1229734

Etude de l’activité électrique des systèmes orageux à
l’aide du modèle Méso-NH
Christelle Barthe
To cite this version:
Christelle Barthe. Etude de l’activité électrique des systèmes orageux à l’aide du modèle Méso-NH.
Physique [physics]. Université Paul Sabatier - Toulouse III, 2005. Français. �tel-00011490�
HAL Id: tel-00011490
https://tel.archives-ouvertes.fr/tel-00011490
Submitted on 30 Jan 2006
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_ `C`_
B6
+
+
+ +
+
_ __ _ _ _ +_
_ _ _ _
+ +
+
_ __ _ _ _ +_
_ _ _ _
+ +
+
_ __ _ _ _ +_
_ _ _ _
+ + + + +
+ + + + +
+ + + + +
distribution
des charges
décharge
préliminaire
t=0
1 ms
+
+
+
+ +
+
_ __ _ _ _ +_
_ __ _ _
_
+ + _+ _+ +
_ _
_ _
_ _
_ _
+
+
+
processus
J et K
+
+
6 6 B
+
+
+
+
+
20.10 ms
20.15 ms
20.20 ms
+
+ +
+
_ __ _ _ _ +_
_ _ _ _
+ + + _
+ +
_
_ __
_ _
_
+
+ +
+
_ __ _ _ _ +_
_ _ _ _
+ + + _+ +
_ _
_
_ _
_ _
_ _
+
+ +
+
_ __ _ _ _ +_
_ _ _ _
62.00 ms
62.05 ms
traceur
obscur
+ + + + +
+ +
+ +
+ +
+
+ +
+ +
+
+
#&% 23-!*?/* 2=%h#7%\9:"23-b4!2"5B"E0;*4+/
+
+ + + _+ +
_
_ _
_ _
+ +
+ +
+
61.00 ms
!!"
+ +
+
_ __ _ _ _ +_
_ _ _ _
+
+ +
+
++
+ +
premier
arc en retour
++
++
+
60.00 ms
40.00 ms
+
1.20 ms
1.15 ms
20.00 ms
+
+
+
+ +
+
_ __ _ _ _ +_
_ __ _ _
_
+ + _
+ _+ +
_
+ +
+
_ __ _ _ _ +_
_ __ _ _
_
+ + + + +
_ _
+
+
+
+
+
+ +
+
_ __ _ _ _ +_
_ __ __ _
+ + _+ _+ +
_ _
_ __
__ _
+ +
+
_ __ _ _ _ +_
_ _ _ _
+ + _ + _+ +
_
+ + + + +
+
+
+
_ __ _ _ _ +_
_ _ _ _
__
+ + + _+ +
__
1.10 ms
d’attachement
+ +
+ +
+
_ __ _ _ _ +_
_ _ _ _
traceur
par pas
+ +
+
_ __ _ _ _ +_
_ __ _ _
_
+ + _+ _+ +
_ _
_ _
_ _
_ _ processus
19.00 ms
+
+
+
+
arc en retour
subséquent
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1
RY COLX =
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4
ã v
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x
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−2 × 10 C
T ≤ T CR
( +64B8:9NB$A7a=( A7s( e ,( KNB$A4BK4;,a=A7!O77(RKBOA"+7D58 ,/45(OqZ2NBO.64;(8iBO6KA" "( [email protected]+<45(Ra=A" 7D;4yxz©H!=,A2/( δq
8 !O4545D;7D;!=.{aOA7 "D54#>.(*D5a=($q45B 8:9BOA7a=(6*8:9NBO.a=*([*\³(µKA7D;01(67!O,ª4BF!=A"01(R
(
7D
|[email protected]
2 × 10−13 C
H
7D T > T CR
δqN I =
−13
T ≤ T CR
−2 × 10 C
}P(64BR0 0 (P%B e*!=.]qN45!OA7`2(*m8*!=454;D5"D5!=..( D;a=(J#a=4B$8 (-KA"D501BOD5A"(F
(
7D
|[email protected]
2 × 10−14 C
H
7D T > T CR
δqN I =
−14
−2 × 10
C
·=~
T ≤ T CR
" )+ ,/.- 0
}-BO.±8 (J+7+7(mKBOACBO0 *+7A7D;CB$+7D5!=.]q45(*A7*7,4p+:B$+7±2/(m45BOYL!=ACB$+7!=D5A"({2( Nf Of¶/fÍ:¶¤bprut'Ov$ws| $Na=,A"(m~x=
7!O.G+{0,4p+CD5K/45D5*KNBOA±,. BO8*+7( ,A±EG,DZ2/ KL( .22/(P4B-+7( 0 KZ*ACB$+7,A7($q2/,8 [email protected]+7( .G,( .c(iBO,145D5EG,D;2(OqO2(
4B-+CBOD54;45(S2(*KBOA"+7D58 ,/45( ¤(J+2/(4;( ,AID;+7( 7"(2/(8:9G,/+7(b f Of¶/fÍ:¶:q/rit=v ã w¦xGW(m"D5a=./(m(J+{4¦\B$01K4;D;+7,2(
2(64B8:9BOA7a=(< 8:9BO.a=*(["[email protected]+2/!=..*mKBOA6
Di,s 2 |vg − vi,s |
= fT (T, LW C)M in 10, 5
D0
v0
",4;+7(¤2]\³,/.(¤[email protected]+C(*A7KL!=4B$+7D5!=.P2(4BS+:B$Y45(2/(SH
k BOBO9NB$79Db
"
T
δqN
I
#
|y~x~O¥@
!#
A7
h +J ZqOrut=t ã 7.VfO¡/ÍukJn¢n>q
fT (T, LW C)
~O¥=¥O¥$wyxZQ{(J+7+C(+CBOY4;([2!O..(45B18:9NBOA"a=(*8:9NBO.a=*(R( .!O.8*+7D5!=.2/(F4B +C(*01KL A7B$+C,A"(F(J+2,8*[email protected]+C( .G,
( . ( BO,R45D;E,/D52($x
( "+¤4;(S2D5BO013J+CA"({2(`45BPK45,/KZ(J+CDp+C(SKNB$A"+CD;8 ,4;(SEG,DD;.G+7( AID;( [email protected]+2NB$.4BP8*!=454;D5"D5!=.
Di,s
|aO4BO8*(!=,.( D;a=(uµq ( +45BFID;+C(*7"([2(68:9G,/+7([2,a=A" 7D;4](*+
8*( 454;(-2(64BRK4;,`KL(*+CDp+C(KNB$A"+CD;8 ,4;(Ox
W( `I$BO45(*,A7{2( vg(J+
[email protected]+ªA"( 7KL( 8J+CDpI=( 0 ( [email protected]+<ru¥ −4 0 (*+mvvRi,s0x_ −1 xNWXB 4;D50 D;+CB$+CD;!=. ru¥R( "+ª2/,(
R4BI$BO45(*,AªBOKKD
BOA70(*.G+7(6v20 (6CB$+7,ACB$+7D5!=.§KL!=,Aª4;( `a=A"!=ª8 A7D;"+CBO,§2/(<./( D5aO(1/b .VfOj:Í:¶/fOn¢nskJlSfOnpo5q]rit 'Ov$w¦x
XH~xL 9A3 N[$ .m.0(*)&@J ,[$ [13AP?=
-"L A2 "%A3(X& #?=8T H% &'HA21 LAYDIw
(* H%$PN* #= >2?[= 13AP?=B(*) 1 #.,J#$-[13"%$- 21(" ?=c1 #=E" #=
7
7
sA 7
_
WXB6+CBOY4;(m2/(kHBOBO9NBO"9DN.(m2!=../(45BI$BO4;( ,A¤2(m4B8:9NB$A7a=(m 8:9NB$.a= (EG,(KL!=,A¤2(*+7( 0 KZ*ACB$+7,A7(*
BO4;4[email protected]+-2/(1¥ ◦ Q c#>=¥ ◦ QPx©s!=,A64Ba=BO010 (F2/( +7( 0 KZ*ACB$+7,A7(F8 !=0 KA7D;7(R(*.G+7A7(#&O¥ ◦ Q«(*+-# ã ¥ ◦ QPq]45B
I$BO45(*,A`2(
8*!=A7A"( [email protected]+C(<( "+m8*( 454;(6D57",(62(64B+:B$Y45(KL!=,Aª,.(6+C(*01KL A7B$+C,/A7(2(
fT (T, LW C)
#&O¥ ◦ QP:x uD]45B+C( 0 KL ACB+C,A"(( "+mD;./ A"D5( ,/A7(6F# ã ¥ ◦ QPqN4B8:9NB$A7a=(6 8:9BO.a=*([(*"+.,/4545($x
·O
" &$& )+ ,-/0
Q{(*+"+C(cKBOACBO0 *+7A7D;CB$+7D5!=.V(*"+YBO7*(c7,/A[4;( 6+CACBiI$BO,¬2/(É/fOÈ=fOj"fOl¡zk§kJlfOnpo{brut=vO$wyxXWXB8:9NBOA"a=(
+CA7BO. A7*(-BO,1a=A7*7D;4N45!=A"7EG,]\_D54( [email protected]+CA"(P(*.c8 !O4545D;7D;!=. BiI=( 8,.(KNBOA"+7D58*,45(ª2(ma=4B$8 (ª!=,c2/(P4B6.(*D5a=(ª( +
2!=./. (<KNBOA
|y~x®~ru
G
m
n
δqN I = kq Di (∆vgi ) (LW C − LW Ccrit )f (∆T )
(*"+,.8*!(Rr 8 D5(*[email protected]+-2(<KA7!OKZ!=A+CD;!=..NBO4;D;+7- [email protected]$4X 'O/qN(J+-4;( ª(J[email protected]+7
(*+ [email protected]+2(F4¦\³!=A"2A7(
kq
m n
2( ã (*+<A7( "KZ(*8*+7D;I=(*01(*[email protected]+ix G ( "+<(µ/K/A7D50 R( .¬¦QPx
(*"+<45(R8*[email protected]+C(*., ( . (iBO,¬45D;E,/D52(
δqN I
LW Ccrit
8 A"D;+7D5EG,(m(*+S(*"+ $/¥/xprPax_0 −3 x
( +S45(m2D5BO013J+CA"(2(4B<KNBOA"+7D58*,45(ª2(a=4BO8*(.G,NB$a=( ,"( |(*.0
(*+
(*"+<4BcID;+7( "7( 2]\³D;01KBO8*+D
A"( i45B$+CDpI=(|(*. 0x_ −1 P(*.G+7A7( 45(RaOA7 "D54(J+<4BKNBOA+CD;8 ,4;(R2( a=4BO8*(Ox
WXB∆v
!=.gi8*+7D5!=.
( "+m,.KL!=4;?. p=0 (-D;77,2(*m+7ACBiI$BO,§2/( É/fOÈOfOj"fOl¦¡zkRkJl{fOnposbrut=vO$w
f (∆T )
|¦~x®~O~=
f (∆T ) = a∆T 3 + b∆T 2 + c∆T + d
B #:rOx('
[email protected]
#&¥x¥=¥O/qO8 #>¥/x¥=·(J+±2
¥/x;ri<"D ( "+(JKA"D50 `( . ¦QPx
×10−5 T − 273.15
KL!=,A
~ 'O/xpr¯·RxzQ{(*+"+C([!O.8*+7D5!=. \³BO..G,45([KZ!O,A,/δq
.([+7( 0 KZ*ACB$+7,A7([2(<#&~∆T
r ◦ QPqL=(*+P
4B 8:9NBOA"a=(
T 8:9BO.a=*(-( +`2!=.8.G,454;(KL!=,A±8 (J+7+C(+C( 0 KL ACB+C,A"(Ox/Q{( KL( .2NB$.G+ qB $.2(PKL( A"01(J+7+7A7(P45B6I$BOA"DB$+7D5!=.
2(-45B<+7( 0 KZ*ACB$+7,A7(2]\³D;[email protected]=( A"7D;!=.2(-KL!=45BOA7Dp+COq Xyk On5kJjkJl±fOnpo]brut=t/r [email protected]+{A7( 0 K45BO8 KNB$A BuIO( 8
∆T
τ
xW\_ EG,NB$+7D5!=.2(+CA7BO."(*A"+m2(68:9NBOA"a=(62(*ID;( [email protected]+m2!=.8
τ = (−21/Tr )(T − 273.15)
BiI=(*8
G
m
n
δqN
I = kq Di (∆vgi ) (LW C − LW Ccrit )f (τ )
" &$ )+ ,1,"+0
τ=
(−21)
(T
Tr
|y~x~O@
− 273.15)
f4545([( +6YNBO7*(R7,A2/( (JKL A"D5( ./8 ( 62(F4BOYL!=ACB+C!=D;A7(<E,/D01(J+7+7( [email protected]+( .1M"(*,,.(R4B$A7a=([[email protected] 01(
2(m+C( 0 KL ACB+C,A"( q2(P8*[email protected]+C(*.,/`( .§(iB$,45D;E,/D52([email protected](P+:B$D5454;( 2(PKNBOA"+7D58*,45(m(*+{2(ID;+7( "7( `2/(P8:9G,/+C($x
W\³(µKA7(*77D;!=.2/( 45B8:9NBOA"a=( 8:9NBO./a= (
|( .¦Q`B§,.(R!=A70 (R7D;01D;4BOD;A7(<8 ( 4;45(2/( 1ªfOj('O¢¡zkJj
∆q
kJl{fOnposb[email protected]·¯w
|y~x®~ ã
m
n
W( `I$BO45(*,A7ª2(*ª8 !=.+:[email protected]+7( q
m n
(J+
∆q = BDi (∆vgi ) δq
B
[email protected]+2!=..*( ª2NBO./4;(-+:BOY4;(iBO,~x/x
kHBOD;4545(2(64BKNB$A"+CD;8
,4;(2(6a=4B$8 ( | 0c
µ
2
r¯·=·
r¯·=· 2
2 ã ≤·=~ ã ·=~
2
~=·$
2≤
«~=·O
L~x³ !#"$+&(*+&
7
?=&#1V[=c1 +&
5
.
uD;+7,NB$+CD;!=.
