluding the
usual Gibbs–Thompson surface energy correction to the melting
temperature.
As the motion of the interface is caused by the drift of the
electrons in the local electric field v = E, the interface moves
with normal velocity
v
n
= n · E
+
, (2)
and outside the streamer the potential obeys the Laplace equation
ϕ = 0. (3)
We discuss the problem defined by Eqs. (1)–(3) in infinite
two-dimensional space, with the electric field becoming constant,
E = −∇φ E
, far from the streamer; such a condition
is realized frequently in atmospheric discharges, e.g., inside
thunderclouds. This far-field condition on φ is different from
the usual source/sink condition in viscous fingering in the absence
of side-walls, or undercooling specification in the quasi-steady
low Peclet number crystal growth problem which have been
extensively studied. However, moving bubbles and fingers in a long
Hele–Shaw channel are indeed subjected to this type of far-field
condition. (For the discussion of the streamer problem equivalent
to the Saffman-Taylor finger, we refer to [38].) Further, in the
crystal growth or directional solidification problem, a systematic
inner-outer analysis for small Peclet number [17,20] shows that
this is an appropriate condition at for the ‘‘inner’’ problem.
A simple steady solution to the streamer equations given above
is a circle translating with constant velocity determined by E
[35,36]. Though such circles differ from proper streamers, which
are growing channels of ionized matter [33,38–40], their front half
closely resembles the head of the streamer where the growth takes
place. It is therefore a question of physical interest whether or not
translating circular solutions are stable to small perturbations and
this is the subject of the present investigation.
The relevance of this analysis for more realistic streamer
shapes is supported by results found in another physical context.
Steadily translating circles also arise in viscous fingering in a
Hele–Shaw cell when surface tension is included (instead of kinetic
undercooling) in the limit when the bubble is small compared
to the cell-dimensions [3]. The linear stability of these bubbles,
including larger non-circular steadily translating bubbles, has
been studied before both for one and two fluids [10,14] and the
results largely mimic those obtained for a finger, though the latter
calculations are mathematically much more involved.
It has been known for a while that in the absence of any
regularization, such as surface tension or kinetic undercooling, the
initial value problem in a Laplacian field is ill-posed [9] in any
norm that is physically relevant to describing interfacial features.
This is reflected in the instability of any steady shape, when the
growth rates increase with the wave numbers of the disturbances.
Ill-posedness makes idealized model predictions sometimes
physically irrelevant (see [15] for a thorough discussion) and
regularization becomes essential.
1
Under most natural conditions of crystal growth, kinetic undercooling is
important in a limit when the Peclet number is not small enough to justify a
Laplacian field approximation; nonetheless, there have been some studies of steady
Laplacian crystal growth with kinetic undercooling effects only [37].
Considering a planar front, one finds that regularization does
not remove the instability against fluctuations of small wave
number. However, for large wave numbers, linear stability analysis
exhibits a basic difference between surface tension and kinetic
undercooling. All large wave number components of a disturbance
decay with surface tension regularization, while for kinetic
undercooling the growth rate saturates to a constant that scales
as `
1
[16]; for streamers, such a saturating dispersion relation is
derived and discussed in [41,42].
For curved fronts, one can pose the question: how does
curvature stabilize, if at all, a disturbance whose wavenumber
is in the unstable regime for a flat interface, either with surface
tension or with kinetic undercooling regularization? With surface
tension regularization, some answers are available in the existing
literature. Arguments have been presented [4,6] that suggest that
a localized wave packet with wave numbers in the unstable
regime
2
advects along the front as it grows; once it reaches the
side of the front where the local normal velocity is zero, the
disturbance stops growing. If the steady shape is closed, as it is
for a circle, the continued advection of the wave-packet towards
the receding parts of the interface will cause the disturbance
to decay eventually.
3
If regularization is small, there is a large
transient growth. Unless the disturbance amplitude is smaller than
a threshold that shrinks to zero with regularization, the transient
exponential growth causes the interface to enter a nonlinear
regime that can destabilize the steady front, even when it is
predicted to be linearly stable. Analysis of approximate equations,
supported by numerical calculations of the full equations support
the above scenario. Similar stabilization should occur for the
kinetic undercooling boundary condition as well, though we are
not aware of any explicit study affirming this expectation.
Note, however, that stabilization of localized wave packets
does not rule out instability to long ranged disturbances. A formal
asympotic study for small nonzero surface tension [12,14] as well
as numerical studies [7] reveal that surface tension stabilizes
precisely one branch of steady solutions for fingers and bubbles in
a Hele–Shaw cell. Similar results follow for a needle crystal [18]
though in the latter case, convective instability of wave packets
caused by significant normal speed along the parabolic front
is believed to cause dendritic structures [1]. These conclusions
have been challenged at times by alternate scenarios (see for
instance [19]) that are based on formal calculations, but with
different implicit assumptions. Such controversy affirms the need
for more rigorous mathematical studies of the stability problem,
even if it is for relatively simple shapes such as the circle in the
present study.
For the kinetic undercooling boundary condition, relying
merely on a numerical study to understand the long time behavior
is fraught with difficulties. One finds a collapsing spatial scale for
large time at the rear of the circle. Analytically, this is found for
= `/R = 1 in [35,36].
4
As will be argued in the present and
the companion paper, the occurrence of this collapsing scale is a
general feature for any > 0. This means that as t , one
must resolve progressively finer scales near the back of the bubble.
Further, calculations for small require resolving a large number of
transiently growing modes. All this underscores the need of some
progress on the analytical side.
2
Localized disturbances refer to those with wavelengths far smaller than the
typical radius of curvature of the steady shape. These can be unstable only if the
regularization parameter is sufficiently small.
3
Surface tension causes localized disturbances to decay as they advect to the
sides even when an interface is not closed but becomes parallel to the direction of
motion as is the case for a finger in a Hele–Shaw cell. However, no decay is expected
for kinetic regularization. This is where a closed interface is different.
4
We recall that R is a measure of the size of the streamer. The precise definition
is given in Eq. (5).

