Streamer branching rationalized by conformal mapping techniques

0.18 MB·English

by  B. Meulenbroek

Streamer Branching rationalized by Conformal Mapping Techniques
Bernard Meulenbroek1, Andrea Rocco1 and Ute Ebert1,2
1CWI, P.O.Box 94079, 1090 GB Amsterdam, The Netherlands,
2Dept. Physics, TU Eindhoven, 5600 MB Eindhoven, The Netherlands
(February 2, 2008)
Spontaneousbranchingofdischargechannelsisfrequentlyobserved,butnotwellunderstood. We
4
recentlyproposedanewbranchingmechanismbasedonsimulationsofasimplecontinuousdischarge
0
0 model in high fields. We here present analytical results for such streamers in the Lozansky-Firsov
2 limit where they can be modelled as moving equipotential ionization fronts. This model can be
analyzed by conformal mapping techniqueswhich allow thereduction of thedynamical problem to
n
finite sets of nonlinear ordinary differential equations. Our solutions illustrate that branching is
a
generic for the intricate head dynamics of streamers in theLozansky-Firsov-limit.
J

1
2 When non-ionized matter is suddenly exposed to rich head dynamics that includes spontaneous branch-
strong fields, ionized regions can grow in the form of ing. Furthermore, our analytical solutions disprove the

] streamers. These are ionized and electrically screened reasoning of [15] by explicit counterexamples. Our ana-
h
channelswithrapidlypropagatingtips. Thetipregionis lytical methods are adapted from two fluid flow in Hele-
p
- a very active impact ionization region due to the self- Shaw cells [18–22]. But our explicit and exact solutions
m
generated local field enhancement. Streamers appear that amount to the evolution of “bubbles” in a dipole
s in early stages of atmospheric discharges like sparks or field [23], have not been reported in the hydrodynamics
a
sprite discharges[1,2],they alsoplay a prominentrolein literature either.
l
p numerous technical processes. It is commonly observed The relation between our previous numerical investi-
.
s that streamers branchspontaneously [3,4]. But how this gations [5,6] and our present analytical model is laid in
c branching is precisely determined by the underlying dis- twosteps. First,numericalsolutionsshowessentiallythe
i
s charge physics, is essentially not known. In recent work same evolution in the purely two-dimensional case as in
y
[5,6],wehavesuggestedabranchingmechanismfromfirst thethree-dimensionalcasewithassumedcylindergeome-
h
p principles. This work drew some attention [7,8], since try[5,6]. Becausethereisanelegantanalyticalapproach,
[ the proposed mechanism yields quantitative predictions we focus on the two-dimensional case. This has the ad-

for specific parameters, and since it is qualitatively dif- ditional advantage that such two-dimensional solutions
2
ferent from the older branching concept of the “dielec- ratherdirectlyapply to,e.g., dischargesin Corbinodiscs
v
2 tric breakdown model” [9–11]. This older concept actu- [24]. Second, we use the following simplifying approxi-
1 ally can be traced back to concepts of rare long-ranged mations for a Lozansky-Firsovstreamer: (1) the interior
1 (and hence stochastic) photo-ionization events probably ofthe streameris electrically completely screened;hence
5
first suggested in 1939 by Raether [12]. Therefore, it the electric potential ϕ is constant; hence the ionization
0
came as a surprise that we predicted streamer branch- front coincides with an equipotential line, (2) the width
3
0 inginafullydeterministic model. Sinceourevidencefor of the screening layer around the ionized body is much
/ thephenomenonwasmainlyfromnumericalsolutionsto- smaller than all other relevant length scales and in the
s
c getherwithaphysicalinterpretation,theaccuracyofour present study it is actually neglected, (3) the velocity of
i numerical scheme was challenged [13,14]. Furthermore, the ionization front v is determined by the local elec-
s
y someauthorshavearguedpreviously[15,16]thatinade- tric field; in the simplest case to be investigated here,
h terministic discharge model like ours, an initially convex it is simply taken to be proportional to the field at the
p
streamer head never could become locally concave, and boundary v = c ∇ϕ with some constant c (for the va-
:
v that hence the consecutive branching of the discharge lidity of the approximation, cf. [25,26]). Together with
Xi channel would be unphysical. ∇2ϕ = 0 in the non-ionized outer region and with fixed
Therefore in the present paper, we investigate the is- limiting values of the potential ϕ far from the streamer,
r
a sue by analytical means. We show that the convex-to- thisdefinesamovingboundaryproblemforthe interface
concave evolution of the streamer head with successive between ionized and non-ionized region. We assume the
branchingisgenericforstreamersintheLozansky-Firsov field far from the streamer to be constant as in our sim-
limit [5,6,17]. We define the Lozansky-Firsov-limit as ulations [6]. Such a constant far field can be mimicked
the stage of evolution where the streamer head is al- byplacingthestreamerbetweenthetwopolesofanelec-
most equipotential and surrounded by a thin electro- tric dipole where the distance between the poles is much
static screening layer. While in the original article [17], larger than the size of the streamer.