#
#
h
ru¥ 13
ã x t=~
ã x³¥ ã × ru¥ 6
·=~×
x³v=¥
·x~ ã
ru¥ 8
~ã×
x³¥=¥
0
/ x 'OÎ
r=x³t=¥
¥/x ã=ã
~x· ã
¥/x·O¥
21=L"13AYUAY&@J4+&(*=&X#&gJ 2 "[13A3?[=& %_
.
~ x®·
~x®·
~x®·
~xv
~xv
21 %_ _
^ 8:9NBOEG,(¬A7*a=D5!=. 2/, =a A7BOK9((*"+BO7"!8*D5 (V,/.( * EG,NB$+CD;!=. 2 8*A7DpIOB$.G+45B 8:9NBOA7aO(¬*8:9NBO.aO (¬( .
!=.8J+CD;!=.2(64BF+C(*01KL A7B$C+ ,/A7((*+m2,8 !O.G+7( .G,(*.( BO
, 45D5EG,D;2(
|( .¦Q`Jx
δq(T, EW )
·ã
S8
−3
Contenu en eau liquide effectif (g.m )
1.5
1.0
S4
S1
CEW
S7
0.5
S3
S6
S5
S9
S2
S9
0
0
−10
−20
−30
Température (°C)
X~x ã
,T$4J+&@#=T1V[13A3?[=N$-+,cH*A2"EX(* H%$PN*X13$-=&Z>@J2$-J4o#=X>2?[= 31 AP?= (*Wm1 2.S,EJ#$ I13"$4
21 ("
7 ( !# &2(I?= 21<!_
7
? =T1 #=E" #=z: -[" UA2"AP(*9 13A >
I(I#,]1 JX*
_< ?"%$ o$ ?"[N*X$- ,]$-J+&T#=c1 9
7 ?=T1 #=E" 2=C-[" UA2"%AP(*m 7 13A > EW
5
7
6
$#A313A2"E _
7
7
7
*( "+§4;(8*[email protected]+C( .G, ( . ( BO, 45D;EG,D52((*Z( 8J+CDp1|ax_0 −3 Jq±8O\_( "+#&#2D5A"(45BKZ!OA"+CD;!=. 2z\³( BO, 45D;E,/D52(
7,/A"!=.2/,(EG,DmKNBOA+CD58*D5KL(BO, a=DpIACBOa=(2, a=A7*7D54¦x>4ª( "+c [email protected]$4P
!#
( +4;(
LW C × Ecg
LW C
8 !O.G+7( .G,(*.( BO,45D5EG,D;2(-(J+
(*"+4y\_(sr 8iB$8 D;+72(68 !=4;45(*8*+CD;!=.§2(*maO!=,/+7+7( 4;(*+7+7( ª2]\_(iBO,7,/A"!=.2/,(
KNBOAm45(6a=A" "D54|*[email protected](<rF2NBO./EPcg45(60 !2/3 45(iJxLQ{([KNBOACB$013J+CA7(6(*"+-,+CD54;D5"614B K45BO8 (<2,8*[email protected]+C(*.,( .
(iB$,c45D;EG,D52(`8 BOA¤45(ª0 8iB$.D5"01(ª2(ª+CA7BO."(*A"+2(m8:9NBOA7aO(2*KZ(*.22, +:BO, 2(8 BOK/+7,A7(m2( a=!=,/+"+C(*
K4;,/2+ pO+ªEG,([2,8 [email protected]+C(*.G,( .(iBO,45D5EG,D;2(-2,.G,NBOa=($xZW(<8 [email protected]+7( .G,(*.(iB$,45D;EG,D52((Jz(*8*+7D;8*A7Dp+CD5EG,(
|( .ax_0 −3 {(*"+m2!=./.6KNBOAª4y\_ EG,NB$+7D5!=.V
EW
CEW
CEW = −0.49 − 6.64 × 10−2 T
|y~x~=·=
Q{(*+"+C( EG,NB$+7D5!=.[2 8*A7D;[email protected]+
( + I$BO4B$Y45(¤KL!=,As2( H8 [email protected]+C(*.G,( .F(iBO,4;D5EG,D;2(¤(*Z( 8*+7D;N8*!=0 KA7D;
( @. +7A7(-¥x®~=~6(*+r=xprPax_0 −3 xTuCEW
D
¥/x~=~ax³0 −3 q
¥/x[email protected](J+{"D
²r=xprPax_0 −3 q CEW
EW CEW EW
rOx;r=x
W- ( ª EG,NB+CD5!O.ª2 8*A7D;[email protected]+m8:9NBOEG,(<A7*a=D5!O.§2,aOACBOK9/(<"[email protected]+mA7 8 BOKDp+C,4; ( `2BO.m45(-+:B$Y45( BO,~x ã x
f. uZrOq±J+:[email protected]+c2!=..EG,]\_D54ª.]\_(JD5+C(KBO12/(A" 7,/4;+:B+c2(4B$YZ!=A7B$+C!OD5A7(cKZ!=,/A12(*1+7( 0 KZ*ACB$+7,A7(*
7,/KZ*A7D5(*,A7(* # 'x³@· o QPq 4;( FI$BO4;( ,A"2(*8:9NBOA"a=( +7ACBO. A" ( "[email protected][email protected]+C(*A7KL!=45*( 4;D5. BOD5A"( 0 ( [email protected]+[
KNBOA+CD;A2(*PI$BO45(*,A72(
# 'x³@· o Q (*+-2/(F+C(*4545(<7!OA"+C([EG,(
¥1KL!=,A
¥ o QPx%-.!=Y+CD5(*[email protected]+
δq
δq T 2!=./8<2BO.ª45(8 BOª2(64y\[email protected]+C(*A7KL!=4B$+7D5!=.12
( uZr=Gq u/q u~R(*
+ u/t/qNA"( "KZ(*8*+CDpI=(*01(*.G+[
δqS1
δqS3
δqS2
δqS9
=
=
=
=
(−2.75 × EW − 0.007) × T
(−57.06 × EW + 12.60) × T
(42.78 × EW − 1.08) × T
0
·=·
|y~x~OÎ@
%,0 A"! 2(64y\_ EG,NB$+7D5!=.
uZr
u~
u/
u ã
u·
u/Î
u '
u/v
u/t
Z~x ã 2"[13AP?=&(*13$-=&Y>
#$21(*
fE,B$+CD;!=.2,+7ACBO./"(
A+ª2(68:9NBOA"a=(
±BO4;( , A"`[email protected]+C(*A7KL!=45*( `4;D5. BOD5A"( 0 ( [email protected]+
E
#>/r ã x ã ¥
f x 'x³[email protected]~
δ ×
E
f x #[email protected]~xÎ ã
ã rut/x ã ¥
δ ×
E
~O¥/x~=~
f x rOx=Î
k ru¥/x³¥@·
δ ×
E
~O×
¥ ã r=x 'OÎ
f x #mr¯~$v/x('$¥
δ E
#~OtO¥=¥/x~=×~
ã Î@~x³t/r
δ ×
E
/x³¥@~[#/r=x 'OÎ
f x ~$Î/x®·$
fOx
δ E
~O¥/x~=~ × f x #{~=~x~OÎ ×
δ E × ¥/x³¥
δ
7
2
-
H$3NS,?"%$9Q,[$ .0J213$#AY&2[13AP?= (* ]
6 "%=T(*#$+&g21_
@_
© !O,A645( -+7( 0 KZ*ACB$+7,A7(*8 !=0 KA"D57(*( [email protected]+CA"( #>=¥ o Q (J+-# ã ¥ o QPq!=.,/+7D545D;7(4;( *E,B$+CD;!=.-I$BO4B$Y45(*
H
#&O¥ o QPx
4;!=. 2[kJnÍ'h¡ kJlfOnpo±b³~O¥=¥/r*wyqZD54(*"+6.*8 ( "CBOD;A7( 2(RA7*2,D;A7(R2(ru¥ 4y\_ EG,NB$+7D5!=.VKL!=A+:[email protected]+",A
7 Dª4y\_!=. 8 !=./7D52/3 A7(,.((sr 8 BO8 Dp+C2(8*!=454;( 8J+CD5!O.Ä2( 1a=!=,/+"+C( 4;(*+"+C( 2(¬r=x¤>4`7(*0Y4;(EG,(45(*
∆q
EG,NB+CD5!O.R2( A" a=D;!=.R8 !OA7A7(*7KL!=.2NB$.G+1V2(§BOD;Y45(*8 [email protected]+7( .G, ( . (iBO, 45D;EG,D52(| u~qu//qSu·(J+
u/Î@R7,/A7( +CD;01(*.G+c4B8:9NBOA"a=(*8:9NBO.a=*(OxQu/BO,.2/( A7B 7,/a=a= A"2(A7*2,D5A"(B$,7"D8*( *E,B$+CD;!=. x
WXBA7*a=D5!=.V./ [email protected]$+7D;IO(R( "+<B$D5."D A7*2,Dp+C( ~O¥ 2(*6I$BO4;( ,A"-KA" 2Dp+C( KBOA0u~c(*+u/q](*+645BcA7*a=D5!O.
KL!=7Dp+CD;IO(( +A" 2,/D;+C(<§ru¥ ¨2( ªI$BO45(*,A7`K/A7 2/D;+C(*ªKNBOo
A u·R(*g
+ u/Î/x
u(
" &$ )+ ,1,1- 0
´ jChuh uÍ kJl`fOnpo brut=t '¯wH!O.G+A7(*KA7D;4;( mA7*7,4p+:B$+7P2(*P(JKL A"D5( ./8 ( P2(F45BOYZ!OACB$+7!=D5A"(-2(1ZfOË¡ 'kJjCÍ
ª
kJlªfOnpobrut=t/r*wyq( .+CACB$."!=A"0cB$.G+45(<8 !O.G+7( .G,( .( BO,4;D5EG,D52/([(Jz(*8*+CDpm|
ª(*.+:BO,2z\BO8*8 A"*+CD;!=.
EW
KNBOA1a=D;IA7BOa=(|
JxWX(+:BO, 2]\³BO8*8 A7J+CD;!=. KNBOA1a=D;IA7BOa=(( "+2 $N./DP8 !O010 (4;(KA"!2/,D;+12(45B
ID;+7( 7"(A7(*4B$+7D;I=RAR
(2]\_D50 KNBO8*+(*+S2,§8 [email protected]+7( .G,( .§(iB$,§45D5EG,D;2(P(Jz(*8*+7D;¤
x
( +
RAR = EW × V RAR
(JKA"D50 ( .ax_0 −2 x_ −1 x}-BO.P "!$#&%'[qN4;(-+:BO,§2]\B$8 8 A"*+7D5!=.KNBOA`aOD;IACBOaO(( "+m2 $N.DXKNBOA
RAR = ρair × rc × Ecg × Vgmean
BiIO( 8
Vgmean =
R +∞
V (Dg )n(Dg )dDg
0 R
+∞
0
|y~x~'=
n(Dg )dDg
7A (*KA7*7( [email protected]+7({4y\_(sr 8 BO8 Dp+C2(±8 !=4;45(*8*+CD;!=.-2(*HaO!=,/+7+7( 4;(*+7+7( X2]\_(iBO,[.G,NBOa=(*,7(±KNBOAX45(aOA7 "D54=(J+
Vgmean
( +m4BID;+C(*7"(<0 !u?=(*..(<2,aOA7 "D54¦x
WXB8:9NBOA7aO(- 8:9NB$.a= (1|¢¦Q`45!=A"{2]\_,.(P8 !=4;45D5"D5!=.1BiI=(*8A7( YL!=.2§(*[email protected]+CA7(2/( ,KNBOA"+7D58*,45(*{2(a=45BO8 (
( +m4BR01*01(PE,/([KL!=,A`45B KBOACBO0 *+7A7D;CB$+7D5!=.12(RZfOË¡ 'kJjCÍRkJl{fOnposbrut=t/r*w
Ecg
∆q = BD m (∆v)n q
|y~x[email protected]
*( "+ 45(¤2DB$013J+CA7(¤2(4BªK45,/HKL(*+7D;+C(¤KNB$A"+CD;8 ,4;(|0cJq
( + 54 BªID;+7( "7(SA"( 4B+CD;IO(¤2]\³D;01KBO8*+{|0x_ −1 Jx
∆v
W ( 8*!(Rr 8 D5(*[email protected]+C q (*+ [email protected]+4;( 0 0 ( HEG,(`KZ!O,AZfOË¡ 'kJjCÍkJlHfOnpobrut=t/r w(*+ [email protected]+A" 8 BOKD;+7,45*
D
m n
B
·OÎ
N2 BO./{45(`+CBOY45( BO,c~xx |( .1¦Q`( +S45B<8:9NB$A7a=(ª+CA7BO."*A7*(PKNBOA±8 !O4545D;7D;!=.][email protected]\_( +S,./(!=.8J+CD;!=.12/(45B
+C(*01KL A7B$+C,/A7(F(J+2,+CqBO,2]\B$8 8 A"*+7D5!=.VKNB$APa=DpIA7BOa=($xZ>4H(JD5+C(,.+CBO,2]\³BO8 8*A7*+7D5!=.KNBOA-a=DpIACBOa=(
8 A"D;+7D5EG,(1|
`BO,2/( 7",-2,/E,/( 4 45BRK45,ma=A"!=7"(FKNBOA+CD;8 ,4;(6I$B 7([8:9NBOA7aO( AmKZ!="D;+7D;I=(*01(*[email protected]+P(J+
RARcrit
( .R2/( 7"!=,2,/E,/( 4yq$( 4;45(I$BP"(S8:9NBOA"a=( A . [email protected]+CD;IO( 0 ( [email protected]+ixOW\³*E,B$+CD;!=.2*8 [email protected]+45(±+:BO,2]\³BO8*8 A7J+CD;!=.