890 S. Tanveer et al. / Physica D 238 (2009) 888–901
The present paper, which is part I of a two-paper sequence,
is devoted to the spectral properties of the linear stability
operator, associated with infinitesimal perturbations of a circle.
In part II [43], we will consider the initial value problem,
presenting analytical and numerical results on the evolution of
both infinitesimal and finite perturbations.
The present paper is organized as follows. In Section 2 we
reformulate the problem defined by Eqs. (1)–(3) by standard
conformal mapping, and we present the PDE governing the time
evolution of infinitesimal perturbations of the circle. This material
has been presented before in [35,36], where also the general
solution of the PDE in the case = 1 has been discussed
in detail. The explicit solution found for = 1 shows that
outside any fixed neighborhood of the rear of the bubble, the
long-term behavior of infinitesimal perturbations is described by
P
n=0
e
λ
n
t
β
λ
n
, where λ
n
is the nth eigenvalue (ordered according
to absolute value) of the linear stability operator and β
λ
n
is the
corresponding eigenfunction.
We then study this eigenvalue value problem for arbitrary
> 0. We show in Section 3 that the linear stability operator,
defined in an appropriate space of analytic functions, has a pure
point spectrum. In Section 4 it is proven that there are no discrete
eigenvalues with non-negative real part, except λ = 0 that
corresponds to the trivial translation mode. A set of discrete,
purely negative eigenvalues is calculated in Section 5 as a function
of ; they smoothly extend the results found previously for
= 1. The results suggests that as 0, the spectrum
degenerates to the trivial translation mode and this limit is
discussed in detail in Section 6. Section 7 contains a discussion of
the eigenfunctions belonging to these eigenvalues, and Section 8
contains the conclusions. Some part of our analysis exploits general
results on the asymptotic behavior of the coefficients of Taylor
expansions. These results are presented in an Appendix.
2. Reformulation by conformal mapping
In this section, we collect results and notations from [35,36] that
will be used in later sections.
2.1. Problem formulation and rescaling
We consider a compact ionized domain D in the (x, y)-plane.
We assume that the net charge on the domain vanishes (i.e., it
contains the same number of electrons and positive ions). The
domain moves in an external field that far from the domain
asymptotically approaches
E
= −|E
|
ˆ
x. (4)
Here
ˆ
x is the unit vector in x-direction, and |E
|sets the scale of E
and thus of the potential ϕ. As length scale we take
R =
r
|D|
π
. (5)
where |D| is the area of D, which is known to be conserved. This
follows from the charge neutrality of the streamer since
0 =
Z
D
dx dy ·E =
Z
D
ds n(s) · E =
Z
D
ds v
n
, (6)
where in the last step we inserted Eq. (2) for the normal velocity of
the boundary. Since
Z
D
ds v
n
=
t
|D|, (7)
the area is conserved, irrespective of the precise charge distribu-
tion in the interior.
5
Also introducing the time scale R/|E
|, we
5
We remark that the argument is straight forward to generalize to three spatial
dimensions. Therefore the volume of a charge neutral object with surface velocity
v E
+
in three spatial dimensions is conserved as well.
rescale the basic equations to the dimensionless form
ϕ = 0, (x, y) 6∈ D (8)
v
n
= n · (ϕ)
+
(9)
ϕ
+
= n · (ϕ)
+
. (10)
The only remaining parameter in the rescaled problem is
= `/R. (11)
The boundary condition at infinity after rescaling takes the form
ϕ x + const for
p
x
2
+ y
2
. (12)
2.2. Conformal mapping
We now identify the physical (x, y)-plane with the closed
complex plane z = x + iy, and we introduce a conformal
map f , t) that maps the unit disk U
ω
in the ω-plane to the
complement of D in the z-plane, with ω = 0 being mapped on
z = ,
z = f , t) =
a
1
(t)
ω
+
ˆ
f (ω, t), a
1
(t) > 0. (13)
We further define a complex potential Φ, t) obeying
Re[Φ(ω, t)] = ϕ(f (ω, t)) for ω U
ω
. (14)
The boundary condition (12) and the Laplace Eq. (8) enforce the
form
Φ(ω, t) =
a
1
(t)
ω
+
ˆ
Φ(ω, t) (15)
with
ˆ
Φ being holomorphic for ω U
ω
.
The two boundary conditions (9) and (10) take the form
Re
t
f
ω∂
ω
f
= Re
ω∂
ω
Φ
|
ω
f |
2
for ω U
ω
, (16)
|
ω
f | Re[Φ] = Re[ω
ω
Φ] for ω U
ω
, (17)
which completes the reformulation of the moving boundary
problem (8)–(12) by conformal mapping.
We will restrict the analysis here to initial conditions
ˆ
f (ω, 0)
holomorphic in some domain U
0
0
U
ω
. In part II [43] of this paper
sequence, we will give evidence that analyticity on U
ω
is preserved
in time, though the distance of the domain of analyticity U
0
t
to
U
ω
shrinks with time. The streamer boundary D, which is the
image of boundary U
ω
under f , t), will turn out to be analytic
and therefore smooth. Similar analytic representations exist for the
entire class of 2-D Laplacian growth, with details depending on the
type of boundary condition, geometry and asymptotic conditions
at infinity. For the classic viscous fingering problem, Polubarinova-
Kochina [44] and Galin [45] use a representation that coincides
with the one given above in the unregularized case = 0.
2.3. Linear perturbation of moving circles
It is easily seen that Eqs. (16) and (17) allow for the simple
solution
f
(0)
(ω, t) =
1
ω
+
2t
1 +
,
Φ
(0)
(ω, t) =
1
ω
1
1 +
ω,
(18)
which in physical space describes circles of radius 1 moving with
constant velocity 2/(1 + ) in x direction. (We recall that the
radius was scaled to unity in Section 2.1.) We note that relaxing

