momentum equation

Transcription

momentum equation

Structures hors équilibre à grande échelle en
plasmas naturels : limites de la MHD (2)
or: "The importance of not being Maxwellian"
Nicole Meyer-Vernet
CNRS, LESIA, Observatoire de Paris
SEMHD
ENS (11 Février 2008)
[email protected]
http://calys.obspm.fr/~meyer
N. Meyer-Vernet SEMHD 2008
1
What fluid (MHD) models are doing:
©
Basically: replace eqs. of motion of
N >> 1 particles
by a few local differential eqs.on a few
macroscopic parameters (n
(n,V,T,..)
continuity equation
momentum equation
energy equation
.....
Infinite hierarchy
of coupled
differential
equations
Closing the hierarchy:
assume nearly (bi-)Maxwellian
OR ad-hoc heat transfer, or ..
Planet. Space Sci.49 247 (2001)
(l ^ H )
Each species: one fluid with one T
(possibly Ty g Tz)
©
2
Niveaux de description Cas classique :
d ~ n-1/3 >> h/mv
v << c
N particules individuelles
fonction de distribution
fluide
état : quelques grandeurs macroscopiques n,
V, P, ... de l'élément de fluide en r, t
évolution : éqs. de conservation (nombre de
particules, qté de mouv., énergie, ..
u équations différentielles locales
en général f défini
par une infinité de
paramètres
yennes
mo
état : f(r, v) à t (proba. de trouver
particules avec position r, vitesse v)
évolution : éq. d'évolution de f(r, v)
mécanique statistique
description
impossible si N trop
grand (quoique ...)
yennes
mo
état : r, v pour chaque particule à t
évolution : éq. mouv. de chaque particule
mécanique analytique/constantes du mouv.
défini par
petit nombre de
paramètres
3
Problème: Fermeture de la hiérarchie: Equation d'énergie
Cas le plus simple: Pression isotrope P,
γ
2
Ø
Øt
V
P ∝ ρ u (Bernouilli)
V$= 2 +
ou:
V2
Flux de chaleur Q g 0 : V $ = 2 +
Q=-
κ =T
si lpm << L ~ T / =T
conductivité thermique classique
électrons
conduction
advection
MAIS :
i
T/L2
nVk B T/L
i
vthe lpm
VL
=0
kB T
−1 m
+
kB T
−1 m
+
G
G
=0
= − = $Q
κ ~ (3/2) n k v
κ g valeur classique dès que
B
l
the pm
u =.Q domine
lpm / L > 10-3
(Scudder & Olbert 1983, Shoub 1983, Canullo & al.1996, Dorelli & Scudder,
1999, 2003, Pantellini & Landi 2001)
Car :
Q déterminé par électrons suprathermiques
électrons suprathermiques non collisionnels (l >> lpm)
4
Collisions : Difference between plasmas and neutral gases
Neutral particles
NEUTRAL PARTICLES
collisional cross-section ~ "physical section"
in order of magnitude
Charged particles :
Coulomb interactions
e
Collision
rL
e
For a (relative) speed v, a significant perturbation
2
requires distance rL such that: 4 e0 rL i m v 2
Coulomb energy
cross section :
C
~ r L2
u free path l (v) ∝ v 4
∝ 1/ v 4
u if v X 3 then l (v) X 100 !
fast particules are collisionless, even when most particles
(the core of the velocity distribution) are collisional
(X ln )
Trajectoire d'une particule neutre
Trajectoire d'un électron en plasma
gazeux
rL<< …r i n −1/3
h
e2
4 0…r kBT
L D/…r = (4
^1
5
Slow electron
u la plupart des interactions
produisent petites perturbations
) −1/2 p 1 u 1 particule subit beaucoup
d'interactions simultanément
simulation n-corps
(Arnaud Beck)
v = 0.6 vth
Beck & M-V (2008)
Fast electron
Γ = 0.02
unité: distance moyenne entre particules
v = 2.3 vth
6
fast particules are collisionless
various acceleration processes
fast particles are not Maxwellian
u fonction de distribution g Maxwellian
Température hors équilibre local ?
