View
2
Download
0
Category
Preview:
Citation preview
C. Negulescu, June 2012
1
Asymptotic preserving schemes (AP) formagnetically confined plasmas
Lecture I
Claudia Negulescu
Institut de Mathématiques de Toulouse
Université Paul Sabatier
C. Negulescu, June 2012
2Introduction/Motivation
Plasma
➠ gas of charged particles: electrons, ions, neutral atoms
➠ π λα σ µα → modulable substance
➠ 99% of the universe: stars, intergalactic medium,magnetosphere, ionosphere, aurora borealis, lightenings
➠ collective effects play an important role, interaction viaelectromagnetic forces
➠ diff. behaviour as neutral gases: short range interactions
➠ far from equilibrium, highly anisotropic, turbulent environ.
Earth plasmas/Applications
➠ fusion reactors (energy production)
➠ fluorescent lamps, plasma displays (light production)
C. Negulescu, June 2012
3Introduction/Motivation
Earth plasmas/Applications
➠ satellites/rockets propulsion
➠ air flow control (aviation)
➠ sterilization, water cleaning (biology, medicine)
➠ semiconductor technology (switches, transistors)
➠ plasma lasers, weapons, gas discharges, ...
Mathematics
➠ modeling(link between mathematicians and physicists)
➠ analysis(ex/uniqueness/long time behaviour/properties)
➠ numerical analysis(error estimates/design of newschemes)
➠ scientific computing(implementation, phys. validation)
C. Negulescu, June 2012
4Nuclear power
➠ nuclear force : binds the nucleons
➠ atomic nuclei mass < sum of the masses of the constitutingnucleons
➠ “lost” mass→ binding energy (E = mc2)
➠ binding energy/nucleon :→ low for light and heavy atomic nuclei→ high for average atomic nuclei (max. forFe)
➠ nuclear reactions to create average atomic nuclei→ production of nuclear energy
➠ two types of nuclear reactions :fusionandfission
C. Negulescu, June 2012
5Thermonuclear fusion
→ Lawson’s criterion for successful fusion:
nτET ∼ 3 ∗ 1021m−3s keV
→ Under these conditions:Plasma tends to disperse and cool down
Requirements for fusion:
• suff. high temperaturesT (10− 15keV )
→ to overcome the repulsive Coulomb barrier
• suff. high densitiesn
→ to ensure that collisions take place
• suff. confinement timesτEτE := W
Ploss= plasma energy content
lossed power
→ to ensure the plasma heating by fusion
products
C. Negulescu, June 2012
6The Sun
Three confinement principles:
➠ gravitational:stars of suff. high mass (Sun) are contractedso by gravitational forces, that the cond. for fusion occur
➠ magnetic:particles are trapped in a strong magn. field
➠ inertial: plasma/fuel capsule is compressed by laser beams
SUN:• highly complexe dynamical system (dynamoeffect/turbulences)
• solar activity has direct impact on Earth’smagnetosphere and ionosphere (solar wind)
C. Negulescu, June 2012
7Magn. conf. fusion reactors
• helicoidal magnetic field linestwisted around toric mag. surfaces
• particle trajectories:gyration around the mag. field lineswith gyration radius :ρ = mv⊥
eB
• macroscopic quantitiesn, T , pare homogenous on a mag. surface
• confinement: equilibrium ofplasma pressure and mag. pressure
β = nTB2/2µ0
<< 1
→ superior bound on the density
C. Negulescu, June 2012
8Instabilities / turbulence
• Plasma dynamics is governed by turbulent transport• Understanding plasma turbulences is crucial for the successof nuclear fusion→ more complexe than in neutral fluids: electromagn. fields
• Degradation of the confinementdue to the high density,temp., pressure gradients
➠ transverse transp. due to collisions between particles (1%)
➠ transverse transp. due to various micro-instabilities (99%)Rayleigh-Benard, Kevin-Helmholtz, ITG/ETG(ion/electron temperature gradient driven turbulence)
• Stabilizing effects:
➠ velocity shear, magnetic shear
➠ sheath conductivity
C. Negulescu, June 2012
9Kinetic description of plasmas
• Vlasov eq.governs the dynamics of the particle distribution fct.
f(t, x, v) (6D phase-space)
∂tfα + v · ∇xfα +eα
m(E + v ×B) · ∇vfα = 0 , α : ions/electr.