KNBOA`aOD;IACBOaO(8 A7Dp+CD;E,/(( "+m45B ",D;[email protected]+7(R
RARcrit = a + bT + cT 2 + dT 3 + eT 4 + f T 5 + gT 6
q
q
q
|y~x[email protected]
q
a = 1.0 b = 7.9262 × 10−2 c = 4.4847 × 10−2 d = 7.4754 × 10−3 e = 5.4686 × 10−4
(*+
x
\_(JKA"D50 (( .ax_0 −2 x_ −1 x
f = 1.6737 × 10−5
g = 1.7613 × 10−7 RARcrit
4
3.5
RAR (g m^-2 s^-1)
3
+
2.5
2
1.5
−
1
0.5
0
0
-5
-10
-15
Temperature (C)
-20
-25
-30
N$-+,cH*A2"E(Q H%[$3N*13$-=&Z>@J#$4J4\M,TU"&ON$-? &+&@ ,$213A "%!g2=o>@?= 31 AP?=
7
(GF $4J213A3?[=o,$<7 NA ; $ N _ S(I+,]1 J(* 6]"%=T(*#$+&@1
_ < ?7 "$
7 5
7
6
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;
7'7
7
X~x·\
,]$-J+& #=c1V[13AP?=
:
*( OWo1 #.,J#$-[13"$4X21G(["m1V["
W ( [ EG,NB$+7D5!=.[7,A .(1"[email protected]+FI$BO45BOY45(*[EG,(1KZ!O,A[2/( [+7( 0 KZ*ACB$+7,A7(*F8*!=01K/A7D5"( <( [email protected]+CA"( #>v o Q (J+
q
#~$ o QPxZ©H!O,A
(J+
qZ45B K4;,PaOA7!="7(FKBOA"+7D58 ,/45(I$BBO8 EG,*A7D;A
o
o
,.(68:9NB$A7a=(6KL−23
!=7Dp+CD;IOC(R< T < −8 C RAR > RARcrit
|y~x³=¥@
q = 6.74RAR − 1.36(−T ) + 10.05
©H!O,A
(*+
q45B[K4;,{aOA7!="7(PKBOA"+7D58 ,/45(ªIOB[7(-8:9NB$A7a=(*A{.*[email protected]$+CDpI=(µ#
0.3 < RAR < 3.3 RAR < RARcrit
0 ( [email protected]+<
|y~x³/r¯
2
q = 3.02 − 10.59RAR + 2.95(RAR )
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29
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2(
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T
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2,a=A7BOK9($q(*+ª,./((J+CA7BOKZ!O4B$+7D5!=.1( "+m.*8 (*7CB$D5A7(<45B[!=D5{KZ!=,/A`45( {YNBO7"( `+C(*01KL A7B$+C,/A7( `(J+ª45(*
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8 !2*( F2BO.F "!$#&%'[qs(*+[45( ,/A<D;./N,(*.8 ( 7,/AF4y\_*IO!=45,+CD5!O.¬*45(*8*+CA"D5EG,( 2,.G,NB$a=(Oxs© B$AFBOD;454;( ,A" q
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|y~x³ ã
x
¿¤¾ ¼[À Á a Ã¿ j.(!OD5cEG,(45( c8:9NBOA7aO( §[email protected]+§J+C" KNBOA" (*KBOAc4;( KA"!8*( "7,.!=.#>D5./2,8*+7D;*q±(*454;( c"[email protected]+
+CA7BO. A7*( [2]\³,/.(cKNB$A"+CD;8 ,4;( 4¦\B$,/+CA"( 4;!=A762(*[K/A7!8 ( "7,F01D;8 [email protected]?"D5EG,(* xs©H(*[email protected]+8*( [KA7!$#
8 (*7",| $Na=,A"(<~xpr¯Jq2/(4B01BO7"(( "+m*8:9NBO.a=*([(*[email protected]+CA7(4;( `KNB$A"+CD;8 ,4;( `EG,D]D;[email protected]+C( A7BOa=D;77(*[email protected]+iq(J+ª2!=.8$q
4B8:9BOA7a=(§*45(*8*+CA"D5EG,(§J+:[email protected]+ A7*KNBOA+CD5(45B7,/A"BO8*(2/(§4BKNBOA+CD;8 ,4;(Oqs,.(8 (*A"+:B$D5.(§EG,[email protected]+7D;+Cc2(
8:9NBOA"a=(m( "+SBO,"7DN+CA7BO."*A7*(Ox/Q{(+:B$, 2(ª+CACB$."(*A"+2(8:9BOA7a=(m( +S8 BO458*,45`(*.[email protected]+C*a=ACB$.G+±2(ªB%e*!=.
(JK4;D58*D;+C(P45(-+:B$,2(+CA7BO.( A"+m2(601BO7"(60,4;+7D5K4;D5mKNBOA`45B A"( 45B$+CD;!=.8:9NBOA"a=(J#2DB$013J+CA7($x
1 Z +∞
QY COLX =
ρ 0
Z
%-.BR4;( ` EG,NB+CD5!O.ª7,DpIOB$.G+7( 6
0
K(Dx , Dy )qy (Dy )ny (Dy )dDy nx (Dx )dDx
|y~x³@·=
G(b)
λb
0
Z +∞
G(f )
ρQ =
q(D)n(D)dD = eN f
λ
0
|y~x³=Î@
1
dλ
ρdr = aN G(b)d b = aN G(b)(−b) b+1
λ
λ
1
dλ
ρdQ = eN G(f )d f = eN G(f )(−f ) f +1
λ
λ
|y~x³[email protected]
ρr =
uD!=.2Dpz*A7(*.8
+∞
Z
+∞
m(D)n(D)dD = aN
|y~x³ '=
D;(8 ( m2(*, EG,NB$+7D5!=./ qN!=.B§
Î=¥
|y~x³[email protected]
\ %-.2/D;ID5"(4BKA"( 0 D53 A"( EG,NB$+7D5!=.KNB$A
r
(*+m4B2(*,D53*01(6KNBOA
∆r
∆λ
= aN G(b)(−b)
r
λ
∆λ
∆Q
= eN G(f )(−f )
ρ
Q
λ
ρ
uD!=.8*!=0YD;.(45(*m2/(
Q

|y~x ã ¥@
|y~x ã r¯
,*EG,NB$+CD;!=.6
∆Q =
f ∆r
Q
b r
|y~x ã ~=
^ D;.7D¦qHD;4¤( "+ KL!="7D5Y/45(c2z\³(µ/K/A7D50 ( A[7D;01K4;( 0 ( [email protected]+[45(1+CBO,2(8:9NB$A7a=(c+7ACBO./" A" (§(*. !=.8*+7D5!=.2,
P
+:B$, 2(01BO7"(+7ACBO./" A" ($x`Q{( KL( ./[email protected]+iq45!=A"7EG,(45( 12( , KBOA"+7D58 ,/45( EG,[email protected]+C*ACBOa=D;7"( [email protected]+c"[email protected]+
KA" 8 D;KD;[email protected]+C(* q 4;( +CBO,2(§8 !=4;45(*8*+CD;!=../(§KZ(*,/I=(*.G+ K4;,R*+7A7(D;[email protected]+C aOA7 B$.NBO4;?G+7D5EG,( 0 ( [email protected]+R",AR45(
7KL( 8J+CA"([2(+CBOD54;45(Ox}-BO./m8*(68iBO&#>4q!=.,/+CD;45D;7(-2(*./!¯?=BO,K/A7J#8iBO4;8 ,4; *x
W( <9G?2A"!=0 *+C*!=A7(*[+7ACBO."KZ!OA"+C(*[email protected]+F8*( <8:9NBOA7aO( <2NBO.<2]\³BO,/+CA"( <KNBOA+CD5(*62,.G,NBOaO(1"!=,<4y\_D5.#
N,(*.8 (P2(P45B[7*2D50 ( [email protected]+CB$+CD;!=.][email protected](4B<+7,A7Y/,45(*.8 ((*+S2/(P4y\³BO2/IO( 8J+CD5!O.]xW¤\³BO2/IO( 8J+CD5!O.( +SA7 BO45D;7*(KNB$A
45(78:9*0cB ©¤}`O
^ ks^ |¦,4p+CD52/D50 ( ."D5!=.NB$4©H!="D;+CDpI=(}( $N.Dp+C(-^`2I=( 8J+CD;!=. kHA7BO.7KL!=A+`^P4;a=!=A7Dp+C90c
EG,DzKL( A"01(J+{45B[8*!=.7(*A"I$B$+7D5!=.2/( S2(*.7Dp+C*S2(P8:9BOA7a=(PKZ!="D;+7D;I=(*2z\³,./(-KNBOA+S(*+{. a=B$+CDpI=( {2]\B$,/+CA"(
KNBOA+ixHW( FKBOA"+7D58 ,/45( <45(*FK4;,FaOA7!="7( F8:9,+C( [email protected]+ B$45!=A"[EG,(4;( [KNBOA"+7D58*,45(*[4;( FK/45,<45*a=3 A"( F"[email protected]+
+CA7BO."KZ!=A+C*( `2NBO./`4BFKNB$A"+CD;(P",KZ*A7D;( ,A"(2,§.G,NBOa=((J+`2NBO.`4¦\³(*.8 4;,0 (Ox}P( `A" a=D;!=.S2(PKZ!=45BOA7Dp+C
!=KKL!=" (R7!O.G+<B$45!=A"P!=A70 (* qZ(*+6,.8:9NBO0 KV 4;( 8J+CA7D;EG,([email protected]+C(*.7(R(*"+<a=*. A" 4y\[email protected]+C(*A"BO8*(2(R8 (*
A7*a=D5!O. x
! !
"
W(V8:9NBO0 K 45(*8*+7A7D5EG,( →
− (*"+7!O45,/+7D5!=. 2]\³,/.( EG,NB$+7D5!=. 2( n<BO,/7§[email protected]+45B7!=,/A78 (V(*"+45B
2(*.7Dp+CI=!=4;,0 D5EG, (2(68:9NBOA"Ea=(+C!O+CBO45(

ρtotal
ρtotal
→
−
∇· E =
0
|y~x ã @
→
−
E
→
− 0
∇V = −
ρ̃
|y~x ã ·=
ρtotal
→
−
GDIV (ρ̃ ∇V 0 ) =
0
|y~x ã Î@
( "+ 45B8*!=.+:[email protected]+C(2D; 45(*8*+7A7D5EG,(2(4¦\BOD;A x WX(KL!O+C(*[email protected]+CD5(*4` 4;( 8*+7A7D;E,/(
( + 2 $N./DKNBOA+CD;AR2,
V
8:9NBO0 K 45(*8*+7A7D5EG,(KBOA6
|y~x ã=ã
→
−
→
−
E = − ∇V
f.§BOD;+ qD;4Z(*"+ª8 !O010 !2/(P2NBO./`4BF",D;+7(-2]\[email protected]+CA"!2,/D5A7(P,.§K7(*,2!$#KZ!$+C( [email protected]+7D5( 4z 45(*8*+7A7D5EG,(
2!=./.