S. Tanveer et al. / Physica D 238 (2009) 888–901 891
the analyticity conditions on f , t) on |ω| = 1, one can
obtain another set of uniformly translating solutions, as recently
discovered [46]. The present paper is restricted to perturbations of
the steady circle that retain the imposed analyticity of the streamer
shapes, and hence analyticity of f (as well as Φ) on |ω| = 1.
As the area is conserved (as shown in Section 2.1), the residue
a
1
= 1 does not change to linear order in the perturbation. We
therefore can use the ansatz
f (ω, t) = f
(0)
(ω, t) + η β(ω, t),
Φ(ω, t) = Φ
(0)
(ω, t) + η
2
1 +
χ(ω, t),
(19)
where η is a small parameter, and β, t), χ(ω, t) are holomorphic
in U
ω
. A first order expansion of Eqs. (16) and (17) in η yields the
following boundary conditions for the analytic functions β(ω, t)
and χ (ω, t) on |ω| = 1:
Re[ω∂
τ
β ω
ω
β] = Re[−ω
ω
χ],
2
Re

ω +
1
ω
ω
2
ω
β
= Re[ω∂
ω
χ +χ],
(20)
where we rescaled time as
τ =
2
1 +
t. (21)
Since the left and right sides of each of the two equations in (20)
are real parts of analytic functions and each is assumed a priori
continuous up to the boundary, they can differ everywhere in ω
by at most an imaginary constant. Evaluation at ω = 0 shows this
constant to be zero for the first of the two equations. Elimination
of χ results in the linear PDE:
L
β = 0 (22)
with the operator
L
=
2
ω
2
1) ω
ω
+
ω
ω
τ
+
τ
ω
. (23)
We note that L
is of similar structure as the operator resulting
from a linear stability analysis of translating circles in the context
of void electromigration [22,24]. The main difference here is the
occurrence of the mixed derivative
ω
ω
τ
.
2.4. Formulation of the eigenvalue problem
To motivate our formulation of the eigenvalue problem, we
note some results on the temporal evolution of infinitesimal
perturbations. In [36], the equation L
β = 0 was solved as an
initial value problem for the special value = 1. It was found
that any initial perturbation β, 0) holomorphic in U
0
U
ω
for
τ is exponentially convergent to some constant. This results
from the expansion
β(ω, τ ) =
X
n=0
g
n
β
(1)
λ
n
(ω) e
λ
n
τ
, (24)
with
λ
n
= n, n N
0
, for = 1. (25)
The coefficients g
n
and the eigenfunctions
6
β
(1)
λ
n
(ω) =
Z
ω
0
x dx
ω
2
x 1
x + 1
λ
n
(26)
6
Note that the exponent λ
n
in (26) is correct while +λ
n
in Eq. (4.20) in [36] is
a typo.
are determined by an expansion of (2 +ω
ω
(ω, 0) in powers of
(1 ω)/(1 +ω). For n > 0 the eigenfunctions (26) are singular at
ω = 1, though β(ω, τ ) is not. The expansion (24) is convergent
in a domain D
τ
expanding in time that eventually includes every
point in
¯
U
ω
\ {−1}. For large τ the region where the expansion is
invalid, shrinks to ω = 1 exponentially. This region is measured
by the new scale η
1
(ω, τ ) = (1 + ω)e
τ
, and the expansion (24)
is valid if η
1
is large. For η
1
O(1) the perturbation for τ
behaves as β, τ ) F
0
1
) + O(e
τ
) where F
0
is some analytic
function of its argument, depending on β, 0).
For an analytic initial condition on U
0
, with a lone branch
point singularity ω
s
in |ω| > 1 not on the positive real axis, the
emergence of this new scale near ω = 1 can be related to the
approach of this complex singularity towards 1 exponentially
in τ for large τ. Asymptotic arguments that will be presented in
part II [43] suggest that this behavior is generic for all > 0. The
analysis is based on the linear the evolution equations for b
k
, where
β(ω, τ ) =
X
k=0
b
k
k
.
For k e
τ
, we find the asymptotic relation
b
k
(1)
k
k
α
h ) exp
[
kf )
]
,
where
f ) = log
1 + Ce
τ
1 Ce
τ
, with C =
ω
s
+ 1
ω
s
1
.
For ω
s
6∈ (1, ), f ) stays finite and approaches 0 exponentially
in τ for large τ . If ω
s
(1, ), f ) increases monotonically
to for τ (0, τ
c
) where e
τ
c
= 1/C. For τ > τ
c
, f )
decreases monotonically and approaches 0 exponentially in τ as
τ . In either case, from the known relation between Taylor
series coefficients and the location of the closest singularity of an
analytic function (see the Appendix), it follows that f ) e
τ
as τ implies that β has a singularity approaching ω = 1
exponentially in τ for large τ . This feature is retained for any other
isolated initial singularities as well, though k
α
is replaced by a
more complicated dependence in k. Since the problem is linear, the
evolution of a distribution of initial singularities can be understood
from the linear superposition principle.
This suggests that for any > 0, as for = 1, β, τ ) has a
collapsing scale (1+ω)e
τ
, and an expansion of the type (24) cannot
be valid in this neighborhood of ω = 1.
Thus, in seeking an eigenfunction by substituting
β(ω, τ ) = β
()
λ
(ω) e
λτ
, (27)
into (22) and (23), it is appropriate to allow β
()
λ
to be singular at
ω = 1. Indeed, substituting the form (27) reduces Eqs. (22) and
(23) to the eigenvalue problem
L(, λ) β
()
λ
(ω) = 0, (28)
L(, λ) =
2
1
2
2
ω
+
(3ω
2
1)
2
1
ω
+λ(1 + + ω∂
ω
). (29)
Evidently this ODE has three regular singular points, namely ω = 0
and ω = ±1. The independent solutions at these points for > 0
are in leading order
β
()
λ
(ω)
ω
0
ω
2/
for ω 0,
(30)
β
()
λ
(ω)
(1 ω)
0
(1 ω)
1/ λ
for ω ±1,
(31)