Par ex. : Température cinétique : 32 k B T = ¶ d 3 v (mv2 /2 ) f(v)
dans repère …v = 0
normalisation ¶ d 3 v f(v) = 1
Lois classiques de la thermodynamique pas valables
Ex: loi 0 : T dépend du "thermomètre" (échelle spatiale, temporelle, degrés
de liberté, temps de relaxation différents..)
Extension si système formé de sous-systèmes
chacun en équilibre - pas vrai ici
c.f. Casa-Vasquez & Jou, 2003
7
Non-Maxwellian distributions: Power-law distributions
probability p(x) ∝ 1/x
Solar
flares
of wars
energy of cosmic rays
words in Moby Dick
citations to papers,
hits on web sites,
copies of books sold
telephone calls
magnitude of earthquakes
diameter of moon craters,
intensity of wars
net worth of Americans
frequency of family names,
number of species in flowering
plants
etc...
2.5
2.2
3.0
2.4
3.5
2.2
3.0
3.1
1.8
2.1
1.9
2.3
Zipf's law, Pareto distribution, ...
Newman 2005
8
Power-law distributions
log[p(x)]
if
α
if
if
xmax
log x
xmin
α<2
α>2
α=2
probability p(x) ∝ 1/x
α
<x> déterminé par xmax
<x> déterminé par xmin
<x> ∝ ln(xmax /xmin)
Fraction F of the total due to the
top fraction X of the population:
(α -2)/(α -1)
F=X
u if α m 2 a small fraction at the
top holds most of the total
Newman 2005
(if α j 2.1, the richest 20% hold 85% of the wealth)
Exemples en astrophysique
9
(en dehors de la turbulence)
Rayons cosmiques
Primaire
gerbe
atmosphérique :
1 proton de 1017 eV
t 108 électrons
énergie des secondaires :
dN/dW ∝ 1/(W+W0)2
W
-2.9
1 particule
par km²
par siècle!
Physics Rep. 327, 109 (2000)
10
?
W
Frequency dN/dW
Solar activity
(s-1 m-2 J-1) Distribution of events
> 2 energy in small events
10-30
Quiet Winebarger 2002
-3
Spicules
Quiet
Krucker &
Benz 1998
Quiet
-2
W
Aschwanden
et al. 2000
Quiet
Aschwanden &
Parnell 2002
Active
Shimizu 1995
SoHO
M-V (2006)
IAU Symp
-50 233
10
W
-1.5
Active
Crosby et al. 1993
< 2 energy in large events
1016
Energy W (Joules) 1026
11
Distribution des objets dans le système solaire
Cumulative interplanetary flux
1 UA
1 molécule
10 km
1 m
M-V 2007 (C.U.P.)
6
-10
dN/dm ∝ m-11/6
(dN/dr ∝ r-3.5 si m ∝ r 3)
Equilibre fragmentation
12
Generic physics for power-law distributions ?
Yule process ("rich" gets richer), new biological species,
Fermi acceleration, etc
Long-range correlations/Critical phenomena- forced
and/or self-organized criticality Bak 1987
Exponential growth:
growth:
αt
dx/dt = α x u x = x0 e
ν
P(t) = dN/dt = ν e- t (Poisson)
Random stop
u dN/dx ∝ x
u t = α−1 ln(x/x0)
-(1+ν /α )
= ν (x/x0)-
<t> = 1/ν
x > x0
ν /α
Fermi 1949, Yule 1924, Reed & Hughes 2002, ...
13
Particules en plasmas naturels
Fonctions de distribution observées: Kappa
f (v) } 1 +
v2
v 2e
−( +1)
magnetospheres
Earth: Bame et al 1967, Vasyliunas 1968, Gloeckler&Hamilton
solar wind
1987, Christon&al 1989
electrons: Maksimovic & al 1997, 2006
ions: Gloeckler & al 1992, Collier & al 1996
VELOCITY DISTRIBUTION OF
ELECTRONS IN SOLAR WIND
Maxwell
power law
(collisionless)
Kappa
function
Jupiter ions:
Krimigis & al 1981, Hamilton & al; 1981,
Kane 1991, Kane & al 1992, Collier & al 1995
electrons: Meyer-Vernet & al 1995, Steffl & al 2004
Saturn: protons: Krimigis & al; 1983
Uranus: Krimigis & al 1986, Neptune:Mauk & al1991
solar corona ?