only simple steady state solutions with parabolic head When the electric field points into the x direction
shape are discussed, we will show here that a streamer and y parametrizes the transversal direction, our two-
in the Lozansky-Firsov limit actually can exhibit a very dimensional Lozansky-Firsov streamer in free flight in a
1
homogeneous electric field is approximated by: the Laplace equation (1) in a given region is equivalent
to finding a complex function Φ(z) that is analytical in
∇2ϕ(x,y)=0, outside the streamer, (1)
the same region and has real part ReΦ(z)=ϕ(x,y).
−∇ϕ(x,y)→E0xˆ, far outside the streamer, (2) (ii) A conformal map from the interior of the unit circle
ϕ(x,y)=0, inside the streamer, (3) totheexteriorofthestreameror“bubble”isconstructed.
Including the point at infinity, the region outside the
v =c∇ϕ , velocity of the boundary, (4)
bound. bound. streamer is simply connected and Riemann’s mapping
(cid:12)
where xˆ is the (cid:12)unit vector in the x direction, and we theorem applies; therefore the mapping exists. Since the
have chosen the gauge such that the potential inside the boundarymoves,the mapping is time dependent; we de-
streamer vanishes. The asymptote (2) implies that the note it with z =f (ω) where ω parametrizes the interior
t
total charge on the streamer vanishes; otherwise a con- oftheunitcircle|ω|<1. Thecompletemapcanbecom-
tribution ∝1/ x2+y2 has to be added on the r.h.s. of posed from a conformalmap ζ =h (ω) that deforms the
t
Eq. (2). p unitdisccontinuously,followedbytheinversionz =1/ζ.
SimilarmovingboundaryproblemsariseinHele-Shaw Since h (ω) is conformalon the unit disc, it is analytical
t
flow of two fluids with a large viscosity contrast [18,19]: and has a single zero which we choose to be at ω = 0.
Lozansky-Firsov streamers and viscous fingers on the Thereforef (ω)=1/h (ω)hasasinglepole∝ω−1 andis
t t
presentlevelofdescriptioncanbeidentifiedimmediately otherwise analytical. Rather than a pole decomposition
by equating the electric potential ϕ with the fluid pres- [20], we choose a Laurent expansion for f (ω)
t
sure p [25,26]. To such problems, powerful conformal
∞
mapping methods [20–22] can be applied. Most work
x+iy =z =f (ω)= a (t)ωk. (5)
with this method is concerned with viscous fingers in a t k
channel geometry, i.e., with boundary conditions on a kX=−1
lateral external boundary that cannot be realized in an
This expansion allows us to identify the exact dynami-
electric system. A few authors also study air bubbles
within a viscous fluid, or viscous droplets in air, mostly calsolutions (10)below. Taking a−1(t)as arealpositive
number makes the mapping unique, again according to
under the action of flow fields generated by one source
Riemann’s mapping theorem.
or one sink of pressure,i.e., by monopoles. On the other
(iii)Now the potentialΦˆ(ω) on the unit disc can be cal-
hand, the approximation (1)-(4) describes streamers in
culated explicitly. Since f (ω) is a conformal mapping,
free space between two electrodes as in [6]. With the t
the function Φ(z)is analyticalif and only if the function
asymptote (2), this is mathematically equivalent to air
Φˆ(ω) = Φ(f (ω)) is analytical. The asymptote of Φˆ(ω)
bubbles in a dipole field. This case has not been studied t
for ω →0 is determined by (2) and (5): for |x|,|z|→∞,
indetail. Itisknownthatanyellipsewiththemainaxes
we have ϕ(x,y) → −E x, hence Φ(z) → −E z, and
orientedparallelandperpendiculartothedirectionofthe 0 0
dipole is auniformly translatingsolutionofthis problem therefore with (5): Φˆ(ω) → −E0a−1(t)/ω for ω → 0.