892 S. Tanveer et al. / Physica D 238 (2009) 888–901
We require the eigenfunctions β
λ
to be solutions of (28) that are
analytic in ω = 0 and ω = 1. This is also the natural choice
from a physical point of view since it is the right half of the circle,
Re[ω] > 0, that corresponds to the physically interesting tip of
the streamer. In general, eigenfunctions cannot be expected to be
regular at all three points. Starting with a function regular at ω = 0,
we cannot generally require regularity at both points ω = ±1 by
adjusting the single parameter λ. As shown in Section 4.3, the only
eigenfunction regular at all three points is the trivial translation
mode
λ
0
= 0, β
()
0
(ω) = const. (32)
As noted above, an operator similar to L
(23) occurs in the
problem of void electromigration, see section 4.1.3 in [24]. Again
an eigenvalue analysis would yield a second order linear operator
with three singular points at ω = 0 and ±1 and therefore the
eigenmodes in general cannot be regular at all three singular
points. It is interesting to note that the authors [24] conclude
that their problem is unstable because the initial value problem
for large time is singular at ω = 1. In the current problem,
the solution [43] of the initial value problem is not singular at
ω = 1; the singularity of the eigenfunctions does not reflect
the true behavior of solution since, as has been pointed out earlier,
there is an anomalous contracting scale e
τ
(1 + ω) near the back
of the bubble. Whether or not there is an analogous contracting
scale for the void electromigration problem [24] remains an
interesting question. This anomalous scale shows up when the
limiting processes lim
ω→−1
and lim
τ →+∞
do not commute for the
solution of the initial value problem.
3. Discreteness of the spectrum
We define λ to be in the spectrum, if the linear operator L(, λ)
does not have a bounded inverse in the class of functions f that
are analytic in an arbitrary compact connected set V U
0
\ {−1}
that contains the whole line [0, 1]in its interior. λ is in the discrete
spectrum if (28) has a nonzero solution β
()
λ
(ω) that is analytic in
any such domain V. We now argue that if λ is not in the discrete
spectrum, then L(, λ) has a bounded inverse, i.e. there is only a
discrete spectrum in this problem.
To determine L
1
, we solve the equation
L(, λ)g = h (33)
for a given h analytic in V, imposing the condition that also
g is analytic in V. The solutions of the homogeneous equation
L(, λ)f = 0 that are regular at ω = 0 or ω = 1
will be denoted by f
1
(ω) or f
2
(ω), respectively. It follows from
Eqs. (30) and (31) that these functions are determined uniquely
up to a multiplicative constant. In the exceptional case where both
independent solutions are regular at ω = 1, λ belongs to the
discrete spectrum, see Section 5. A standard calculation shows that
Eq. (33) is solved by
g) =
1
C(, λ)
Z
ω
0
dω
0
G(ω, ω
0
) h
0
) + a
1
[h] f
1
(ω), (34)
where
G(ω, ω
0
) =
ω
0
2/
(1 ω
0
)
1/λ
(1 + ω
0
)
1/+λ
×
f
2
(ω)f
1
0
) f
1
(ω)f
2
0
)
, (35)
and the coefficient a
1
[h] is a functional of h
0
). C(λ, ) does not
vanish since otherwise the Wronskian f
1
ω
f
2
f
2
ω
f
1
vanishes
identically and λ is part of the discrete spectrum. It is easily seen
that Eqs. (34) and (35) render g) analytic in ω = 0, and this
condition eliminates any contribution of the form a
2
[h] f
2
(ω).
Analyticity at ω = 1 is enforced by a proper choice of a
1
[h].
To make the analysis explicit, in addition to f
2
(ω), we introduce
another solution to L[, λ]f = 0 by requiring
f
3
(ω) = (1 ω)
1/λ
ˆ
f
3
(ω), (36)
where
ˆ
f
3
(ω) is analytic at ω = 1. Using this form of f
3
(ω), we
exclude the case
1
λ Z
+
, that will be discussed later. Writing
f
1
(ω) as
f
1
(ω) = c
2
f
2
(ω) + c
3
f
3
(ω), (37)
we find that G, ω
0
) from Eq. (35) takes the form
G(ω, ω
0
) =
c
3
ω
0
2/
(1 + ω
0
)
1/+λ
×
"
f
2
(ω)
ˆ
f
3
0
)
1 ω
0
1 ω
λ1/
ˆ
f
3
(ω)f
2
0
)
#
.
Evidently the first part in the square brackets for ω 1 yields
a regular contribution to g(ω) from Eq. (34). The contribution to
R
G h that is singular in ω = 1 has the form
c
3
f
3
(ω)
Z
ω
0
dω
0
(1 ω
0
)
λ1/
H
0
),
where
H
0
) =
ω
0
2/
(1 + ω
0
)
1/+λ
f
2
0
) h
0
) (38)
is regular at ω
0
= 1. If Re λ
1
> 1, we can write
c
3
f
3
(ω)
Z
ω
0
dω
0
(1 ω
0
)
λ1/
H
0
)
= c
3
f
3
(ω)
Z
1
0
dω
0
(1 ω
0
)
λ1/
H
0
)
+c
3
ˆ
f
3
(ω)
Z
1
ω
dω
0
1 ω
0
1 ω
λ1/
H
0
). (39)
The second part is regular at ω = 1 and the singular first part is
canceled by the choice
a
1
[h] =
Z
1
0
dω
0
(1 ω
0
)
λ1/
H
0
). (40)
We note that this result is valid also for λ =
1
+ n, n N,
where f
3
(ω) instead of being of the form (36) shows a logarithmic
singularity.
If n > Re λ
1
> n 1, n N, we carry through n
subtractions of H
0
) at ω
0
= 1, defining
H
0
)
n
= H
0
)
n1
X
j=0
H
j
(1 ω
0
)
j
, (41)
so that
H
0
)
n
const (1 ω
0
)
n
. A short calculation shows that
the singular part of
R
G h is canceled by the choice
a
1
[h] =
Z
1
0
dω
0
(1 ω
0
)
λ1/
H
0
)
n
+
n1
X
j=0
H
j
λ
1
+ j + 1
. (42)
The expressions above clearly remain valid when
1
Re λ = n,
except when
1
λ = n, a positive integer.
When
1
λ = n is a positive integer, from well-known
theory [47] for regular singular points, instead of (31), the