V0
KNBOA
0
iB I=(*8
x (*"+4;(M7B$8 !=YD;( .[2({45Bª+7ACBO.!=A701B$+7D5!=.c|IO!=45,0 (¤2({45B01BOD54;45(iJq (J+ A"( K A" "( [email protected]+C(
ρ̃ = ρdref J J
2!=./8<45B0cB$77(62]\³BOD;A`7( 868*[email protected]+C( .G,2NBO.m,.(601BOD54;45(OxG^D5."Dyq4¦\³*E,B$+CD;!=.§2(n<B$,7ª2(Jρ̃ID;( [email protected]<
+
Î/r
n6}P
( +X4y\_!=KZ*ACB$+7( ,Az2D;IO( A7aO( .8*(±2BO.X45(?"+C3*01(¤2(8*!!OA72!=./. (*8 ,AID;45D;a=.( z.!=.#!=A"+79!=a=!=.BO4yx
©A7*7( [email protected]+7 (<BOD5./7Dyq8 (J+7+7( EG,NB$+7D5!=.(*"+ª8*!=01KBOACBOY/45(8 ( 4;45(,/+CD;45D5" (mKZ!O,AS45(8iBO4;8 ,4L2(45B[KA"( 7"D5!=.zx
© B$AR8 !=." EG,(*.G+ q¤45("!=4;IO( ,AR2(§K/A7( "7D;!=. ( 454;D5K/+7D5EG,(1"+:B$.2NBOA"2 2/( "!$#&%' KZ(*,/+ *+7A7(§,/+7D545D;7
KL!=,AªA7(*.GIO!u?=( AP45(K"( ,2!#>KL!O+C(*.G+7D5(*4 0 x %-.A"(*+7A7!=,/IO(F4;(8:9NBO0 K*45( 8J+CA"D5EG,(<( .,/+CD;[email protected]+ª4y\_ EG,NB#
V
+CD;!=.~x ã · ux
W( F8*!=.2Dp+CD;!=.[BO, 45D;01Dp+C( [email protected]+RB$2NBOK/+7 ( BO, +7ACBODp+C( 0 ( [email protected]+<2, 8:9BO01K 4;( 8*+7A7D;E,/(Ox %.¬BOD;+
4y\[email protected]?KZ!O+793 "(VEG,(4B4;D50 D;+C(",KZ*A7D;( ,A"(V2, 2!=01BOD5.(8 !=A"A7(*7KL!=.2£ 4y\_D5!=./!=7K9/3 A7((*+45B45D;01Dp+C(
D5. A7D;( ,A"(< 45B ",A"BO8*(<+7( A"A7( +CA"(OxZWX(*4;D50 D;+7( S",KZ*A7D;( ,A"([(J+PD;./ A"D5( ,/A7(62,2/!=0cB$D5.("[email protected]+-2!=./8
8 !O.7D;2 A" ( ±8 !O010 (S*[email protected]+±2( , 8*!=.2,/8*+C(*,A7±KNBOABOD;+7 [email protected]©HBOA¤8 !=./7 EG,(*[email protected]+iq45(`KL!O+C(*.G+7D5(*4 4;( 8*+7A7D;E,/(
( +`8 !O."[email protected]+`BO,1A7!O.G+7D53*A7( 7,KL A"D5(*,A7(*S(*+{D5./*A7D5(*,A7(* q(*+{45(m8:9NBO0 K 4;( 8*+7A7D;E,/(P( +`!=A+C9!=aO!=.NBO4
R8 (*",A"BO8*( *x %-.BOKK/45D5EG,(2/!=.862( m8 !O.2D;+7D5!=./S2(<%(*,0cB$.
|y~x ã '=
→
−
→
−
→
−
−
n · ∇V 0 = →
n · E fw
( "+1,. I=(*8*+C(*,A ,.D;+CBOD5A"(.!=A"0cB$4`V4BV7,ABO8 ($q(*+ −−→ (*"+ 45(§8:9BO01K 4;( 8J+CA7D;EG,(2(YZ( BO,
+C(*01K/ xG©H!O,A4;( ¤45D;01Dp+C(* 4B+C A7BO45(* q=!O. BODp+¤4y\[email protected]?KL!O+793 "(EGE,f(mw 45(`8:9NB$01K1 45(*8*+7A7D5EG,(ª(*"+±A7(*4B`I=(*A7
CBRI$BO45(*,A`2(6YL(iBO,§+C(*01K/ xW(68:9NBO0 K2/([YL(iBO,§+7( 0 KmB KL!=,A`(µKA7(*77D;!=.
→
−
n
→
−
→
−
E f w = E 0 exp −KE H
z − zs
H − zs
|y~x ã [email protected]
( "+ [email protected]/[email protected]~xpru¥
−
→ (*"+IO( A+CD58 BO4/(J+¤2D5A"D5a=¤I=(*A7¤45(YNBO O7!=.R0 !2/,45(±( +ªru¥=¥ Fx³0 −1 x
E0
(*"+`45B[9NBO,+C( ,/A{01BD501BO45(ª2,2/!=0cB$D5.($q(J+
4B[9BO,/+C(*,AS2,cA7( 4;K
D5(JE&x^,c45D50 D;+7( ¤4B+C
H
zs
KL!O+C(*.G+7D5(*4] 4;( 8*+7A7D;E,/(<2/!=D;+`2/!=.86!=YL D5Aª§
∂V 0
=0
∂n
0 −1 x
A7BO45(* Gq 45(
−6
|y~x ã t=
4 W \B¯M"!O,/+¤2( *8 45BOD5A"2NBO.45BPKNBOA7BO0 *+CA"D57B$+CD;!=.F2(*!=ACB$a=( (*"+¤D5./2D5"KZ(*.CBOY/45(Ox%f. (Jz(J+iq=D;[email protected]+
2(*, (*Z(*+701B¯M"( ,/A7m@D545 45D50 D;+7( [email protected]+4¦\BO0 K4;D;+C,/2({2,18:9NBO0 K1*45(*8*+CA"D5EG,(ªKA"!2,/D;+¤2NBO./4;(`.,BOa=(OqG(J+
D54;{A"( 2D;"+CA"D5Y,/( [email protected]+ª45(*`8:9NBOA7aO( mR4y\[email protected]+C*A7D5(*,A{2/,.G,NB$a=(OxGuDz!=..]\³D;[email protected]+CA7!2,Dp+SKNB$`45( `*8 45BOD5A"S2NBO.`8*(
+>?KZ(`2/(`KNBOACB$01J+CA7D;CB$+7D5!=.zq4;({8:9NB$01K 4;( 8J+CA7D;EG,(`B$+"+C( D;[email protected]+2( I$BO4;( ,A"7,/A7A7 BO45D;"+7( q+CA"3 4B$A7a=(*01(*.G+
7,/KZ*A7D5(*,A7(*PB$,"( ,D;42(6A7,K+C,A"(<2/(<4¦\B$D5AS8*4BOD;A x&%.2D;"+CD;.a=,(2/( ,8 B$+C*a=!=A"D5( ª2]\_ 8*4BOD;A7-
4;( ª 8*4BOD;A7SD;.G+7ACB#.G,NBOa=(*F|.!O+7 ªQ`ªEG,D.z\B$+"+C(*D5a=.(*[email protected]+KBOª45(67!=4¦x>45`8*!=.+CD;+7,( [email protected]+( [email protected]"!=.
v=¥ [email protected]· 2(645BF+C!O+CBO45Dp+C-2(*ª 8 45BOD5A"`7,[email protected]+ª4BR4B$+7D;+7,2(Ox
4;( 6 8*4BOD;A7.G,NBOa=(µ#>"!=4m|.!$+C [X
Q n<JxXW( < 8*4BOD;A7<X
Q n "[email protected]+FKL!="D;+CDp-!=,¬. a=B$+CDp7,DpIOB$.G+<45B
8:9NB$A7a=(<.( ,+CACB$45D5" (*m2BO.m45(.G,NBOa=($xLWX(*-X
Q n )"[email protected]+PYL(iBO,/8 !=,K0 !=D5.`./!=0YA"( ,EG,(<45(*
Q nP#mD54z7(6KA7!2,Dp+ª( [email protected]"!=.,/.X
X
Q n KL!=,Aru¥ X
Q nP#:x
GÑ
Jß"Ô¦×³ÓyÖ ¯ ÙÑ OØµÓyÑÒ Ó¦×ÑÙ§ß"Ù³Ñ7Ü>Ó¦Ôy×®á*âÑ
→
−
→
−
GDIV (ρ̃2 ∇V 0 ) = GDIV (ρ̃ E )
V0
Ø ¯ Ó¦×ÑÒ Ó ÖµÙØJÔ ¬Ñ"Ò)Ôyß¦ØJÙ:ÖµÒ ÓÙ ß"á âÖ:Óy×³Ø*Ò
Î@~
→
−
GDIV (ρ̃ ∇V ) =
¸S¹ m¹º
£½ a ¾gaÃ XÆ¾ Á VbP¾ ½ a &¿
W(R8:9NBO0 KV2(A7,/K/+C,A"(R2D5*45(*8*+CA"D5EG,([( "+62/(R4y\_!=A72A"(F2(=¥=¥=¥c x_0 −1 BO,V./D;I=( BO,V2(45Bc0 ( A x
Q ( KL( ./[email protected]+iq=4;( K/A7! $N4;2(S8:9BO01KR*45(*8*+CA"D5EG,(S0 ( ",A7*2NBO. 45( .,BOa=( 2z\³!=A7BOa=($q$0 0 ({*\³D;45 "[email protected]+
{
KL!=.8*+7,( 4; [email protected]+m,.([BO0 K45Dp+C,2/(-0cB¯/D;0cB$45(P2/(<4¦\³!OA72A"(2([45B8 (*.G+CBOD5./([2(6 x_0 −1 xN>4"( 0Y/45( A7BOD;+
2!=./8`E,/(m4;(`8:9NBO0 K 2(`A7,K+C,A"(m2/D5 4;( 8J+CA7D;EG,(S0 (*[email protected]+( .-M"( , 2(*¤ 45(*8*+7A7!=.2/(ªB$D5Y4;({ ./( A7aOD5(<|
=¥m( 6s./({KL( A70 (*+"+C(±KNBOH2]\³(µK45D;E,/( A 4;(¤2 8*45( ./8:9( 0 ( [email protected]+2(* 8*4BOD;A7*x .f0+ !¤fOjµl¢¶ÈFkJlZÇSfOj uÍ[email protected]∼
·w
[email protected]+*+C,/2D5[4y\_ 0 D57"D5!=.2(<ACBi?O!=.
# M","+7([email protected]+P4;([2/ 8 4;( .8:9/( 0 ( [email protected]+2(*P 8*4BOD;A7*xN>[email protected]+P0 [email protected]+CA"
EG,(<2( 45(*8*+7A7!=.ªA"( 4B+CD;ID;"+C(*[| r( 6ªBO8 8* 4; A7*KBOAª45(68:9NBO0 K*45(*8*+CA"D5EG,([ 4¦\³D;[email protected]+C A"D5(*,A`2,
.G,NBOa=(KL!=,/I$BOD;( [email protected]+`J+CA7(64¦\³!OA7D5aOD5.∼
(m2(8 (J+7+7( 0 D57"D5!=.c2(ACBi?=!O. #x jm.§ 8 45BOD5A7(*ACBODp+`2!=.8D;.D;+7D5
45!OA77EG,( 2(*6 45(*8*+7A7!=.6*.( A"a=*+7D5EG,( [BO8*8 4; A" 6KNBOA4;(R8:9NBO0 K *45( 8J+CA"D5EG,(R2( 4y\_!=A7BOa=(RB$+"+C(*D5a=.(*[email protected]+
2(*P ./( A7aOD5( m7T, [email protected]+C(*-KZ!O,AªKA7!2,D;A7(62(F./!=,/I=( BO, 4;( 8J+CA7!O.(*.8 BO78 BO2(FKBOA8*!=454;D5"D5!=.§BiIO( 8
2(*0 !=4; 8 ,/45( c2]\B$D5A x{W\³D;!=.D57B$+CD;!=. 2(*B$+C!=0 ( §2z\B *!O+C(V(J+2]\_!u?a=3*.( 4;!=A7§2/(V8 ( 8 !O4545D;7D;!=.
KA"!2,/D;+12( A7Bi?=!=. # b 1SËjC%k u¢l +:¶£kJlRfOnpo5qmrut=t=~ .0f + !¤fOjµl¢¶È kJl<ÇSfOj uÍ@[email protected]~w¦x±©H!=,/AEG,]\³,/.(
+C(*454;(§BiIOB$4BO.8:9/(*45(*8*+CA"!=.D;E,/(B$D;+R4;D5( ,zqD54{B$,/+ EG,(4;(§8:9NBO0 K *45(*8*+CA"D5EG,(§"!=D;+ ",KZ*A7D;( ,A ,.
8:9NBO0 K 2/(A7,/K/+C,A"(
|&Y/A7(iB$O(*IO( . $N(*452&Jx>4m8 !=A"A7(*7KL!=.2 B$, 8:9NBO0 K *45( 8J+CA"D5EG,(. 8*( 77BOD5A"(
KL!=,Am8 !=0 KZ(*.7(*AP45B1KLE( beA"+7([2z\³*.( A"a=D5([45!=A"2(*P8*!=454;D57D;!=.ªBiI=(*8D5!O.D57B$+CD;!=.BuIO( 84;( m01!O45 8*,45(*ª2(
4y\³BOD;A x$WX({8:9NBO0 K *45(*8*+CA"D5EG,(`2(`2/ 8 4;( .8:9/( 0 ( [email protected]+±( "+¤D;7", 2/( +CA7BiIOB$,R2*
( .VfOjCÍ:¶/fOn¢nZkJl fOnpobrut=t=·$Y/w]
Ebe = ±167ρA (z)
BiI=(*8
ρA (z) = 1.208exp
−z
8.4
|y~x·O¥@
*\³(µ/K/A7D50 (V( . x_0 −1 (*+4¦\B$4;+CDp+C,2/( V(*.£G0x -,(*4545(
45(8:9NBO0 K£*45(*8*+CA"D5EG,(2/(VA7,K/+7,A7(
EG,(67!=Dp+`4y\³BO4p+CD;+7,2($q
( +mD5./*A7D5(*,EA`beBO,8:9NB$01K 4;( 8J+CA7D;EG,(9NBOYDp+C,(*454;( 0 ( [email protected]+S.*8 (*7CB$D5A7(6KL!=,A`45B
A7,/K/+C,A"(62D5*45( 8J+CA"D5EG,(OEx be.VfOjCÍ:¶/fOn¢nHkJl{fOnpoHb[email protected]·[email protected]+m,/+CD;45D5"6~OR7!O.2NBOa=(*ª2([8:9BO01K*45( 8J+CA"D5EG,(<
4y\[email protected]+7 A7D;( ,A{2]\³!OACBOa=(*`KZ!O,A`01!O.G+7A7(*A`E,/(
45D;01Dp+C(mYD5(*.45( SI$BO4;( ,A"`2,8:9NBO0 K*45(*8*+CA"D5EG,(2NB$.