This means that the pole of Φˆ(ω) at the origin of the
[27]. The time dependent solutions of [28] do not apply
unit disc ω = 0 corresponds to the dipole of Φ(z) at
to streamers since the boundary condition on the mov-
z → ±∞. This dipole generates the field and the inter-
ing interface is different. Refs. [23] and [29] study how
facial motion. In the remainder of the unit disc, there
andwhen cusps in the interfaces ofdroplets andbubbles
areno sourcesorsinksofpotential, henceΦˆ isanalytical
emerge when these are driven by multipole fields. But
there. Furthermore,attheboundaryofthe streamer,we
for bubbles in a dipole field, again only the steady state
haveϕ=0 from(3)or ReΦ=0, resp. The boundaryof
ellipse solutions are given [23].
the streamer maps onto the unit circle, so Re Φˆ(ω) = 0
In the present paper, we therefore apply conformal
for |ω| = 1. Using the asymptotics at ω → 0 and an-
mapping methods to the evolution of “bubbles” in a
alyticity in the remaining region, the unique and exact
dipole field in a Hele-Shaw experiment and proceed be-
solution for the potential is
yond the steady state ellipse solutions. We identify the
general structure of time dependent solutions of (1)-
1
(4). The analyticallyderivedexactsolutionsshowhow a Φˆ(ω)=E0a−1(t) ω− . (6)
(cid:18) ω(cid:19)
streamerheadcanbecomeflatter,concaveandbranchas
observednumerically[5,6]. Ratherthanapoledecompo-
(iv) The velocity v = c∇φ (4) determines the mo-
sition[20],wederiveadecompositionintoFouriermodes bound.
tionoftheinterface. Thisinterfaceisthetimedependent
of the circle and calculate an equation for the non-linear
mapf (ω)ofthe unit circleω =eiα parametrizedby the
dynamical coupling of their amplitudes. t
angle α ∈ [0,2π). Therefore Eq. (4) determines the dy-
In detail, this is done in the following steps:
namics of f eiα . According to [18,20], it is
(i)Thespatialcoordinatesareexpressedbythe complex t
(cid:0) (cid:1)
coordinate z = x+iy. According to standard complex
analysis,finding arealharmonicfunctionϕ(x,y) solving Re −i∂αft∗ eiα ∂tft eiα =cRe i∂αΦˆ eiα . (7)
h (cid:0) (cid:1) (cid:0) (cid:1)i h (cid:0) (cid:1)i
2
The problem (1)-(4) is symmetric under reflection on First, it is now easy to reproduce the uniformly pro-
the x-axis. We assume the solutions to have the same pagating ellipse solutions of [23,27]as the solutions with
symmetry. Therefore all ak(t) have to be real. The po- N =1: for|a1|=6 |a−1|, the equations reduce to ∂ta−1 =
sition (x,y)(α,t) of the point of the interface labelled 0 = ∂ta1 and ∂ta0 = 2E0c a−1/(a1 −a−1). These so-
by the angle α at time t can be read directly from the lutions correspond to ellipses whose principal radii are
Laurentexpansion(5) by insertingω =eiα; ithas essen- oriented along the axes. These radii maintain their val-
tially the form of a Fourier expansion of the unit circle ues rx,y = a−1±a1 (assuming a−1 > a1 > 0) and move
wherethecircleanditspositionarecreatedbythemodes with constant velocity v =−E c (r +r )/r . The
ellipse 0 x y y
k =−1 and 0: Lozansky-Firsov-parabola can be understood as limit
∞ ∞ cases of such uniformly propagating ellipses.
x(α,t)= a (t) coskα , y(α,t)= a (t) sinkα, In contrast to N ≤ 1, all solutions with N ≥ 2 have
k k
kX=−1 kX=−1 nontrivialdynamics. Itcanbetrackedbyintegratingthe
ak(t) real , a−1(t)>0 , 0≤α<2π . (8) N+2ordinarydifferentialequations(11)numericallyand
then plotting the boundaries (10) at consecutive times.