- Solar wind suprathermal electrons
remnants of coronal ones? Olbert 1981
- Production of suprathermal particles
(temperature grad., waves, turbulence)
Roussel-Dupré 1980, Owocki & Scudder 1983, Vinas & al
2000, Vocks 2002, Vocks & Mann 2003
l ~ 260 AU l
Y. Zouganelis 2006
velocity
~ 16 AU l ~0.2 AU
- Observational inferences: Dufton et al. 1984,
Owocki & Ko 1999, Pinfield et al. 1999, Esser & Edgar 2000, Chiuderi &
Chiuderi-Drago 2004, Doyle et al. 2004, Ko 2005
to the contrary Ko et al. 1996
and claims
14
Generating Kappa distributions
Find a way of producing a Kappa
distribution in particular cases
From turbulence
Roberts & Miller (1998), Vinas & al. (2000), Leubner
(2000), Vocks (2002), Vocks & Mann (2003)
velocity distribution
Particular
f (v) } 1 +
v2
v 2e0
−( +1)
Kappa
Maxwell
Power
law
-1
10
1
v/v010
Difficulty : self-consistency!
Rather easy to generate f(v) from given wave spectrum ... BUT
derive wave intensity from non-linear wave kinetic equation
Yoon & al. (2006) (beam-plasma with 1/nLD3 >5 10-3)
Particles (l >>T/=T) propagating down =T u suprathermal tail
(c.f. Roussel-Dupré (1980))
see also Treumann 1999 (Turbulent quasi-équilibrium)
see also Collier 1993 (Levy flights)
15
General
Rappel : Maxwell-Boltzmann
"à la Jaynes" (1957) : théorie de l'information
Distribution f (v) la plus probable sachant que
moyenne <g(v)> = U
¶ d 3 v f(v) = 1
On minimise S = −k ¶ d 3 v f(v) ln(f)
"entropie" = "ignorance"
(tous les états également probables)
u f (v) ∝ exp[- g (v)]
g (v)=mv2/2 u 1/ = 2U/3
Généralisation : Hypothèses à modifier ?
Cimetière de Vienne
Note: S=k log(W) : c'est Planck!
(Pais, 1982)
entropie additive
énergie constante
16
General
Note: another way of generating a Maxwellian
c. f. Baranger 2002
System: constant total energy U: bath + probe
Energy:
U-
Let: Proba. bath energy E: (E) with
(E) ∝ E
u for E j U
= U
with
~ effective number of
degrees of freedom
u P( ) = (U - ) ∝ (U - )
u P( ) ∝ (1 - /U) = (1 -
for E = U
'/ =
/ ) t e
d∝
-
But if >> 1, P = Kappa function with
Boltzmann's factor
<0
finite bath
17
General
Tsallis entropy and Kappa distributions
Tsallis 1988-1998, 2004
Boltzmann-Gibbs entropy S = kB < ln(1/f) > additive
Tsallis entropy
Sq = kB < lnq (1/f) > non-additive
1-q
lnq (f) = (f -1)/(1-q) …g = ¶ dx f q g/ ¶ dx f q
x p = ep ln(x) ~ 1+p ln(x)
d ln(f)
q d1
p d0
Minimize S with <E> = U
u f(E)∝ expq(- E)
u Kappa dist. with
expq(x)=[1+(1-q)x]
= 1/(q-1)
1/(1-q)
d exp(x) 1-D
q d1
Physical meaning of q ? Non-additive entropy*, i.e. possible bias in
the distribution .. BUT should be derived from first principles
(* ex.: merging two indep. turb. systems u change R u change turbulent behavio
Presently not a genuine theory but "a description which works"
c.f. Balmer formula for spectral lines, Ptolemy and Copernicus epicycles
...