Ebe
45(.G,NB$a=(Ox
^PKA"3 §BiIO!=D5A 2 $.D4;(2!=01BOD5.(KL!=,A 45(*E,/( 4m45(2 8*45( ./8:9( 0 ( [email protected]+§2]\_,. *8 45BOD5A ( +KL!=7"D5Y4;(Oq
!=. 8 BO458*,45(45(8:9NBO0 K *45( 8J+CA"D5EG,((*. +C!O,[email protected]+C*x %. 8 !=0 K/+:B$YD54;D57(4;( KZ!=D;[email protected]+Cc! # 45(8:9NBO0 K
4;( 8J+CA7D;EG,(F(*"+-",KL A7D;( ,APBO,8:9NB$01K*45(*8*+CA"D5EG,(<2(A",K/+C,/A7(F0F,4;+7D5K4;D5-KNB$AP,.B$8*+C(*,A-¥xt KZ!=,/A
+C(*.D5A[8 !=0 K/+7(c2(14BI$BOA"DBOYD;45Dp+C"!=,#0cB$D5454;(R2,Ä8:9NBO0 K *45( 8J+CA"D5EG,(O
x f. (*Z(*+ q !=. ./(KZ(*,/+RKNB$
2J+C( A"01D;.( ABiI=(*8-KA7*8 D;7D5!O.1EG,(*[email protected]+B["!=.0 !2,/45(ª45(ªK4;,{KA"!8:9/(P2,10cBD;0,0KNB$A70 DL8*( ,
[email protected]@+<,.0 !2,/45(F2/,¬8:9BO01K*45( 8J+CA"D5EG,(8*!=01KBOACBOY/45(OxzW(RKZ!=D;[email protected]+2(R2*8 45(*.8:9(*01(*.G+<2(R4¦\³*8 45BOD5A
( +m8:9!=D5"DXBO,9NBO7BOA72KNBOA"01D]4;( `[email protected]+7mEG,DA" KL!=.2(*[email protected]+P 4B8*!=.2Dp+CD;!=.§KA7*8 *2( [email protected]+C($x
% &Á a &À±¾¾ j.(<KA"( 0 D53 A"(6KNBOA"+7D5(62(64B KNBOA7BO0*+CA"D7B$+CD;!=.2( 8*4BOD;A *\BOK/K,D5(7,/A45BR+79 !OA7D5(62,45( BO2(*A
YD;2D5A"( 8*+7D5!=./.( 4<b®ÌFfÍukR
3 ¢jqmrut=Î=¥qmrit=v=$w¦x{W(8 BO.NBO4m2(4¦\*8 45BOD5A*[email protected]+§8 !O.2,8J+C( ,/A qS,./(8:9NBOA"a=(
I$B *+7A7(VD5./2,D;+7( 45B7 ,ABO8 (V2/, 8iBO.NB$4PK45BO8
2BO. §
45(8:9NBO0 K 45(*8*+7A7D5EG,(2, .,BOa=(OqEG,DPI$B
a*. A"( AF,/.KL!O+C(*.G+7D5(*4" ( 8*!=.2NBOD;A7($x %-.B2!=./8c,. KZ!O+7( [email protected]+CD;( 4¤8 !O. " [email protected]+4B7 ,ABO8 (12(14¦\* 8 45BOD5A
EG,D ( " +645B1A 7 ,4p+:[email protected]+7(2,KZ!$+C( [email protected]+7D5( 4HBO0Y/D[email protected]+45DF
BO,8:9NB$01K* 45( 8J+CA"D5EG,(2,V.G,NB$a=((*+-2,V8:9NBO0 K
4;( 8J+CA7D;EG,(7 (*8 !=./2NBOD5A"(OxWXB0 !=YD;45D;+ 2( KNBOA+CD;8 ,4;(F
.G,NB$a=( , " (* [email protected]++CA"!=K BOD5Y/45(Oqs8 (" [email protected]+ 45( 4;( 8J+CA7!O. ` 45D;YA7( q/K/A7!2,D;+ S KNBOA{D5!=.DC B$+7D5!=.145!=AS 2(8*!=454;D7 D;!=. ± 2NBO. ` 4;(-8iB$.NBO4yqEG,D7 !O.G+mR4¦\³!=A"D5a=D;.(
2( <
0 !=,/IO( 0 ( [email protected]+6 2( 8:9NB$A7a=( BO,7 (*D5.2,¬8iBO.NB$4yx u!=, 6 4y\³BO8*+7D5!=.V2/,¬8:9BO01K 4;( 8J+CA7D;EG,( BO0[email protected]+iq
¸S¹ m¹y¸
¿¤Á
Î=
8 ( *45(*8*+CA"!=. 7( 2 K/4BO8*( [email protected]+F2z\³,./(1(µ+7A *01Dp+ 2( 4y\ 8 45BOD5A-4¦\BO,+CA7($q8 ( EG,D¤8 A *(1,/.7,/A7K4;, 62(
8:9NBOA"a=(KL!7Dp+CD;IO(-2]\_,.8 pO+ (*+`. [email protected]$+7D;IO(2(4y\³BO,/+7A7(OxQ{( KL( [email protected]+iqN4B[8:9NBOA7aO(a=45!=YBO45(2,8iBO.NB$4z(+
.G,454;([email protected]+ E,/(§4y\ 8 45BOD5A<.(+7!=,8:9(§KNB R45("!=4yxQP\³( "+ 8 (§0 !=,/IO( 0 ( [email protected]+ 2( 45(*8*+7A7!=. R4;D5YA"([E,/D
(+c4¦\³!OA7D5aOD5.(2,KZ!O+7( [email protected]+CD;( 4 7(*8 !=./2NBOD5A"(OxZ}-BO. P45(<8iB 2]\³,/. 8*4BOD;AmEG,D B$+7+7( D;.G+45(7!O4yq4B 8:9NBOA"a=(
.(J+7+C(( +S2Dp *A7( [email protected]+7(P2( A"!qK,D "EG,(Oq/BOKA73 {8*[email protected]+:BO8J+`BiI=(*8-45( 7!=4¦qG45(ªKZ!$+C( [email protected]+7D5( 4L2,8 BO.NBO4 "( ACBFBO,
KL!O+C(*.G+7D5(*4]2, 7!=4¦x
}-BO. [.!O+7A7( 01!23 4;(Oq4;( 8:9BO01K 4;( 8*+7A7D;E,/(1,/+7D54;D RKL!=,A62 *8 A"D5A7( 4BKA"[email protected]+CD5!O.¬2/,45( BO2(*A
YD;2D5A"( 8*+7D5!=./.( 4H( "+<4;(R8:9NBO0 K *45(*8*+CA"D5EG,( BO0FYD[email protected]+(J+6.!=.¬45(8:9BO01
K 4;( 8J+CA7D;EG,(+C!O+CBO4yxz%!=, .(
+C(*.!=. <2!=./8 KB [8*!=0 K/+C( 2,¬8:9NBO0 K *45(*8*+CA"D5EG,( 45!8iBO4S|8*A KNBOA64;( 68:9NBOA"a=( <D5.2/,D;+7( 62NBO. <45(
8iB$.NBO4¢µqZ(J+62,VA7(*./!=A"8 ( 0 ( [email protected]+62,a=A7BO2D;( [email protected]+-2(RKL!O+7( [email protected]+CD;( 4 (J+2!=./8R2,8:9NBO0 K 45(*8*+7A7D5EG,(BO,V(µG#
+CA 0 D;+ {2/,§45(iB$2( A*x uDZ!=.I=(*,/+`+C(*.D5AS8*!=0 K/+C(2,8:9NBO0 K 45(*8*+7A7D5EG,(P45!8iBO4KL!=,[email protected]$+7D5!=.
2,45(iB$2( A*qLD;4]BO,/+m8iB$458 ,/45( Aª8*(F8:9NBO0 K 8:9NBOEG,(F(J+CA 0 D;+ 62( 7(*a=01(*[email protected]+ixZQ{( 45B 7D;a=.D $N(62!=.8<A µ#
7!O,2A7(P4y\ EG,NB$+7D5!=.2/(©H!=D "!=.]q/(J+`KNBOA8 !=. EG,(*[email protected]+,+CD54;D "( A±45( "!=4;IO( ,A{2(-KA"( "D5!=.§2/( &%'
8:9NBOEG,
( "( a=0 ( [email protected]+RA7B¯M =,/+ OxW(8 /+ ( . +7( 0 K R2(8 BO458*,4{( "+ BO4 =A <IACBOD;01(*.G+(J =A7Y/D;+:B$.G+ q (J+
. =, BiI =. mKA J A <BOaOD5A ",Aª45( 7(*,D542/(<K/A [email protected]$+CD =.]x
W =A "E,/(( .§,.K=D;.G+{2,§.G,NBOa=($q4;(P8:9NB$01K*45( 8J+CA"D5EG,(P2/(*ID5(*.G+ 7,K A"D5( ,/A`F,.(IOB$45( ,/A 7(*,D54
q{2(*, "( a=0 ( [email protected]+§2/(VK=45BOA7Dp+=KK *(7(VKA=KNBOaO( [email protected]+2NBO. §4;(2/D5A7(*8*+7D=. 1KNBOA7BO454;3 45( c(J+
Etrig
[email protected]+CD;KNBOA7BO454;3 45( `BO,8:9NBO0 K 4;( 8J+CA7D;EG,([01BD50F,0x/W([4;(iBO2(*AK"D;+CDpHI$B7([2 K45BO8 (*AP 4y\[email protected]+*A7D;( ,A
2(45B =.(2(F8:9NBOA7aO(. a=B$+CDpI=(Oq+:B$.2D EG,(45([45( BO2( A. a=B$+CDpI$B 7(2 K45BO8 (*A-2NBO. 4B =.([2(
8:9NBOA"a=(K 7Dp+CDpI=(OxHW(8:9NBO0 K 4;( 8J+CA7D;EG,( J+:[email protected]+R2D;A7D;a 12/(§4BA a=D =. 2(c8:9NBOA7aO(K 7D;+7D;IO(1I=(*A R45B
A *a=D O.§2(68:9NBOA7aO(. [email protected]+CD;IO(Oq4;(45( BO2( A`K "D;+CDpX( +ªKNBOACB$45453*45(PBO,8:9NBO0 K *45(*8*+CA"D5EG,($q/+:BO./2D `EG,(645(
45( BO2(*A`. [email protected]+CD;H( [email protected]+CD5KBOACBO4;453*45(BO,8:9BO01
K *45( 8J+CA"D5EG,(1| $NaO,A7([~x '=Jx
−
+
−
−
−
−
−
−
−
−
−
−
−
E
+
+
+
+
+
+
+
+
+
$4JPNAP?= $-?"[N* $4 ,T$4J+& #=c1 "=] ?=] H%$PN*J4\,? &#AZ13A ; #.0#=T1 219W $4JPNAP?= 2" &+A313"J'
" (*+&'&@?"& "%=] ?[=] H%$PN*J4 =]JPNII13A ; 2.)2=c1 _ <: ; 7 1 #"%$ H%., J#! 13$#A25"L+&#1X(I?[= (AZ6 $+AN*J ; #$+&0!
& _< $ ?5A $ ?["N*9$4 ,T7 $4J+&@#=T1 ,?AZ=c1w(*(*J !#= H#.7 0#=c1 7(*9-F J AZ$ 7 21<W013$ 1V?[AY$47 g(*X4F J AY$#&#1
6.)[1 J#$#A3[ 7 AY&@J4,; $M#&UAWN= +&=T?[AY$4+&@_<G#& $-= H7 +&w,7 ? &+A313A +&S& w,T$-?+7, N*2=5 c1(I?[= (I=E7 &SW ?=]9(* 7 H%[$3N*
;
7
7
=]JPN*[13A ; 21 F +&#1-F AZ= ; #$+&@O,?"%$9!+& $-6 = H+&7 o=]JPN*[13A ; +&@_
5
7
6
7
X¤~x 'i <G
W( 4;(iBO2/( AYD;2D5A"( 8*+7D=./.( 4 7( KA OKNBOa=(RKNB$A 7( aO01(*.G+ <[email protected]+6EG,( 45(R8:9NBO0 K 4;( 8*+7A7D;E,/( 4¦\³(µ
+CA 0 D;+ [2,2( A".D5(*A 7(*a=0 ( [email protected]+-( "+ ",K *A7D;( ,APBO, 7(*,D54s2([KA [email protected]$+7D=.
xz©=,A-A7B¯M=,/+7( A,.