Substituting the mapping function (5) and the poten- Examples of such dynamics are shown in the figures.
tial (6) into the equation of motion for the mapping (7),
andassumingthe a (t) to be real,we obtainforthe evo-
k 2 2 2 2
lution of the amplitudes a (t):
k
x x x x
∞
′ ′
k ak′(t)∂tak(t) cos (k−k )α 1 1 1 1
k,kX′=−1 (cid:0) (cid:1)
=2E0ca−1(t) cosα. (9)
0 0 0 0
A closer investigation shows that this equation has
an important property: suppose that the streamer
boundary can be written initially as a finite series
N a (0) eikα, a (0) 6= 0. Then at all times t, the −1 −1 −1 −1
k=−1 k N
Pinterfaceisdescribedbythesamefinitenumberofmodes
N y 0.5 0 −0.5 y 0.5 0 −0.5 y 0.5 0 −0.5 y 0.5 0 −0.5
z(α,t)= a (t)eikα, (10)
k
kX=−1
i.e., the a (t) with k > N stay identical to zero at all
k
times t > 0. Sorting the terms in (9) by coefficients of
coskα, the equation can be recast into N +2 ordinary
differential equations for the N +2 functions ak(t) FIG. 1. Upper panel: evolution of the interface in equal
time steps up totime t=0.1/(E0c) with initial condition
N−m a) z0(α,0)=e−iα+0.6·eiα−0.08·e2iα,
(k+m)ak+m ∂tak+k ak ∂tak+m b) z(α,0)=z0(α,0)−5·10−3·e8iα
kX=−1h i c) z(α,0)=z0(α,0)+3·10−3·e8iα
=2E0ca−1 δm,1 for m=0,...,N +1, (11) d) z(α,0)=z0(α,0)−4.5·10−7·e30iα,
and lower panel: zoom into theunstable head of Fig. d.
where δ is the Kronecker symbol. Eq. (11) is equiv-
m,1
alent to a matrix equation of the form A({a (t)}) ·
k Fig.1showsfourcasesofthe upwardmotionofa con-
∂t a−1(t),...,aN(t) = 0,2E0ca−1(t),0,...,0 ,where ically shaped streamer in equal time steps. The initial
(cid:16) (cid:17) (cid:16) (cid:17)
the matrix A depends linearly on the {a (t)}. conditions are almost identical. On the leftmost figure,
k
Eqs. (10) and (11) identify large classes of analytical an ellipse is corrected only by a mode e2iα to create the
solutions with arbitrary fixed N. These solutions reduce conical shape. This shape with N = 2 eventually de-
the dynamical moving boundary problem in two spatial velops a concave tip, but only after much longer times
dimensions of Eqs. (1)–(4) exactly to a finite set of ordi- than shown in the figure. In the other figures this coni-
nary differential equations for the nonlinear coupling of cal shape is perturbed initially by a minor perturbation
the amplitudes a (t) of modes eikα, 0 ≤ α < 2π. These with wavenumber 8 or 30, corresponding to N = 8 and
k
equations are easy to integrate numerically or for small 30 in (10) and (11). The amplitude of the perturbation
N evenanalytically. Wewillusethisformtodiscussnow is chosen such that a cusp develops at time 0.1/(E c).
0
generic solutions of Eqs. (1)–(4) as the simplest approx- Depending on the sign of the amplitude, the cusp devel-
imation of a streamer in the Lozansky-Firsovlimit. ops on or off axis — where we stress that we are not
3
interested in the cusp itself, but in the earlier stages of
evolution. Notethatourreductionofthemovingbound-
ary problem to the set of ordinary differential equations
(11) assures that the evolving shape is a true solution of
the problem (1)–(4). Figs. 1b and 1d demonstrate that [1] V.P.Pasko,U.S.Inan,T.F.Bell,Geophys.Res.Lett.25,
spontaneous branching is a possible solution.
2123 (1998); E.A. Gerken, U.S. Inan, C.P. Barrington-
Leigh, Geophys. Res. Lett. 27, 2637 (2000); V.P. Pasko
et al.,Nature416, 152 (2002).
[2] E.R. Williams, Physics Today 54, No. 11, 41-47 (2001).
[3] W.J. Yi, P.F.Williams, J. Phys.D:Appl.Phys. 35, 205
(2002).
[4] E.M. van Veldhuizen, W.R. Rutgers, J. Phys. D: Appl.
Phys. 35, 2169 (2002).
[5] M.Array´as,U.Ebert,W.Hundsdorfer,Phys.Rev.Lett.
88, 174502 (2002).
[6] A. Rocco, U. Ebert, W. Hundsdorfer, Phys. Rev. E 66,
035102(R) (2002).
[7] P. Ball, Nature(April2002),
http://www.nature.com/nsu/020408/020408-4.html.
FIG.2. Evolution of thetip of an elongated “streamer” in [8] P.R. Minkel, Phys. Rev.Focus (April2002),
equal time steps up to time t = 0.1/(E0c); initial condition http://focus.aps.org/v9/st19.html.
z(α,0)=e−iα+0.9·eiα−0.03·e2iα−1.2·10−5·e12iα. [9] L. Niemeyer, L. Pietronero, H.J. Wiesman, Phys. Rev.