Citations
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Journal ArticleDOI
S Sander Nijdam1, van de Fmjh Ferdi Wetering1, R Blanc2, R Blanc1  +3 moreInstitutions (3)
Abstract: Positive streamers are thought to propagate by photo-ionization; the parameters of photo-ionization depend on the nitrogen : oxygen ratio. Therefore we study streamers in nitrogen with 20%, 0.2% and 0.01% oxygen and in pure nitrogen as well as in pure oxygen and argon. Our new experimental set-up guarantees contamination of the pure gases to be well below 1 ppm. Streamers in oxygen are difficult to measure as they emit considerably less light in the sensitivity range of our fast ICCD camera than the other gases. Streamers in pure nitrogen and in all nitrogen–oxygen mixtures look generally similar, but become somewhat thinner and branch more with decreasing oxygen content. In pure nitrogen the streamers can branch so much that they resemble feathers. This feature is even more pronounced in pure argon, with approximately 10 2 hair tips cm −3 in the feathers at 200 mbar; this density can be interpreted as the free electron density creating avalanches towards the streamer stem. It is remarkable that the streamer velocity is essentially the same for similar voltage and pressure in all nitrogen–oxygen mixtures as well as in pure nitrogen, while the oxygen concentration and therefore the photo-ionization lengths vary by more than five orders of magnitude. Streamers in argon have essentially the same velocity as well. The physical similarity of streamers at different pressures is confirmed in all gases; the minimal diameters are smaller than in earlier measurements. S Online supplementary data available from stacks.iop.org/JPhysD/43/145204/mmedia (Some figures in this article are in colour only in the electronic version)

161 citations


Journal ArticleDOI
Abstract: Positive streamers are thought to propagate by photo-ionization whose parameters depend on the nitrogen:oxygen ratio. Therefore we study streamers in nitrogen with 20%, 0.2% and 0.01% oxygen and in pure nitrogen, as well as in pure oxygen and argon. Our new experimental set-up guarantees contamination of the pure gases to be well below 1 ppm. Streamers in oxygen are difficult to measure as they emit considerably less light in the sensitivity range of our fast ICCD camera than the other gasses. Streamers in pure nitrogen and in all nitrogen/oxygen mixtures look generally similar, but become somewhat thinner and branch more with decreasing oxygen content. In pure nitrogen the streamers can branch so much that they resemble feathers. This feature is even more pronounced in pure argon, with approximately 10^2 hair tips/cm^3 in the feathers at 200 mbar; this density could be interpreted as the free electron density creating avalanches towards the streamer stem. It is remarkable that the streamer velocity is essentially the same for similar voltage and pressure in all nitrogen/oxygen mixtures as well as in pure nitrogen, while the oxygen concentration and therefore the photo-ionization lengths vary by more than five orders of magnitude. Streamers in argon have essentially the same velocity as well. The physical similarity of streamers at different pressures is confirmed in all gases; the minimal diameters are smaller than in earlier measurements.

114 citations


Journal ArticleDOI
Alejandro Luque1, Ute Ebert2, Ute Ebert3Institutions (3)
TL;DR: The intrinsic stochastic particle noise triggers branching of positive streamers in air at atmospheric pressure, and it is concluded that the ratio of branching length to streamer diameter agrees within a factor of 2 with experimental measurements.
Abstract: Branching is an essential element of streamer discharge dynamics. We review the current state of theoretical understanding and recall that branching requires a finite perturbation. We argue that, in current laboratory experiments in ambient or artificial air, these perturbations can only be inherited from the initial state, or they can be due to intrinsic electron-density fluctuations owing to the discreteness of electrons. We incorporate these electron-density fluctuations into fully three-dimensional simulations of a positive streamer in air at standard temperature and pressure. We derive a quantitative estimate for the ratio of branching length to streamer diameter that agrees within a factor of 2 with experimental measurements. As branching without this noise would occur considerably later, if at all, we conclude that the intrinsic stochastic particle noise triggers branching of positive streamers in air at atmospheric pressure.

76 citations


Journal ArticleDOI
Ute Ebert1, Fabian Brau2, Gianne Derks3, Willem Hundsdorfer  +8 moreInstitutions (7)
Abstract: Streamer discharges determine the very first stage of sparks or lightning, and they govern the evolution of huge sprite discharges above thunderclouds as well as the operation of corona reactors in plasma technology. Streamers are nonlinear structures with multiple inner scales. After briefly reviewing basic observations, experiments and the microphysics, we start from density models for streamers, i.e. from reaction–drift–diffusion equations for charged-particle densities coupled to the Poisson equation of electrostatics, and focus on derivation and solution of moving boundary approximations for the density models. We recall that so-called negative streamers are linearly stable against branching (and we conjecture this for positive streamers as well), and that streamer groups in two dimensions are well approximated by the classical Saffman–Taylor finger of two fluid flow. We draw conclusions on streamer physics, and we identify open problems in the moving boundary approximations.

55 citations


Cites background from "A moving boundary problem motivated..."

  • ...Details can be found in [83] where the present figure appears as figure 1....

    [...]