Eprop
7(*a=0 ( [email protected]+SB$,c8iB$.NBO42(m4y\ 8 45BOD5A*q=.18iBO4;8 ,4;(m4¦\B$.a=45(`EG,(mBOD;+¤4;(`I=( 8J+C(*,AS8:9NBO0 K *45(*8*+CA"D5EG,(BiI=( 845( I=(*8*+7( ,Am2/,"? "+73 0 (2(P8 OA72 =./. ( x^ KNBOA"+7D5A±2(-8=. 7D;2 A7B$+CD=. 7,A{45({BO.a=4;(*q =.cKZ(*,/+`2 Î ã
$N.D;A 7D/45( "( a=0 ( [email protected]+I$B 7(ªKA=KNBOaO( A "( 4O.R4BIO( A"+7D58 BO45($q 7(*4=.R4¦\³9 =A7D [email protected]+:B$45( =,7( 4=.R4B2DBOa =.NBO4;(Oq
(*+m2 O.8=.KZ(*,/+m2 *+7( A70 D5./( A`45( ª8 =A72 O.. ( ª2(64B $N.2, 7( aO01(*.G+ x
W( +7ACBODp+C( 0 ( [email protected]+<2( *8 4B$D5A .G,NB$a=( =4±B*+ 7D;01K4;D $ $x f+:B$.G+[2 =./. 1EG,( 45( [D=. 6(*+<4¦\³(Jz(J+
8 OA =.NB[.( [email protected]+`KNB ±+CACB$D;+ 2BO. S4;(0 2/3 45($qG45(m8:9NBO0 K 4;( 8*+7A7D;E,/( O, S45(m.G,NBOa=((+S+CA=KcBOD;Y45(
K =,AKL( A"01(J+7+CA"(S4BKA [email protected]$+7D =.[2, 8iBO.NB$4/2(S4¦\*8 45BOD5AM", 7EG,]\³BO, =4¦xOW(S4;(iBO2/( A( "+¤2 =./8`8 =../( 8*+ BOA+CD $8 D5(*454;( 0 ( [email protected]+`BO
, =44 =A 7EG,]\_D54zB$+7+7( [email protected]+ª4¦\BO4p+CDp+C,2(2([~0 KNBOA`ACB$KKOA"+mBO, O4yq(J+ 7BRK=45BOA7Dp+
2 J+C( A"01D;.(m8 ( 4;45(2,X
Q nRx/©HBOA{8 =. *EG,( [email protected]+iq 7Dz,/.45( BO2( A. a=B$+CDpB$+"+C( D;[email protected]+S45( O4yqG4y\ 8*4BOD;A±A",4;[email protected]+
( +m,.XQ n :q/(*+ 7D,.45( BO2(*A`K "D;+7D;HB$+7+7( D;.G+ª4;( =4¦q =.BR,.XQ n x
WXB K/A [email protected]$+CD =.2,45( BO2(*ASYD52/D5A7(*8*+7D =../( 4z( "+ "+ =KK (64 OA 7EG,(4;(8:9NBO0 K 45(*8*+7A7D5EG,(64¦\³(µ x
+CA" 0 D;+ -2,2( A".D5(*A "( a=0 ( [email protected]+m2(JID;( [email protected]+mD5./ *A7D;( ,AªBO
, "( ,D;42(6KA [email protected]$+7D =.
Eprop
¿a/¿! Á ½ a &¿ % ¿¤¾gaÃ XÆ¾ Q{( KL( [email protected]+iqz4=A "EG,]\ O . = Y "( A"IO(,. *8 45BOD5A*q =. 7([A7(*.28 O01K/+7(FEG,]\_D54X.]\_(+KNB ,.D5EG,(*01(*[email protected]+
8 O. "+7D;+C, [
2]\_,.8 BO.NBO4IO( A"+7D58 BO4yx %-.A"( 01BOA7EG,([K45,/+#pO+m,.( "+CA",8*+7,A7(<I=(*A"+CD;8iBO4;([Y/ACBO.8:9 ((J+P2( (J+7( . " D= . ¤
9 O A7D = [email protected]+:BO4;(H D50 K= A+:[email protected]+7( x ]Íuh¡N5ÍPkJlnÍJ¡zkJjbrut=v'w O.G+±B$.NBO4;? S2(K/9 O+ Oa=ACBOK/9D5( 2]\ 8*4BOD;A |y~O}-(J+ = [email protected]+0 = [email protected]+CA ¬
EG,(¬45( 8*4BOD;A = [email protected]+,.(¬a = 0* +7A7D;(ACBO8J+:BO4;( B 8 D ( ,/.(
2D;01(*. 7 D= .ACBO8J+:BO4;(2( r=x³ ã x
W\ = Y 7 ( AI$B$+CD= . 2( 0 D7 DO . 'qKBOA¤45( * 8 45BOD5Aª
b¶/fhkJlXÌ<jCk:¶/Ê ykJn>qrut=t=Î Ì<jCk:¶/Ê ykJnzkJlfOnpo5q
~O¥=¥O¥ ª
5 5Íuh¡ kJlfOnpo5q¤rut=tOt ]¶Nh73 fÍ§kJlfOnpo5qH~O¥=¥/r*w{BA J I 4c
45B.NB+C,A"(cYD;2D5A"( 8*+7D= ./.( 4;45(R2( F
2 8:9NBOA"a=(s
[email protected]+7ACB .,BOa=(¤BOD;. 7 D=EG,(±4B +CA7,/8*+C,/A7(m2(*,<.DpI=(iB$,62(±8 (+>?KZ(±2]\ 8*4BOD;A x¯Q{(J+7+C(" +7A7,8J+C,A"(
BOKKBOACBl;+8 45BOD5A"( 0 ( [email protected]+ " ,A¤4B $a=,A7(6r=xpr¯·xOW(m4;(iBO2/( AYD;2D5A"( 8J+CDO ..( 4.]\_(" +±2 O .8`KNB 7 ,Tr 7 [email protected]+±K= ,/A
A7(*KA 2,D;A7(14B +CA",8*+7,A7(§a=4= YNBO4;( (*+R4¦\³(µ+7( . " D= .9 = A7D = [email protected]+CBO45( 2( * 8 45BOD5A xX>4{( " +RD;01K= A"[email protected]+<2(
A7(*KA 2,D;A7([BO,0 D5(*,4;( 8 45BOD5A
K/,D" E,/(F45(68:9(*01D;.EG,]\_D54 " ,D;IO( [email protected]+PI$B12 * +C(*A70 D5.(*A45B EG,NB$.G+7D;+ 2(S8:9BOA7a=(.( ,+CACB$45D ($q$45BmA * EG,( .8*(Oq=45B4 8 BO45DC B$+7D= .(J+4y\_(J+7( . " D= .2/( * 8 4B$D5A * xO},KO [email protected]+ 2/({I,(
2(45B18:9D;01D;(FB+C0 7 K9 A7D;EG,(OqZ4B1KBOACBO0* +7A7DC B$+7D= .2( 8 45BOD5A Bc,/.VD50 KNBO8*+ " ,A-4B1KA 2,8J+CDO .
2( % % x<u( 4O . RfO¡ kJlfOnpobprit=t=vOY/w¦qz4;(F+:B$,V2( % %¨K/A 2,D;+PKZ(*,/+6*+7A7(R( " +CD;0 F
KNBOA-,./D;+F
2(
x
4O .a=,(*,A2(ª4y\ 8*4BOD;A x$© BOA8= . EG,( [email protected]+ q = A"8 (`( " +±2(ª8 = . " +:B+C( AEG,(ª45B-4= .aO,( ,A+ O +:B$45(S2(`4¦\* 8 45BOD5A
( +m,. * 4* 01(*.G+ª8*46
K= ,A`( +CD;01(*AªBO,01D;( ,4BRKA 2,8J+CD= .§2(<% % KNB$Am4;( * 8 4B$D5A * x
¸S¹ m¹ `
½ À±Æ ½
x
-yk3ckJÈ@kJj§kJlfOnpobrut=v ã w [email protected]+F0 [email protected]+CA1EG,]\_,. 0 23 4;( "+ 8:9NB +CD;E,/(c2( A7,K+C,A"(c2D 45(*8*+7A7D5EG,(
8 O.2,Dp+.NB$+7,A7(*4545(*01(*[email protected]+-2( "+7A7,8J+C,A"(ACBO8J+:BO4;(*xGÌFf±fÍ f kJl.fOl¢ÍJËËj"fcb³~O¥=¥=¥$w [email protected]+¤BOKK/45D5EG, 8 (ª0 2/3 45(2(ªA7,K+C,A"(m2/D 4;( 8J+CA7D;EG,(`4y\³B$+C0 "K93 A"(S(*+ [email protected]+¤2 O.8`[email protected]+CA2/,D;+45B-. O+7D=.2/(ªA7BO8*+CBO45(
2NBO. -45(*,AP0 2/3 45(<2(K/A =a=A"( 7D O.2,V45( BO2(*APK=,A "D50,/45( Am4B + =A+C, 7D;+ [(J+45( PYA7BO.8:9(*01(*[email protected]+
2,45( BO2( A*xQ{( m2(*,0 23*45( \³BOKK,D;( [email protected]+ 7,Am4B0 0 (KA 8 2,A"(
45B $Na=,A"(<8*A %l;+ *E,/( [email protected]+CD;( 454;( 0 ( [email protected]+[| $a=,A7([~x³[email protected]
,./(<K/A =YNB$YD54;D;+ 2 KL( ./[email protected]+P2/,
K O+C(*[email protected]+CD5(*4]4 8iB$4z( +-B 8 D (6 8:9NBOEG,(<45D5BOD =.K "D5Y4;(
£ 8:9NBOEG,
( *+CBOKZ(6(J+ J+:[email protected]+m2 =.. (<4BR2D +CA"D5Y,/+7D =.§2/(<K/A =YNB$YD54;D;+ $q/,./(<4;DBOD =.§( "+m8:9 =D 7D;(
BO,9NB CBOA"2(J+A7B¯M =,/+ (<BO,0 O+CDp&q¯M =D;[email protected]+`,.K [email protected]+ª2(64B $a=,A7(<R,.. =,/IO(iBO,K [email protected]+ix
W\BO2BOK/+:B+CD O. 2(8*(*+7+7(VKA 8 *2,A"(IOB . =, KL( A"01(J+7+CA"(2(VA"( KA 7(*[email protected]+C( A45B + =A+C, 7D;+ 2( 8*4BOD;A -(J+62]\³B$+7+7( D5./2A7(R2( 6A aOD =. 2/,¬.G,NB$a=( 2NBO. 4;( 7EG,(*454;( 4B2/( . "D;+ 2(R8:9BOA7a=( ( "+ 4;(*I *(
01BOD # 4;(8:9NBO0 K 4;( 8*+7A7D;E,/(§( "+ BOD;Y45($x %-.ÄI$B2 =.8§8 O0YD5./( A,./(B$KKA 8:9(§ACB$8*+:B$45(2( 8*4BOD;A {4B[+7( 8:9.D;EG,(P x D;452 qHD5A7(µ2( .0f +21h4j 3 fO¡kJl{fOnpo]b³~O¥=¥r wy]x j.( =D SEG,(4;(P4;(iBO2(*ASYD;2D Î@·
XN~x³v\
$#AZY! .)?=c13$-=c1 "%=]w,?$213AP?=L(*o(*J H%[$3N*R_1<G+& #$ !+& = &Q$- ,T$-J#&@#=c1 2=c1<!+&<,?AY=T1Z&
(*oN$+AZZg=]?= ?=c=] 1 J+&
(*J H%$PN* _"<:+& #$ +&Q7 ,]#AZ=&)21O!+&7 UAW7 N[=]+&6 ,]#A7 Z=]+&mAZ=T(A2"2=c1 H#.mAZ=
7
7
7 AZ$-J (
(* C(*J H%$PN* _9<G#&0&@VN.0#=c1Z& #=7 z,?AZ=c13AZY!J+&07 $4 ,T7 $4J+&@2=c1 #=c1o+&=T?" ; #Z+& UA3AY&2?[=&Q,? &+&+A #&@_ 8:
6
z=E&@#Z:7 21 _
@_
5
A7(*8*+7D=../( 4B $N.D2( "({KA=KNBOaO( A q4y\ 8*4BOD;A]I$BPKO,/I=D;A "({YA7BO.8:9(*A(*+ \ *+7( .2A"(S9 =A"D [email protected]+:BO4;( 0 ( [email protected]+ix
W( <K [email protected]+EG,DI [email protected]+FJ+CA7( A7B$+7+CBO8:9 [BO,¬8iBO.BO42 =DpI=( [email protected]+=Y D;A62/(<8 A7Dp+C3*A7( ",A[45B2( . 7D;+ 2(
8:9NBOA"a=(R(*+ ",A45(FKO+7( [email protected]+CD;( 4 45(*8*+7A7D5EG,($x]W\ 8 45BOD5A8 ( 7( 2( "(RKA =KBOa=( AP4=A7EG,]\_D54B+7+C(*[email protected]+2( A a=D =. 2(BOD5Y4;({2(*. 7Dp+ `2(`8:9BOA7a=( O,R2(A a=D=. 2NB$. 45( 7EG,( 4;45( 45(S8:9BO01K*45( 8J+CA"D5EG,( *\ =K/K"(
CB KA [email protected]$+7D =.zx
%O+7A7( 78:9 01B 2/(<Y/ACBO.8:9/( 0 ( [email protected]+P8=0 K=A+C(-K/45, "D5( ,/A *+CBOKL([| $NaO,A7(<~x³[email protected]µx f.KA"( 0 D5(*AS45D;( ,]q
=.8 =. +CA",D;+m,.([0cB+CA7D;8 (2 *8 A"D;[email protected]+P45B +7ACB¯M"(*8*+ =D5A"(<2/(F4y\ 8*4BOD;AS(*+,.([0cB+CA7D;8 ((*.GID;A =../( 0 ( [email protected]+
EG,DA7(*8 ( . 7(-4;( ±K =D;[email protected]+ ¤A7(*01K4;[email protected]+±2(±8 O.2D;+7D=. ",A±4B62( . "D;+ 2(m8:9NBOA"a=(P(J+ ",A±45(ªK$+C( [email protected]+7D5( 4
4;( 8J+CA7D;EG,(OxLQ{( m8 =.2Dp+CD =. O.G+m4;( 7,/D;[email protected]+C( [
(
|Q(i, j, k)|
|V (i, j, k)|
≥
≥
Qexcess
|V0 |
|y~x·r¯
8 =A7A"(7K=.2BO,7(*,D54z2(68:9NBOA7aO(K=,/A`45(`Y/ACBO.8:9/( 0 ( [email protected]+<q(*+ V (+4;(K$+C( [email protected]+7D5( 4Z0 u?O( .