Lett. 52, 1033 (1984).
InFig.2 the ionizedbody is longerstretchedandonly [10] A.D.O. Bawagan, Chem Phys.Lett. 281 325 (1997).
the tip is shown, again at 6 equidistant time steps. The [11] V.P.Pasko,U.S.Inan,T.F.Bell,Geophys.Res.Lett.28,
streamerbecomes slowerwhen the headbecomes flatter, 3821 (2001).
[12] H. Raether, Z. Phys.112, 464 (1939) (in German).
since the electric field then diminishes. Eventually, the
[13] A.A. Kulikovsky,Phys. Rev.Lett. 89, 229401 (2002).
head becomes concave and “branches”.
[14] U. Ebert, W. Hundsdorfer, Phys. Rev. Lett. 89, 229402
In summary, the solutions of the moving boundary
(2002)
problem (1)–(4) demonstrate the onset of branching
[15] S.V. Pancheshnyi, A.Yu. Starikovskii, J. Phys. D: Appl.
within a purely deterministic model. They show a high Phys. 34, 248 (2001).
sensitivity to minor deviations of the initial conditions. [16] A.A.Kulikovsky,J.Phys.D:Appl.Phys.34,251(2001).
Astreamerinthe Lozansky-Firsov-limitistherefore also [17] E.D.LozanskyandO.B.Firsov,J.Phys.D:Appl.Phys.
very sensitive to physical perturbations during the evo- 6, 976 (1973).
lution, and simulations in this limit are just as sensitive [18] P.Ya.Polubarinova-Kochina,Dokl.Akad.Nauk.S.S.S.R.
to small numerical errors. But perturbations during the 47, 254-257 (1945).
evolution are not necessary for branching. [19] G. Taylor and P.G. Saffman, Q. J. Mech. Appl. Math.
12. 146 (1959).
Our analysis applies to streamers in the Lozansky-
[20] D. Bensimon, L.P. Kadanoff, S. Liang, B.I. Shraiman,
Firsov-limit, i.e., to almost equipotential streamers that
and C. Tang, Rev.Mod. Phys. 58, 977 (1986).
are surrounded by a very thin electrical screening layer. [21] D.BensimonandP.Pelc´e,Phys.Rev.A33,4477(1986).
Thislimitisapproachedinourprevioussimulations[5,6]. [22] K.V. McCloud and J.V Maher, Phys. Rep. 260, 139
These results raise the following questions that are (1995).
presently under investigation: 1) When does a streamer [23] V.M. Entov, P.I. Etingof and D.Ya. Kleinbock, Eur. J.
reach this Lozansky-Firsov-limit that then generically Appl. Math. 4, 97 (1993).
leads to branching? 2) The formation of cusps should [24] G. Schwarz, C. Lehmann, and E. Sch¨oll, Phys. Rev. B
be suppressed by some microscopic stabilization mecha- 61, 10194 (2000); G. Schwarz, E. Sch¨oll, V. Nov´ak, and
nism. Is the electric screening length discussed in [5,30] W. Prettl, Physica E 12, 182 (2002).
[25] U. Ebert, W. van Saarloos and C. Caroli, Phys. Rev.
sufficient to supply this mechanism? 3) If this stabiliza-
Lett.77,4178(1996);andPhys.Rev.E55,1530(1997).
tionistakenintoaccount,cananinterfacialmodelrepro-
[26] U. Ebert and M. Array´as, p. 270 – 282 in: Coherent
duce numerical and physical streamer branching quanti-
Structures inComplexSystems(eds.: Reguera,D.etal.),
tavely? 4) How can the motion of the back end of the Lecture Notes in Physics 567 (Springer,Berlin 2001).
streamer be modelled appropriately (rather than assum- [27] S. Tanveer, Phys.Fluids 29, 3537 (1986).
ing the velocity law v ∝ ∇ϕ (4) everywhere)? How can [28] Q. Nie, S. Tanveer,Phys. Fluids 7, 1292 (1995).
it be incorporated into the present analysis? [29] Q.Nie,F.-R.Tian,SIAMJ.Appl.Math.62,385(2001).
Acknowledgment: B.M. was supported by CWI [30] M. Array´as, U. Ebert, arXiv.org/abs/nlin.PS/0307039,
Amsterdam, and A.R. by the Dutch Research School Phys. Rev.E [to appear].
CPS,theDutchPhysicsFundingAgencyFOMandCWI.
4