DOI
03 Feb 2011
Abstract: Streamers are rapidly extending ionized fingers that can appear in gasses, liquids and solids. They are generated by high electric fields but can penetrate into areas where the background electric field is below the ionization threshold. Streamers occur in nature as a precursor to sparks and lightning, but also independently as sprites (large discharges high above thunderclouds) or St. Elmo’s fire. Their main applications are gas and water cleaning, ozone creation, particle charging and flow control. Streamers are very efficient in creating active chemical species as no energy is lost in heating of the background gas and surrounding materials. Furthermore, as streamers are the first phase of sparks, they are relevant for any application of sparks, e.g., in the ignition process in a combustion engine or a discharge lamp. Finally, streamers can occur in high voltage applications, like switch-gear. In this thesis, a number of aspects of the physics of streamers are investigated experimentally. In our study, we have created streamers by applying a high voltage pulse to a wire or sharp tip that is located 40 to 160 mm above a grounded plate. These experiments were conducted inside a vacuum chamber at various pressures between 25 and 1000 mbar, with various gasses and gas mixtures, most of high purity (up to less than 0.1 ppm contaminations). We create the voltage pulses by two different high voltage pulse sources. The C-supply can give pulses between 5 and 60 kV with a minimum risetime of about 15 ns and an exponential decay of varying duration. The newly built Blumlein pulser creates quasi-rectangular pulses with an amplitude between 20 and 35 kV, a duration of about 130 ns and a risetime of about 10 ns. Both pulse sources can produce pulses of positive and negative polarity but have primarily been used with positive polarity.First, the interaction of individual streamer channels and the streamer branching angles are analysed by stereo-photography. Then insight into the propagation mechanism of positive streamers (i.e., against the electron drift direction) is gained by changing the gas composition and the repetition frequency of voltage pulses. Finally, morphology, channel diameters, propagation velocities and spectra of laboratory streamer discharges in a variety of gasses and gas mixtures are studied. Some of these studies are used as a "simulation" of sprite discharges on earth as well as on other planets. Interaction and branching of streamers Pictures show that streamer or sprite discharge channels emerging from the same electrode sometimes seem to reconnect or merge even though their heads carry electric charge of the same polarity; one might therefore suspect that reconnections are an artifact of the two-dimensional projection in the pictures. We have used stereo-photography to investigate the full three-dimensional structure of such events. We analyse reconnection, possibly an electrostatic effect in which a late thin streamer reconnects to an earlier thick streamer channel, and merging, a suggested photo-ionization effect in which two simultaneously propagating streamer heads merge into one new streamer. We find that reconnections as defined above occur frequently. Merging on the other hand was only observed with a double tip electrode at a pressure of 25 mbar and a tip separation of 2 mm, i.e., for a reduced tip distance of p . d = 50 mmbar. In this case the full width at half maximum of the streamer channel is more than 10 times as large as the tip separation. We have also investigated streamer branching with the stereo-photography method and have found that the average branching angle of streamers under the conditions that were investigated is about 42° with a standard deviation of 12°. The role of photo- and background ionization in streamer propagation Positive streamers in air are thought to propagate against the electron drift direction by photo-ionization whose parameters depend on the nitrogen:oxygen ratio. Therefore we study streamers in nitrogen with 20%, 0.2% and 0.01% oxygen and in pure nitrogen and argon. Our new experimental set-up guarantees contamination to be below 0.1 ppm for our purest nitrogen. Streamers in pure nitrogen and in all nitrogen/oxygen mixtures look generally similar, but become thinner and branch more with decreasing oxygen content. In pure nitrogen the streamers can branch so much that they resemble feathers. This feature is even more pronounced in pure argon, with approximately 102 hair tips/cm3 in the feathers at 200 mbar; this density can be interpreted as the density of free electrons that create avalanches towards the streamer stem. It is remarkable that the streamer velocity is essentially the same for similar voltage and pressure in all nitrogen/oxygen mixtures as well as in pure nitrogen, while the oxygen concentration and therefore the photo-ionization lengths vary by more than five orders of magnitude. This is supported by recent modelling results byWormeester et al. in 2010. To study the effects of background ionization on streamers, we have used two methods: variation of pulse repetition frequency (0.01–10 Hz) and addition of about 9 parts per billion of radioactive 85Kr gas to pure nitrogen. We found that higher background ionization levels lead to smoother and thicker streamers. This is similar to the effect of increased photo-ionization close to the streamer tip, created by increasing the oxygen concentration. Again, we do not see any major effects on streamer properties, except that initiation probabilities go down significantly in pure nitrogen with low (0.01 Hz) repetition frequency. At 200 mbar, the estimated background ionization level from the 85Kr was about 4 ?? 105 cm-3, which corresponds to the theoretical level in non-radioactive gas at a pulse repetition frequency of about 1 Hz under similar conditions. This fits with the observed variations in streamer morphology as function of repetition frequency for both pure nitrogen and the nitrogen-krypton mixture. Furthermore, we have found that streamers do not follow the paths of streamers in preceding discharges for pulse repetition frequencies around 1 Hz. This can be explained by the combination of recombination and diffusion of ionization after a discharge pulse which nearly flattens any leftover ionization trail in about 1 second. Streamers in other gasses and streamer spectra In order to get more insight in positive streamer propagation, we have studied more than just nitrogen-oxygen mixtures. We have studied pure oxygen, argon, helium, hydrogen and carbon dioxide. Each of these gasses has different properties like ionization levels, excitation levels, cross sections and electronegativity. Furthermore, we have studied streamers in binary gas mixtures that simulate the atmospheres of Venus (CO2–N2) and Jupiter (H2–He). Streamers in these gasses, as well as in air are physically similar to large scale sprite discharges on the corresponding planets. Therefore, the results of our measurements can be used to better equip (space) missions that study sprites on earth and other planets and can help in the interpretation of the observations of these missions. For all gasses and mixtures, overall morphology, velocities, diameters and emission spectra have been investigated. We have found that it is possible to create streamers in all gasses. Streamer diameters are more or less the same for all gasses, except for pure helium and the Jupiter atmosphere where minimal streamers are respectively 3 and 5 times thicker than in the other gasses. The physical similarity between streamers at different pressures has been confirmed for all gasses that enabled us to measure streamer diameters; the minimal diameters in air and other nitrogen-oxygen mixtures are smaller than in earlier measurements. Streamer velocities are even more similar; for a given combination of pressure and pulse voltage all propagation velocities are within a factor 2. Streamer brightness on the other hand is very different for the different gas mixtures. Streamers are brightest in nitrogen-oxygen mixtures, nitrogen, argon and helium and dimmest in oxygen, CO2 and the venusian mixture. The difference between the brightest and dimmest gasses is about three to four orders of magnitude in the optical range. Streamer spectra from molecular gasses are characterised by molecular bands. In gasses containing a significant amount of nitrogen (including the venusian mixture), the nitrogen second positive system dominates the emission spectrum. In contrast, spark-like discharges in the same gasses are dominated by radiation from neutral and ionized atoms. Spectra in atomic gasses (argon and helium) are different: the argon spectrum contains mainly atomic argon lines, but the helium spectrum also contains many lines of impurities, while we have no indication that the gas purity is below specification. The reason for the many impurity lines in helium are the high excitation and ionization levels of helium compared to the impurities. These high levels (and low cross sections for electron-atom collisions at low energies) may also explain the large diameter of streamers in pure helium.