2,8iBO.NB$4s4;(iBO2/( A x %-Am45(6. =0Y/A7(<2([Y/ACBO.8:9/(-2(<4y\ 8*4BOD;A`2 =pD +OY *D5Am 0,/.(F4=D2(6+>?KL([A7BO8*+CBO45(
b -yk3ckJÈ@kJjkJlSfOnpo5q]rit=v ã w
|y~x·=~=
N (d) ∼ dχ−1
# N ( "+¤45(`. =0YA"(S0cB¯/D;0,0 2(`YA7BO.8:9( ,./(`2D +:BO.8*( d 2, K [email protected]+2z\³D;.D;+7DB$+7D =.<2(`4y\ 8*4BOD;A q
(*+ ( +64B2D;01(*. 7D =.A7BO8*+CBO45([2(R4y\ 8*4BOD;
A =,2D;01(*. 7D =.2( 'BO, "2 =A"HqZ+7( 4;45(EG,(§r
kJ~ l
2NBO. χ,.(a =
0 *+7A7D5(1~2D;01(*. 7D =. <(*+ ~
2NBO. F45(c8 B R2D50 ( . "D =. b®Çm kJlχjCh
ÇªkJl¦jCh fNqzrutOt=$w¦x>4BO,/+{2 =.8P2D;ID "( A45(m2 =01 BOD5.χ(m2 ( "D50,/4B$+7D =. ( . "K93*A7( <| =,8*( A78*45( S( .~O}-
8 (*[email protected]+CA *( 6BO,K =D;[email protected]+P2]\_D5./D;+CD5B$+CD =.§2(4¦\ *8 45BOD5A[| $a=,A7(~x³[email protected]JxLW(ACBi? =.2(8*( "K93*A7( ( "+ *[email protected]c45B
K4;, `KZ(J+CDp+C(A =4;,/+CD =
. 7KNB+CDB$45(-0,/4;+CD;K45D (mKNBOAª,.( [email protected]+CD;( A*x
}-BO. ,. KA7(*01D;( A+7( 0 K q =. I$BV(*Z( 8*+7,( A 45( YACBO./8:9( 0 ( [email protected]+ 12(ACBO./a rOq8O\_( "+ & 2D5A"(§45( YA7BO.8:9(*01(*.G+ ª( [email protected]+CA"(45(-4;(iBO2(*ASYD;2D5A"( 8*+7D =./.( 4(*+ª4;( SK =D;.G+ `BO2M7B$8 ( [email protected]+ ª8 ( 4;,D 8 D¦x&%-.§A"( 8:9(*A78:9(
BO4 =A S+ =,/+C( ª45( `. =,/I=(*454;( ªYACB$.8:9( mK "D5Y4;( 6 =.(*Z( 8J+C,(<,.2 8 BO4BOaO(2(64B01B$+CA"D58 (8 [email protected]+C( [email protected]+ 4B+7ACB¯M"(*8*+ =D5A"(§2(4y\ 8 45BOD5A[2NBO. R+ =,/+C( 45( R2/D5A7(*8*+7D =. |&ruvV( . =}µxW(A ",4;+CB$+ ( +1,/.(
Qexcess
Î=Î
(n+1.5) mesh_size
Positive branches
(n+0.5) mesh_size
4
3
3
4
1
3
2
2
1
1
3
E
2
1
2
3
1
2
2
1
3
2
Negative branches
Xª~x³t #
<GL!-I(*2$
AP(AZ$- 13A3?[=E=]#X+&#1g$4 ,T$4J+&@#=T1 Ji#= =T?AZ$ _+O&@),]$ ?+, N*
,[$#13AZ$L(" ,?AZ=c1g(
6
7
(*J #= H#.0#=c1K $-?A =T?[AY$4 ^[_ =C$-?"[N* ?= (AY&#13AZ=N"Eg!+& $ = H+&,? &#AZ13A ; +& 2"A& S,T$-?+, N*2=c1S(I=E&
"%=]7 ?7 =](* H%[$37 N*= ; JPNI[13A ; 21 2= !#" 5 !+& $-= H#&S=]JPNII136 A ; #&M7 (I?=c1GW,T$-?+,RNI[13AP?= &@G>2[AZ1G(*=&O"=
$4JPNAP?= H$3NJ47 ,? &#AZ13A ; #.0#=c1 _5 <G#& H*6 A O$4+&5 #=c6 1V?"%$-J#7 &9(:F "%= #$ !oAY=T([A2"E#=c1 !Q=cA ; 4" (* $-= H#.0#=c1 _
7
7
7 7
6
7
01B$+CA"D58 ([email protected]+C(*[email protected]+F45B+CACBuM"( 8*+ =D5A"( 2(14¦\*8 45BOD5A(*+[45([email protected]+ [B$2M7BO8 (*[email protected]+[BO,EG,( 4 [email protected]+B$+"+CA"D5Y, ,.. =0YA"(P8 OA7A7( 7K=.2NB$.G+`BO,. =0FYA7(2(-45D5BOD=. ¤K "D5Y4;( ",A{8*(`[email protected]+{FKNBOA+CD;A2(4y\ 8*4BOD;A x
uDG4y\ =.<A7(J+CACB$.8:9(Sm8 (J+7+C(0cB+CA7D;8 (45Bª+7ACB¯M"(*8*+ OD5A7(¤2(4y\ 8*4BOD;A](*+HEG,]\ O.4Bª8 [email protected] =45,(BiI=(*8S4B`01B$+CA"D58 (
( [email protected] =..(*01(*[email protected]+iq =. =Y+CD5(*[email protected]+{,./(P01B$+CA"D58 (ª[email protected]+C( [email protected]+S45( K=D;.G+ EG,DLKL( ,/IO( [email protected]+`*+7A7(YACBO./8:9 *x %-.
8iB$458 ,/45(RBO4 =A 4BcK/A =YNB$YD54;D;+ <2(RYA7BO.8:9(*01(*[email protected]+[2/(R8 ( [email protected]+ qzEG,D ("+62 $N.D;(R8 =0 0 (F4y\[email protected]( A "(
2,. =0Y/A7(2(4;DBOD =. {K 7D;Y45( 7,Aª8*( mK =D;[email protected]+ S2(6a=A"D54;45(Ox&%-.§IOB BOD;. 7DzYNBO4Bi?O( A`45(2 =0cB$D5.(2(
7D;0,45B$+CD =.KNB$A"+CD;A62,K [email protected]+<2]\_D5.Dp+CD5B$+CD =.( .,/+CD;45D [email protected]+645( "K93 A"( 2 $N.D;( <K45, [9NB$,/+ixs© O,A
8:9NBOEG,( "K93*A7(Oq =.8 BO458*,45(45(m. =0YA"(P2(PYACBO./8:9( `8 =01K/A7D "( ( [email protected]+CA"(-4
B 7K93*A7(8 =. 7D;2 A (-(J+S45B
xN>4]?§B B$4 =A S2( ,K 7D;YD54;D;+ -
7K/93 A"([D;01
0 2D5B$+C(*01(*[email protected]+SK4;, `KZ(J+CDp+C(
Nexistantes
=D;+R4;(§. =0Y/A7(§2(YA7BO.8:9( c8*(*+7+7(2/D "+CBO.8*(§( "+ 2 >M7
7,K A"D5( ,/A O, *[email protected]`B$, . =0YA"(
01BD50,/0 2(6YA7BO.8:9( 2 8 A"D;+mKNBOA`45B4 =DZACBO8J+:BO4;(
=D;+ª4;(. =0YA"(2(6YACB$.8:9( 4B2D "+:B$.8 (
( "+mD;./ A"D5( ,/Aª
x
d
}-BO. F45( KA"( 0 D5(*A<8 B qsBO,/8 ,. YACBO./8:9( 0 ( [email protected]+.z\³( "+
A7B¯M =,/+ OqX(*+ =.¬KNNB (d)
7(4
B 7K/93 A"
( 7,[email protected]+C(Ox
}-BO. 4;( 7( 8 =.28iB *qL4;( ªK [email protected]+ `2/([a=A"D54;45(PEG,D "( A O.G+mA"( 45D mR4y\ 8*4BOD;A [email protected]+m8 (*,EG,D [email protected]+m4BRK45, a=A7BO.2(KA =YNBOY/D545Dp+ m2(-YA7BO.8:9(*01(*[email protected]+ix %-.Y =A".(-4;( SKA =YNBOY/D545Dp+ S4y\³BOD;2(P2]\_,
. 7(*,D54 7,K A"D5( ,/A
(J+`2]\_,
. 7(*,D54]D;./ A"D5(*,A
(*+ =.§8 BO458*,45(45(-. =0YA"(2(K [email protected]+ 2 [email protected]+ª4BFKA =YNBOY/D545Dp+ pnew
2(6YA7BO.8:9(*01(*.G+( + 7,K A"D5( p,/old
A7(<
xN©4;, 7D;( ,A m8 B ª2( $Na=,A"(6KZ(*,/NI=(*seuil
.G+ 7(<KA 7(*[email protected]+C( A<
pnew
r=x
( "+ a=BO4]BO,§. =0YA"(2(-K [email protected]+ {. 8*( 7BOD5A"( ªK =,A`BOA"A7DpI=( ASBO,§. =0Y/A7(-01BD50,/0 2(
Nseuil
YA7BO.8:9( ~x
( + ",K A7D;( ,Am
Nseuil
N (d) − Nexistantes
Î '
/ x
(+D;./ A"D5(*,A`
x
Nseuil
N (d) − Nexistantes
W(-8iB r("+S45(-K4;, "D50 K45(< =.§Y/ACBO.8:9/(45({K=D;[email protected]+S2 O.G+S4BFKA=YNBOY/D545Dp+m(+ 7,K A"D5(*,A7(6
(*+ =.1KNB "([45B"K93*A7( ",D;[email protected]+7(Ox}-BO. 45(ª8iB S~qG8 =0 0 ( O.B[K/45, ±2(PK=D;.G+ ¤K7D;Y45( EG,p(new
2(
YA7BO.8:9(*01(*.G+ BO,/+ =A7D q=.BO,/a=01(*[email protected]+C(<45(7( ,/D54s2(FKA=YNBOY/D545Dp+
2(
O~qN(*+ =.
pnew
|pnew − pold |
A *I$BO4;,([4;(6. =0YA"([2/([K [email protected]+ [email protected]$.G+P,.(<KA OYNBOYD;45D;+ 7,/K *A7D5(*,A7(F18*([. O,/I=( BO, 7(*,D54¦x© =,/AP45(
+CA =D "D53*01(-8 B q8$\³( "+4y\[email protected]=(*A 7( =.BOYNBOD "(<4;( 7(*,D542(<KA =YNBOYD;45Dp+ 2/(
O~x %-.
pnew
|pnew − pold |
KA 8 2(PBOD5. "DNKNBOA±2D58:9 O+ =0 D5(`K =,A =K+CD50 D "( A4;(m. =0YA"(2(mYA7BO.8:9( `<,/.(P2D "+CBO.8 (m2 =./. ($x
uDK/45, "D5( ,/A 6
K [email protected]+ -2/( aOA7D54;45( [email protected]+64B0 0 (RKA =YBOYD54;D;+ [2( YA7BO.8:9(*01(*[email protected]+F(J+6E,/( 45B2D; A"( .8*(
( [email protected]+7A7(4;( P2/( ,2( A".D5(*A "( ,D;4 2(KA =YNBOYD;45Dp+ ( +D5. A7D;( ,A"(c,/.(FKA =YNBOY/D545Dp+ 60 D5.D;0,0q45( K [email protected]+ ¤8 =..(*8*+ [email protected]+{8:9 OD 7D ¤2(mB e =.1BO4 B$+ OD5A7($x j./(m =D 45( YA7BO.8:9(*01(*[email protected]+ {(Jz(*8*+7, {<8*(*+7+7(
2D "+:B$.8 ($q =. KNB "(45
B "K93*A7( ",D;[email protected]+C($x %-. KB "(VBO, YA7BO.8:9(*01(*.G+ 2(ACBO.
a 7,K A"D5( ,/A
4 OA 7EG,(64;( `45D;01Dp+C( {2,.G,NBOa=( O.G+B$+"+C(*[email protected]+C( O,EG,(<K4;, ªBO,8*,.Y/ACBO.8:9/( 0 ( [email protected]+P.z\³( "+K 7D;Y45($x
Q{( 78:9 01B2(PYA7BO.8:9(*01(*[email protected]+`KZ(*A70 (*+{2 =.82]\ OY/+C(*.D5A,.(-(µ+7( . "D =.c9 =A7D [email protected]+CBO45(m2(P4¦\ *8 45BOD5A
EG,D`.(KL( ,/+ *+7A7( =Y/+7( .G,(BiI=( 845(§Y/D >4;(iBO2/( A "( ,4¦x¤}P(K45, *q4y\[email protected]+CA 2/,8*+7D =. 2/(45B2D50 ( . 7D O.
