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Leader-streamer model of blue jets - phenomenon of lightning type in the upper atmosphere above thundercloud

Yu.P. Raizer1, G.M. Milikh2 and M.N. Shneider3

1. The Institute for Problems in Mechanics, Russian Acad. Sci

2. Maryland University, USA

3. Princeton University, USA

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Blue jets (BJ) were discovered by Wescott, Sentman et al., 1994 , USA

Scheme of BJ observation; Pasko et al., 2002, Puerto-Rico

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Video recording BJ by Pasko et al., 2002

Time between frames = 0.033 s; BJ lifetime ~ 0.3 s

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Fractal streamer model of BJ Petrov and Petrova, 1999; Pasko and George 2002. •The average field necessary for

positive streamer is ЕS0≈5 kV/cm for normal air density N0

• There is no time in the model

• But electrons disappear due to attachment to O2, for τа~N-2 ~10-5s

at h =18 km whereas BJ should be supplied by current during 0.3 с

• Similarity law is supposed:

Δ≈7.2km ЕS ~N ~ e-h/Δ,

• It is impossible to manage by means of the streamers only, without participation of a leader

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Leader-streamer model of blue jets Raizer, Y.P., G.M. Milikh, and M.N. Shneider; Geophys. Res. Lett., 2006, 33, L23801; J. Atmos. & Solar-Terr Phys., 2007, 69, 925-938.

-transfers the high potential U~30-50 MV outside the cloud up to h ~ 30 km, - here τа ~ 10-2 s >> τа(18km) and plasma conductivity is kept much longer,

- streamers require field ЕS << ЕS(h=18 km).

Effect of a leader:

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BJ origination inside thundercloud

Scheme of the thundercloudcharges

Distribution ofthe cloud and“bi-leader”potentials alongan altitude h

• In non-conductive cloud BJ leader appears together with another one of the opposite polarity (bi-leader)

∙ BJ bi-leader appears in the point В where a field E= -dUc/dh is mах

•ВJ leader propagates upward from the point B

• UR = 150 МV for the cloud charge QC = 50 C and its radius RC =3 km

h

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Condition of “unlimited” growth of upward streamer in the exponential atmosphere

• Field necessary for streamer is /0

hSS eEE

•Streamer, born at altitude hL, can grow up to “infinity” if its source (leader tip) has potential

L

L

h

hSSLSLS eEEEdhEU /

0,

• Streamer can run up to “infinity” from the altitude

• Unrealistically high potential U = 350 MV is required for streamer to runaway from the height h=18 km

km2.7/0 heNN

)/ln( 0 UEh SL

• hL= 25-30 km for real U ~ 30 – 50 MV

(similarity law)

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Modeling growth of the individual streamer belonging to BJ

The streamer propagates in the exponential self-consistent field, created with participation of all BJ streamers

•Equations of a streamer channel, as long line without self-inductivity:

U(x,t) –potential, I(x,t) – current, q(x,t) – linear charge, R1(x,t) , C1 ≈ const – linear resistance and capacity, U0(x)=U0(0) exp (-x/Δ) –the self-consisted (“external”) potential

1 1 00, , ( )q I U

IR q C U Ut x x

The set of engaging equations:

121( , ) ( ) ( ) ( , )m e eR x t r x e x n x t

2 20~ 2.7 10 / ( ), m /V se N N x

rm - streamer radius, μe-electron mobility, ne- electron density, C1 ≈ 7.9 pF/m

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• Equations of electron kinetics; the approximate solution:

t = t - ts(x)

d /d v [ ( )]s s t sx t U x

=

• Equations for the streamer velocity vS , tip radius rm and electron density behind tip ne0 , following from the streamer theory :

vS = 5.3 104Ut m/s, Ut=Ut – U0(xS)[kV]; vS=0 for Ut < 5kV

rm= 3 10-5 Ut (N0 /N) m, ne0 = 1.0 1020( N / N0)2 m-3

)( St xUU

• Equation of streamer tip motion (хS = streamer length):