34 citations


Cites background from "A moving boundary problem motivated..."

  • ...These concepts are further elaborated in [46, 86, 119, 137, 208] and were discussed in section 2....

    [...]


References
More filters

Book
01 Jan 1902
TL;DR: The volume now gives a somewhat exhaustive account of the various ramifications of the subject, which are set out in an attractive manner and should become indispensable, not only as a textbook for advanced students, but as a work of reference to those whose aim is to extend the knowledge of analysis.
Abstract: This classic work has been a unique resource for thousands of mathematicians, scientists and engineers since its first appearance in 1902 Never out of print, its continuing value lies in its thorough and exhaustive treatment of special functions of mathematical physics and the analysis of differential equations from which they emerge The book also is of historical value as it was the first book in English to introduce the then modern methods of complex analysis This fifth edition preserves the style and content of the original, but it has been supplemented with more recent results and references where appropriate All the formulas have been checked and many corrections made A complete bibliographical search has been conducted to present the references in modern form for ease of use A new foreword by Professor SJ Patterson sketches the circumstances of the book's genesis and explains the reasons for its longevity A welcome addition to any mathematician's bookshelf, this will allow a whole new generation to experience the beauty contained in this text

8,955 citations


"A moving boundary problem motivated..." refers background in this paper

  • ...When 1 − λ = n is a positive integer, from well-known theory [47] for regular singular points, instead of (31), the...

    [...]

  • ...where the general solution near ω = 1 can be written as [47]...

    [...]


Book
01 Oct 1991
Abstract: 1. Introduction.- 1.1 What Is the Subject of Gas Discharge Physics.- 1.2 Typical Discharges in a Constant Electric Field.- 1.3 Classification of Discharges.- 1.4 Brief History of Electric Discharge Research.- 1.5 Organization of the Book. Bibliography.- 2. Drift, Energy and Diffusion of Charged Particles in Constant Fields.- 2.1 Drift of Electrons in a Weakly Ionized Gas.- 2.2 Conduction of Ionized Gas.- 2.3 Electron Energy.- 2.4 Diffusion of Electrons.- 2.5 Ions.- 2.6 Ambipolar Diffusion.- 2.7 Electric Current in Plasma in the Presence of Longitudinal Gradients of Charge Density.- 2.8 Hydrodynamic Description of Electrons.- 3. Interaction of Electrons in an Ionized Gas with Oscillating Electric Field and Electromagnetic Waves.- 3.1 The Motion of Electrons in Oscillating Fields.- 3.2 Electron Energy.- 3.3 Basic Equations of Electrodynamics of Continuous Media.- 3.4 High-Frequency Conductivity and Dielectric Permittivity of Plasma.- 3.5 Propagation of Electromagnetic, Waves in Plasmas.- 3.6 Total Reflection of Electromagnetic Waves from Plasma and Plasma Oscillations.- 4. Production and Decay of Charged Particles.- 4.1 Electron Impact Ionization in a Constant Field.- 4.2 Other Ionization Mechanisms.- 4.3 Bulk Recombination.- 4.4 Formation and Decay of Negative Ions.- 4.5 Diffusional Loss of Charges.- 4.6 Electron Emission from Solids.- 4.7 Multiplication of Charges in a Gas via Secondary Emission.- 5. Kinetic Equation for Electrons in a Weakly Ionized Gas Placed in an Electric Field.- 5.1 Description of Electron Processes in Terms of the Velocity Distribution Function.- 5.2 Formulation of the Kinetic Equation.- 5.3 Approximation for the Angular Dependence of the Distribution Function.- 5.4 Equation of the Electron Energy Spectrum.- 5.5 Validity Criteria for the Spectrum Equation.- 5.6 Comparison of Some Conclusions Implied by the Kinetic Equation with the Result of Elementary Theory.- 5.7 Stationary Spectrum of Electrons in a Field in the Case of only Elastic Losses.- 5.8 Numerical Results for Nitrogen and Air.- 5.9 Spatially Nonuniform Fields of Arbitrary Strength.- 6. Electric Probes.- 6.1 Introduction. Electric Circuit.- 6.2 Current-Voltage Characteristic of a Single Probe.- 6.3 Theoretical Foundations of Electronic Current Diagnostics of Rarefied Plasmas.- 6.4 Procedure for Measuring the Distribution Function.- 6.5 Ionic Current to a Probe in Rarefied Plasma.- 6.6 Vacuum Diode Current and Space-Charge Layer Close to a Charged Body.- 6.7 Double Probe.- 6.8 Probe in a High-Pressure Plasma.- 7. Breakdown of Gases in Fields of Various Frequency Ranges.- 7.1 Essential Characteristics of the Phenomenon.- 7.2 Breakdown and Triggering of Self-Sustained Discharge in a Constant Homogeneous Field at Moderately Large Product of Pressure and Discharge Gap Width.- 7.3 Breakdown in Microwave Fields and Interpretation of Experimental Data Using the Elementary Theory.- 7.4 Calculation of Ionization Frequencies and Breakdown Thresholds Using the Kinetic Equation.- 7.5 Optical Breakdown.- 7.6 Methods of Exciting an RF Field in a Discharge Volume.- 7.7 Breakdown in RF and Low-Frequency Ranges.- 8. Stable Glow Discharge.- 8.1 General Structure and Observable Features.- 8.2 Current-Voltage Characteristic of Discharge Between Electrodes.- 8.3 Dark Discharge and the Role Played by Space Charge in the Formation of the Cathode Layer.- 8.4 Cathode Layer.- 8.5 Transition Region Between the Cathode Layer and the Homogeneous Positive Column.- 8.6 Positive Column.- 8.7 Heating of the Gas and Its Effect on the Current-Voltage Characteristic.- 8.8 Electronegative Gas Plasma.- 8.9 Discharge in Fast Gas Flow.- 8.10 Anode Layer.- 9. Glow Discharge Instabilities and Their Consequences.- 9.1 Causes and Consequences of Instabilities.- 9.2 Quasisteady Parameters.- 9.3 Field and Electron Temperature Perturbations in the Case of Quasisteady-State Te.- 9.4 Thermal Instability.- 9.5 Attachment Instability.- 9.6 Some Other Frequently Encountered Destabilizing Mechanisms.- 9.7 Striations.- 9.8 Contraction of the Positive Column.- 10. Arc Discharge.- 10.1 Definition and Characteristic Features of Arc Discharge.- 10.2 Arc Types.- 10.3 Arc Initiation.- 10.4 Carbon Arc in Free Air.- 10.5 Hot Cathode Arc: Processes near the Cathode.- 10.6 Cathode Spots and Vacuum Arc.- 10.7 Anode Region.- 10.8 Low-Pressure Arc with Externally Heated Cathode.- 10.9 Positive Column of High-Pressure Arc (Experimental Data).- 10.10 Plasma Temperature and V - i Characteristic of High-Pressure Arc Columns.- 10.11 The Gap Between Electron and Gas Temperatures in "Equilibrium" Plasma.- 11. Suslainment and Production of Equilibrium Plasma by Fields in Various Frequency Ranges.- 11.1 Introduction. Energy Balance in Plasma.- 11.2 Arc Column in a Constant Field.- 11.3 Inductively Coupled Radio-Frequency Discharge.- 11.4 Discharge in Microwave Fields.- 11.5 Continuous Optical Discharges.- 11.6 Plasmatrons: Generators of Dense Low-Temperature Plasma.- 12. Spark and Corona Discharges.- 12.1 General Concepts.- 12.2 Individual Electron Avalanche.- 12.3 Concept of Streamers.- 12.4 Breakdown and Streamers in Electronegative Gases (Air) in Moderately Wide Gaps with a Uniform Field.- 12.5 Spark Channel.- 12.6 Corona Discharge.- 12.7 Models of Streamer Propagation.- 12.8 Breakdown in Long Air Gaps with Strongly Nonuniform Fields (Experimental Data).- 12.9 Leader Mechanism of Breakdown of Long Gaps.- 12.10 Return Wave (Return Stroke).- 12.11 Lightning.- 12.12 Negative Stepped Leader.- 13. Capacitively Coupled Radio-Frequency Discharge.- 13.1 Drift Oscillations of Electron Gas.- 13.2 Idealized Model of the Passage of High-Frequency Current Through a Long Plane Gap at Elevated Pressures.- 13.3 V - i Characteristic of Homogeneous Positive Columns.- 13.4 Two Forms of CCRF Discharge Realization and Constant Positive Potential of Space: Experiment.- 13.5 Electrical Processes in a Nonconducting Electrode Layer and the Mechanism of Closing the Circuit Current.- 13.6 Constant Positive Potential of the Weak-Current Discharge Plasma.- 13.7 High-Current Mode.- 13.8 The Structure of a Medium-Pressure Discharge: Results of Numerical Modeling.- 13.9 Normal Current Density in Weak-Current Mode and Limits on the Existence of this Mode.- 14. Discharges in High-Power CW CO2 Lasers.- 14.1 Principles of Operation of Electric-Discharge CO2 Lasers.- 14.2 Two Methods of Heat Removal from Lasers.- 14.3 Methods of Suppressing Instabilities.- 14.4 Organization of Large-Volume Discharges Involving Gas Pumping.- References.