18
Super-statistics (variation in "temperature")
System not isolated
Non-infinite heat bath
Temperature fluctuations u Tsallis (Kappa) distribution if
has gamma (chi2) distribution
[1+(q-1) 0E]
-1/(q -1)
= ¶ d e − E f( )
< >=
Wilk & Wlodarczyk 2000,
Beck 2002, Cohen 2004
- E
statistics (f( )) of a statistics (e
q = < 2>/< >2
0
)
u q not arbitrary: determined by temperature fluctuations
"Small heat bath" statistics (Almeida 2001)
implicit in Boltzmann (q =1): temperature of heat bath
independent on its energy U (infinite heat capacity) inverse of
if temperature depends on U, then: q -1=d(1/ )/dU heat capacity
u q not arbitrary: determined by variation of temperature with
energy (q>1 : positive heat capacity; q<1 : negative heat capacity)
19
Collisionless particles in a potentiel Φ
Particles in an attractive force (gravity + electric field)
f(W) = f0(W +
)
ln [f(W)]
e
potential
energy
u With Maxwellian: T constant
Kappa
f0(v) ∝ [1+v2/ v02]
u f(v,z) ∝ [1+(v2+2
-
/m)/ v02]
u f(v,z) = t f0(vt )
t = 1+2
T∝t
-
-1/2
/m v02
u With Kappa distribution:
T increases with potential
energy Scudder 1992
-
ln [f0 (W)]
Liouville theorem
top
slope has
decreased
Maxwellian
Kappa
N. M-V 2001
bottom
W
energy
W
No heating: velocity filtration instead
(attractive force lets suprathermal
particles escape)
20
Collisionless particles in a potential Φ
Kappa f0(v) ∝ [1+v2/ v02]
-
u f(v,z) = t f0(vt-1/2)
Temperature: T ∝ t
u f(v,z) ∝ [1+(v2+2Φ/m)/ v02]
-
t = 1+Φ/[ (mv02/2)]
1/2−κ
Density: n ∝Τ
Define: kBT0 = mv02/2
u kBdT/dΦ = 1/
Maxwellian :
t∝
dT/dΦ t 0
c.f. "small heat bath" statistics : q -1=d(1/ )/dU
=1/kBT
= 1/(q-1) systems with long-range interactions
polytrope law p ∝ nγ
γ = 1−1/(κ−1/2) <1
21
Application sur une structure grande échelle
Io
Europa
Jupiter
IoIo
Satellites Galiléens
Ganymede
Io: marées
gigantesques
rotation = révolution
orbite eccentrique
résonance orbites
Europa, Ganymède
NASA/JPL
Callisto
NASA/JPL
R ~ 1800 km, r from
Jupiter ~ 4 105 km
u Volcanisme
NASA/JPL
22
Particules x ionisation x pick-up par champ magnétique
x Tore de plasma
Schneider & Trauger 1995
Visiteurs du tore
Ω
Champ
magnétique Force centrifuge
Jupiter
Jupiter
M. Moncuquet
Ulysses: exploration en latitude
z
Potentiel : Φc = 3mΩ2z2/2
23
Champ électrique de polarisation e potentiel total Φtot
Maxwellienne
f 0(v) ∝ exp (-mv2/2kBT )
Kappa
f 0 (v) } 1 +
mv2/2
( −3/2)k BT 0
f(v,z) = t −(
t= 1+
+1)
f (v,z) = f 0(v) exp [-Φtot (z)/kBT ]
−( +1)
f(v,z) } 1 +
mv2 /2+ tot (z)
( −3/2)kBT 0
−( +1)
f 0 (vt −1/2 )
tot (z)
( −3/2)k B T 0
T ∝ t ∝ n−1/(κ−1/2)
Note: anisotropic Kappa :
Ty increases with altitude
(Moncuquet et al.,2002)
Sum of several
maxwellians : T increases
with altitude (M-V et al., 1995)
24
polytrope law p ∝ n
γ = 1−1/(κ−1/2)
Confirmation par mesures
spectres UV
electron temperature
γ
Ulysses in Io plasma
torus
T∝n
-1
= 0.48 e
M-V et al. 1995
= 2.4
electron density
Cassini UltraViolet Imager Spectrometer
Hubble Space
Telescope
Imaging
Spectrograph
Steffl et al. 2004
Retherford et al. 2003
25

momentum equation

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