• Electromagn. fields are calculated selfconsist. fromMaxwell’s eq.
∇·E =1
ε0ρ , −
1
c2∂tE+∇×B = µ0j , ∇·B = 0 , ∂tB+∇×E = 0
Convenience:
➠ Plasmas are only slightly collisional⇒ far from equilib.
➠ Necessary to treat Landau resonances, trapped particles,
turbulence, plasma oscillations
Difficulties: High dimensionality→ 6D phase-space
C. Negulescu, June 2012
10Fluid description of plasmas
• Plasma dynamics described by means of fluid variablesn(t, x), u(t, x),
T (t, x), satisfying theconservation lawscoupled toMaxwell’s eq.
∂tnα +∇ · (nαuα) = Snα , α : ions/electr.
mαnα [∂tuα + (uα · ∇)uα] = nαeα(E + uα ×B)−∇ · Pα +Rα ,
3
2nαkB [∂tTα + (uα · ∇)Tα] = −∇ · qα − Pα : ∇uα +Qα ,
Advantages:
➠ Reduce the dimensionality to3D space
Drawback:
➠ Fluid closures are not always adequate
➠ Fluid models over-estimate the transport level
C. Negulescu, June 2012
11Hybrid Kinetic/Fluid models
Plasma dynamics is characterized by multi-scale phenomena
Kinetic modelsFluid models
Hybrid models
τpe τce τpi τci τa τcs τei
Lρe ρi
τa: Alfen wave period
τcs: Ion sound period
τei: Electr-ion collision time
τpe,pi : Inv. electr./ion plasma freq.
τce,ci: Electr./ion cyclotron period
λD : Debye length
ρe,i: Electr./ion Larmor radius
c: sound speed
δe δi
δe,i = c/ωpe,pi : Electr./ion skin depth
ωpe,pi: Electr./ion plasma frequency
λD
C. Negulescu, June 2012
12Multi-scale problems
Exhibit a large varity of space and times scales
Cosmic
Hemisphere
wind
Time
Wind
Ray
15km
Ionosphere
Troposphere
100km
SouthernHemisphere
NorthernS
ea level
Ev. of natural plasma bubble
at altitude∼ 700km
C. Negulescu, June 2012
13Multi-scale problems
A small-scale numerical simulation is out of reach
➠ requires mesh-sizes dependent on small scale param.ε ≪ 1
➠ excessive computational time and memory space are needed tocapture small scales
It is not always of interest to resolve the details at the smallscale. Multi-scale strategies are much more adequate!
➠ homogeneisation, domain decomposition, multi-grids, multi-scalemethods based on wavelets or finite elements, multi-scalevariational methods
Essential feature of these methods
➠ capture efficiently the large scale behavior of the solution, withoutresolving the small scale features
C. Negulescu, June 2012
14Asymptotic Preserving schemes
Difficulty: Resolution of multiscale pb. can be very difficult, ifthe pb. becomes singular, as one of the parametersε → 0
➠ (P ε) sing. perturbed pb. of sol.fε
➠ the seq.fε converges towardsf0, sol. of a limit pb.(P 0)
➠ the limit pb. (P 0) is different in type from the initial(P ε)
➠ standard schemes would require∆t,∆x ∼ ε for stability
Definition: A schemeP ε,h is AP iff it is convergent forh → 0uniformely inε, i.e.
P ε,h P ε
P 0,h P 0
ε→
0
ε→
0
h → 0
h → 0
C. Negulescu, June 2012
15Asymptotic Preserving schemes
AP-procedure:
➠ requires that the limit problem(P 0) is identified andwell-posed
➠ requires a sufficient degree of implicitness (not obvious)
➠ consists in trying to mimic at discrete level the asymptoticbehaviour of the sing. perturbed pb. sol.fε
Advantages:
➠ gives accurate and stable results, with no restrictions onthe computational mesh
➠ enables to capture automatically the Limit modelP 0, ifε → 0 (micro-macro transition)
➠ no more coupling needed, ifε(x) is variable
C. Negulescu, June 2012
16Kinetic models and specific limit regimes
• Fundamental kinetic model: Vlasov/Boltzmann equation
∂tf + v · ∇xf +q
m(E + v ×B) · ∇vf = Q(f)
Several small scales/parameters occur, leading to diff. regimes:• Hydrodynamic scaling[Filbet/Jin; Dimarco/Pareschi]
∂tf + v · ∇xf =1
εQ(f)
➠ 0 < ε ≪ 1: mean free path (Knudsen nbr.)