/0

/0

e

1 (1 e )

a

a

te

e te a

nn

n

8 2 13 307.7 10 ( / ) s, 2.8 10 m /sa N N

ts(x) –the moment when the streamer tip passes the point xτa – the characteristic electron attachment time, β – the coefficient of electron-ion recombination

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Set of equations is reduced to the equation of non-linear diffusion of potential

1 1

2 200

0

1χ , ,

t

( , )( , ) 190 ( ) ( ) m /s

( ) ( )e

te

U U

x x R C

N n x tx t U x U x

N x n x

Boundary condition at x=xs: the field at the channel front should maintain current I(xs) = q(xs)vs . Hence:

kV280( )

mtx x o

U N

x N

Boundary condition at x=0: U(0)=U0(0) = the leader tip potential. Initial condition: the short streamer “germ” at x=0 is taken

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Results of computations Checking the similarity law ES/N=const, for the uniform atmosphere

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Growth of a streamer in the exponential self-consistent fieldwith Е/N= 1.28 10 – 23 kV m2 exceeding slightly critical one ES/N

Streamer was born (х=0) at the height 25 km in the field 11 kV/m, ( ES = 10.4 kV/m), U0(0)=80 MV

Distributions of potential along the streamer at the instants when its lengths are L = 5, 10…50 km.

Distributions of the electron density

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Distributions of the current

• Essential: although the whole charge of streamer is + its back part is –• Cause: impossibility to supply the long streamer by + from outside due to loss of conductivity of its back part • But the space charge in the + streamer zone should be + on the average• It is achieved by continuous emission of new + streamers in the leader tip

Distributions of the charge per unit length

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Diagram illustrating overlap of the streamer segments with different states, in the streamer zone of a positive leader and blue jets Schemes show state in streamers emitted at the leader head (x=0) for time t1─t0 =LS / vS. t1 is the moment when the streamer, born at moment t0 , reaches the leading front of the streamer zone (x=LS)

Segments of (+) and (–) charge per unit length of a streamer

Segments of high and lowconductivity

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Thus a leader is an indispensable part of blue jets:

• it transfers upward the high potential of the thundercloud • it emits streamers with high frequency and they participate in the generation of the necessary self-consistent field

• owing to presence of many streamers of different age + segments of the young streamers neutralize (with excess) ─ segments of the old streamers

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The main problems for theory • To describe mathematically how the self-consistent field is formed in the streamer zones of a laboratory leader and blue jets

• What does determine the blue jets velocity v?

• One can assume that the observing v ~ 100 km/s is the known minimal streamer velocity (it should not depend on N). Streamers of the streamer zone of a laboratory leader are weak and have such velocity. This problem is closely connected with the unresolved first one.

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AppendixSubstantiation of the above-presented streamer modeling by comparison of

red sprites computations with observations Streamer model of red sprites ( Raizer, Milikh and Shneider, 1998 )

• Red sprites are usually associated with positive cloud-to-ground lightning

• Cloud charge Q= ─70C at h= 5km and its images generate U0 (5km) = ─240MV andthe field E=0.5V/cm at hi =80km (N=4 1014 cm-3) sufficient for ionization

• Plasma patches generated by electromagnetic pulses of intercloudlightning could serve as streamer nucleoli (Valdivia, Milikh et al, 1997)

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The recent high-speed images, Cummer et al., 2005

( The first scientific study – Sentman, Wescott et al.,1994)

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Results of computation of the red sprite’s downward streamer

Distributions of potential along streamerat different instants. Dashed curve U0– potential of theexternal field

Distributions of currentDistributions of linear charge

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Evolution of streamer velocity vS

and length l

Distributions of electron density at different instants

• Computed vSmax= 1.2 109 cm/s, maximal length lmax = 32 km, altitude and time of the streamer stop, h =48 km and 6.7 ms, are in agreement with results of the video recording