ACB$8*+:B$45(-KL( A"01(J+2/(<A"( KA 2/,D5A"(4BF+ OA"+C, 7Dp+ (*+m4;( `YA7BO.8:9( 2( *8 45BOD5A x
6 ! ! #
W 7a \ "! Xyk On5kJj[kJl&.f0+21hj43 fO¡Äbrut=t ã w#$"!
. f0+21hj3 fO¡kJlfOnposb³~O¥=¥/r*wyxW%&' "a " "() #)*+! , ! ' "a
V
,<- , q xN} 1
excess
|y~x·O@
|q(i,
j, k)| > qexcess
,.*/'(7a $01 2 3Y a! a 341 !2¦\ *x
q(i, j, k)
} -) -! * 5 67 Oa Oq8{9 :07;< ! = >! ? §a =Y 4!
y\ * @ A $x %? ! B' "a C ( C! ¦\
*( q1$ #C]\"' &1 Oq
?! ,5Y /' 7a 2,! δq . & " & ! y\ * x
(
q(i,j,k)
δq(i, j, k) = |q(i,j,k)|
× (|q(i, j, k)| − qexcess)
δq(i, j, k) = 0
neut
δqneut =
X
i,j,k
δq(i, j, k) ×
,. 3Y ,! *a 34 "! y\ * xW5' 7a !,;Y !D71 q')'
*/ ! = )! :( a OY' ! y\ * 6|QE 01 34 µ-
N
!
|y~x· ã
1
N
dq(i, j, k)
= δq(i, j, k) − δqneut
dt
"
34
|y~x·=·=
(±.¦\F , %! B ' 1! Gy\ * HI D -|9! Gy\ !J! =¥=¥=¥=¥ <Jq
.! +y\F *xLW $ C .3 =Y "01 B. K:! 3 , qL M! 8
!'( [email protected] '(! &¦\*(%D J!]\FOY !5 8M M: Zx%5,! ,,5Y ! G O $ ! J*P' "a % #7N(ZqNQ*P ,M,, % K7! 3 ,*x
W.34
3
Î=v
W5' 7a +(3Y *(1 <5*/ 7a ! ¦\*( q734 P#.;Y 01 7N(3- */ 7a 2! . "! 3 3- * ,01 +/' "a ! y\ *x % ! a ! !
=' P "*/ ( ! .
,8*4' 7a 5! : ;
' "a ! ,5Y ! / * , $q# &7N()
AD ,L`5*5 ( ]xW5' 7a +,! ,,5Y C3- * Ama ! J-D( GOY " C' 7a ! .*D( A J *'(7a
,! ,,5Y xQ
W 5'(7a , D7 5
:(,&&' "a .! . K7! 3 $ x
,Q. .: $C. " ! = C! +3 3- * ,$q "7N(?
+' , A ' X&
*P ]q $*@ 8! P' 7a &,! ,,5Y ,,H&
/' 7a 2!.1K7! 3 x
W/N C! P ' @P K7! 3 $ &! A !) , 3- &! C* ,$ )sr-N ! '(, *x
W ,$ Rr-N + ,.! . 01 2 K7A!]F\ K7! 3 $ [
W5' 7a
σ(i, j, k) =
2 ,.!
Z
+∞
0
πD(i, j, k)2 N (i, j, k)dD
& ,LI2X
|¦~x·OÎ@
$
dq(i, j, k)
σ(l, i, j, k)
= [δq(l, i, j, k) − δqneut ] (1 − α) PN species
dt
σ(l, i, j, k)
l=1
?sr-N 1
|y~x·'=
3 07&! L3-J! ,3/8P$K:&! &[email protected] 3- $ ! G 7a $01 .!' K ,I34 $a | $Na ,~x;r=ruJxW%' 7a !]\FOY ! ' " 7
C ! ;&q:3 &y\ ( !;1'Oa +Am$2/ ,'(. , J
! p'01 !5! Py\3- , J! I M 801]\ 5'(34 ,01 P P ,a !' +7 Oa Ox j q%' 7a 1 ' /a 2 B1K7! 3 C! 1D @
3- K 01 *q7$ , G' J2 D $ [email protected] ND7$xGW ,07C
' 3- $,01 -y\ "! 1'$a B! $D7 , P4 BD( , yq "
! $ x/ A,@ 3- . !]\ ! Y ! ,$ ! 1 .,I.! Oa ( -SY' B 34 yxX^ $' ! 4 , 3- *) , = '(-! * q! Y 1<a < 34 , : , 01 -07 = Y <? O $ 5! ! ! .
3 7 ! [email protected]!2, ! 07OxW% & ' 7a " = P C a5! ' ! ¦\
*( x} & -Y :"! 3B @CK72!D:! . 7a q' D (*$Y , 01 7
< 01 2 K7A+!z\F K7! 3 $ Ox
W\$ A;01 , -) 7 !'Oa ?! 3 7 ! 07 , -* D( !'( ! 3 7 ! I)- ' I,3- ! 5' 01 ' 3
, ( ;34 K OxW%/ ! 5*45a +!Ra ~$} !
Q %©Okªv/r , ) 7$Y ! ?! Y * !; )'(,) D D , p 3-/! 9 =! a 3-/L34 12! K , [email protected]<Y A,34$ +!-3-,,5
ND( + : , B
3- ! 1!-D7 07Oq%! 5D $'2*)*( ,I $
4 ,)* !K7' 3-01 OqH* 3- 1K , 01 $5y\ O q34 +
P!]\ ! /
D( OY /
! /
7 , * a ! ¦\ Oa $xXWQ 3/ * ! , K I3-5 D p ! N
=}FqA 3-3-+ /3/ = A, Oq A3-C!]\ ! y\ D W%
Î=t
07Oq J, a ,!' & *Oa
ND 5* 3- K 01 /y\ OxL}C C! N C ! 3 $01 /3=1
! ! @!
5Y=1y\ $( 5! 1'Oa 34B!]\ xQ 3-3- D M!'( J&' ,Oq 3 3- , P 7 G ! ; , G ' 3-3- ! & 3-34
2 : ,.21'Oa /! C /! +! /01 01 3-1*qL! [a ( ! C , ! ,Y , m/ @! *2 K ,01 ! ) : "
P! , / 3 ( ]x}C , ! D ! F$B*L \F K C! ; , ' 3 $ , " ! , ;Y @! 7 x©8(& *
q7! ! & . 3 , "01 7 , " ! p
A 2 #p 4!'( +
4! 3- 5!]\ * [
QXn xHW\F34 ! 3 N 3- ! p
! /( , ! O xZWQ-! I3-B'(,+! 3 40783 = ! /! 5Y=
. 34-!]\ * * q #]q' 343- ?A.¦\F3 Oa q1# , . C!]\ D( &¦\F34 !
Y ( 3- q B! D $'B* =Y )! , 34 D7 & D:+
! * 3 ( , ! O((G! HY 3- *
x= 1M3- , [email protected]/ = #M ! ;Y 3-C* =Y ! 34 3 8 CD $ 3- & ! 3 /
34 K x % @! 3/ + /
N 5
!; 5
! ) D $ ! N ( 4 ! 5) 3- 341!:K: 3401 OqM3- 1K , 01 ? $01 )! 3 7 ! IOqH :
@ 3-' N , 01 [email protected]!P! p 1 , K I3-*x j. ,3-I,. =D 2! +D( !' )! 3 :
! I2ND B! = Y , ,D(( , [a 2t F-* 3/ * )!
y\ Oa .! ru¥,ªrutOt=Î =
Y , D ! \uTk9f¤Tª^ %[email protected] P &y\ 7 N 4! A , . N( +! 3 07&07. &* 3B *( +! "* :! 5!]\ i 7K:!
!]\F (
!' " Oa *x
34
,@$ '(* @! &¦\ Oa
!]\ 4
18! &D7
'O¥
Branchements de rang n
Branches positives (puis branches négatives)
Points possibles tels que |q| > qexcess et | ϕ| > ϕ0
n=n+1
Calcul de la probabilité de branchement p pour ces points
Pas de branchement possible
n=n+1
oui
Σp = 0
oui
non
non
Sphère de rayon A*r
A*r = Rmax
Nexistants = nombre de points entre les sphères de rayon (A−1)*r et A*r
oui
Nexistants > Nmax
Pas de branchement possible
A=A+1
non
Définition de 2 seuils de probabilitép:new et pold
Nseuil = nombre de points tels que p > pnew
Nseuil = Nmax −Nexistants
Nseuil < Nmax − Nexistants
Nseuil > Nmax − Nexistants
Augmentation du seuil de probabilité
pstep = pnew − pold
2
Diminution du seuil de probabilité
pstep = − pnew − pold
2
Nseuil tel que p > pnew + pstep
oui
0.75
( Nmax − Nexistants )
<
Nseuil
<
( Nmax − Nexistants )
non
pmax = max(p) et pmin = min(p)
p max = pmin
non
oui
pold = pnew
pnew = pnew + pstep
Tirage aléatoire de ( Nmax − Nexistants ) points
Branchements de ces points
A=A+1
non
A*r = Rmax
oui
6 HJ#.)C(*9,T$#AZ= A , (*+& $ [= H#.0#=T1Z&@_
+&#1M"%= #=c13AV#$X,? &+A313A >
+&#1M0,TU"*&X,E213AZ1
7
7
6
7
A
5
r
$4J+&2?U"13A3?=L&V,[13A3[
#&#1<!9$ DI?=z.) AZ.m". (+&Q&V,cH/2$-+& 21
+&21<!X=]?. $-o.) AZ.m". (
N&2max
$-= H+&g"1V?$#A & J4+&5 R0max
W (AY&#1V=
_<G; +&["%13$4+& $-J ; A3[135 AP?=&o
?=c1(*J $#A31 +&g(I6 =E&Q!1 ; 1 _
6
7
7 A×r
6 6
7
;
X~xpru¥
'r
Séparation et échange de charge
Processus
microphysiques et
dynamiques
Transport des charges
E > Ebe
Production
de NOx
T+dT
Calcul du champ électrique
non
oui
Leader bidirectionnel
E > Eprop
Extension verticale
du canal de l’éclair
oui
non
Schéma pseudo−fractal
Extension horizontale
du canal de l’éclair
Neutralisation partielle des charges
XZ~xpr=rg76
7
HJ#.) (*O,T$+AZ= A ,g(" .)? (["!\J#! 13$#A2"E\(I[=&
7
7
'=~
J+&2?
_
!"# " ! &$ %
(' ) "
7 %9W8Ok:35+0<* ; 7 * = 8 /> )k - ,/. 2?0214 35 0 46 /@ NW /* ,
BA
DC
!FE/GIHJEOË "LKNMFE/OQPSRT02141 ã/U @WVA XW
Y.Z,/[ 0/@\,4]^ @]W <*
_W
'/[
`C '/[/[
/abã ÎC
,4,^ A /@W dc Qe
f6 gA /* [email protected] j
_W n* i> ^ j[[email protected]@k[5u0 Y @ gA XW oWSa j 02[ (* l0m,l^
ã
__WW ([email protected] j p > l* q02. @ /a n*
[email protected] s3 /@
WVA /* +> IE/t
/EXu5E ÍquvHwRQ0x1 '/3 U
Q>[email protected] f z,h{la V>
|>
=0 Y,-R®Ç}KNMJ~ |KNM
Ç:KNMg~ XEIa02141/. U [email protected] W
>
[email protected]
n*
(* jC02[4[/[ @
u
/@
LWVA
Q> * * ha
+* s* /@]k 0 $
Y*
46

*
* BA 5
/@
; 5!
N 5! D !
, Q
,,D N(
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d(ρCg )
∂(ρCg )
∂(ρCg )
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∂t
d(ρCl )
∂(ρCg )
∂(ρCl )
∂(ρCl )
∂(ρVl Cl )
=
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|M T +
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dt
∂t
∂t
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|M T
∂t
∂t
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