➠ in the limit ε → 0, one gets the compressible Euler eq.
➠ AP-scheme: Decomposition of the source term in stiff-and non-stiff part
Q(f)
ε=
Q(f)− P (f)
ε+
P (f)
ε
C. Negulescu, June 2012
17Kinetic models and specific limit regimes
• Drift-Diffusion scaling[Klar; Lemou/Mieussens]
∂tf +1
ε(v · ∇xf +∇xΦ · ∇vf) +
1
ε2Q(f) = G
➠ 0 < ε ≪ 1: mean free path
➠ in the limit ε → 0, one gets the Drift-Diffusion model
➠ AP-scheme: Micro-Macro decomp.f = ρM + εg
• Vlasov-Poisson quasi-neutral limit[Belaouar;Crouseilles;Degond;Sonnendrucker;Navoret;Vignal]
∂tf + v∂xf + ∂xΦ∂vf = 0
−λ2∂xxΦ = 1− ρ
➠ 0 < λ ≪ 1: rescaled Debye length
➠ AP-scheme: Reformulation of the Poisson equation
C. Negulescu, June 2012
18Kinetic models and specific limit regimes
• Vlasov-Maxwell quasi-neutral limit[Degond/Deluzet/Doyen]
• High-field limit, strong magn. fields[Bostan, Golse, Saint-Raymond]
∂tf +1
εv(p) · ∇xf −
1
ε(E + v(p)×
B
ε) · ∇vf = 0
➠ 0 < ε ≪ 1: rescaled cyclotronic period
➠ in the limit ε → 0, one gets the guiding-center approx.
➠ asymptotical analysis:• Study of the dominant operatorT := (v(p)× B) · ∇v
• Projection of the eq. onker T = averaging along thecharact. flow associated toT
➠ construction of AP-scheme mimics this asymp. analysis
C. Negulescu, June 2012
19Fluid models and specific limit regimes
• Euler-Poisson quasi-neutral limit[Crispel/Degond/Vignal]
∂tn+∇ · (nu) = 0
∂t(nu) +∇ · (nu⊗ u) +∇p(n) = n∇Φ
−λ2∆Φ = 1− n
➠ 0 < λ ≪ 1: rescaled Debye length
• High-field limit, Euler-Lorentz[Brull;Degond;Deluzet;Mouton;Sangam;Vignal]
∂tn+∇ · (nu) = 0
∂t(nu) +∇ · (nu⊗ u) +1
τ∇p(n) =
1
τn (E + u× B)
➠ 0 < τ ≪ 1: rescaled gyro-period
C. Negulescu, June 2012
20Fluid models and specific limit regimes
• Low Mach-nbr. limit [Degond/Tang; Cordier/Degond/Kumbaw]
∂tn+∇ · (nu) = 0
∂t(nu) +∇ · (nu⊗ u) +1
ε2∇p(n) = 0
➠ 0 < ε ≪ 1: rescaled Mach-nbr.
➠ in the limit ε → 0, one gets the incompressible Euler eq.
➠ AP-scheme: Stiff term is decomposed as
1
ε2∇p(n) = α∇p(n) +
1− αε2
ε2∇p(n)
• Highly anisotropic temperature eq.[Lozinski/Mentrelli/Narski/Negulescu]
∂tT −1
ε∇|| · (K||∇||T )−∇⊥ · (K⊥∇⊥T ) = 0
C. Negulescu, June 2012
21Outline of the talk
• Kinetic models
➠ Boltzmann eq. in drift-diffusion limit
➠ Boltzmann eq. in hydrodynamic limit
➠ Vlasov-Poisson eq. in quasi-neutral limit
➠ Vlasov eq. in high field limit, variable Larmor radii
• Fluid models
➠ anisotropic elliptic eq. (electric potential)
➠ anisotropic, nonlinear, degenerate parabolic eq.(temperature)
Recommended