4,174 citations


Reference BookDOI
Abstract: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool.

3,934 citations


"A moving boundary problem motivated..." refers background in this paper

  • ...(102) As is well known in asymptotics [48] the coefficients in the asymptotic relation (99) may jump when χ crosses a Stokes line emerging from a turning point....

    [...]


Journal ArticleDOI
Abstract: A variety of non-equilibrium growth processes are characterized by phase boundaries consisting of moving fingers, often with interesting secondary structures such as sidebranches. Familiar examples are dendrites, as seen in snowflake growth, and fluid fingers often formed in immiscible displacement. Such processes are characterized by a morphological instability which renders planar or circular shapes unstable, and by the competing stabilizing effect of surface tension. We survey recent theoretical work which elucidates how such systems arrive at their observed patterns. Emphasis is placed upon dendritic solidification, simple local models thereof, and the Saffman-Taylor finger in two-dimensional fluid flow, and relate these systems to their more complicated variants. We review the arguments that a general procedure for the analysis of such problems is to first find finger solutions of the governing equations without surface tension, then to incorporate surface tension in a non-perturbative manne...

612 citations


"A moving boundary problem motivated..." refers background or result in this paper

  • ...Similar results follow for a needle crystal [18] though in the latter case, convective instability of wave packets caused by significant normal speed along the parabolic front is believed to cause dendritic structures [1]....

    [...]

  • ...Themotion of interfaces in a Laplacian field is of general interest and has been a subject of intense study over many years (see for instance the reviews [1,2])....

    [...]


Journal ArticleDOI
Abstract: This review is an expository treatment of the displacement of one fluid by another in a two-dimensional geometry (a Hele-Shaw cell). The Saffman-Taylor equations modeling this system are discussed. They are simulated by random-walk techniques and studied by methods from complex analysis. The stability of the generated patterns (fingers) is studied by a WKB approximation and by complex analytic techniques. The primary conclusions reached are that (a) the fingers are linearly stable even at the highest velocities, (b) they are nonlinearly unstable against noise or an external perturbation, the critical amplitude for the noise being an exponential function of a power of the velocity for high velocities, (c) such exponentials seem to dominate high-velocity behavior, as can be seen from a WKB analysis, and (d) the results of the Saffman-Taylor equations disagree with experiments, apparently because they leave out film-flow phenomena.

506 citations


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