[PPT] - AN INTRODUCTION TO THE NUMERICAL MODELING OF FUSION PLASMAS Herv e PowerPoint Presentation

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AN INTRODUCTION TO THE NUMERICAL MODELING OF FUSION PLASMAS

Herv´ e Guillard

INRIA Sophia-Antipolis & LJAD, UMR 7351, 06100 Nice, France

CEMRACS 2013

Herv´ e Guillard (INRIA & LJAD) 22/08/2013 1 / 36

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Outline

1

Fusion Plasmas

2

Kinetic Models

3

Fluid models

4

The MHD limit

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Outline

1

Fusion Plasmas

2

Kinetic Models

3

Fluid models

4

The MHD limit

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Deuterium-Tritium Reaction

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Nuclear reactions

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Fusion on earth ???

Essentially two studied technologies : Inertial confinement Magnetic confinement

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Inertial confinement technology

relies on fast and violent heating of fusion targets.

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Target Ablator heating

Ablator is illuminated by powerful - X rays and reaches a plasma state

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Ablator expansion & DT Compression

Ablator is accelerated outwards By rocket effect DT is accelerated inwards (v ∼ 300 km/s)

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Deuterium - Tritium Compression

DT ice is compressed up to 1000 times its initial density ( 20 time lead density)

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Hot spot Ignition

Central gas is heated up to several 106 of degrees When ρR ∼ 0.3g/cm2 the gas is self heated by alpha particles Fusion reactions start and propa- gate into DT ice Note : 1 mg D-T → 340 MJ : Fu- sion is equivalent to combustion of 10 kg coal

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The other way to fusion : magnetic confinement

Objective : confine hot D-T plasma by strong magnetic fields Toroidal coils Poloidal coils

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Tokamaks :

magnetic field lines vacuum chamber Tokamaks design in the 1950s by Igor Tamm and Andrei Sakharov

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ITER Tokamak

ITER (International Thermonuclear Experimental Reactor) in construction in Cadarache, construction begins in 2010, first plasma in 2020, first fusion plasma in 2027 840 m3 150M OK 13 Tesla = 200 000 x earth magnetic field

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Numerical simulations

Necessary for Equilibrium computation : steady state and control of the machines prediction of the possible occurrence of instabilities

magnetic instabilities hydrodynamic instabilities

determination of the value of key parameters e.g transport coefficients due to turbulence understand and explain physical phenomena Very large number of different numerical models

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Outline

1

Fusion Plasmas

2

Kinetic Models

3

Fluid models

4

The MHD limit

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Kinetic models

Vlasov or Boltzmann eq for each particle (ions, electrons, neutral) ∂fs ∂t + divx(vfs) + divv(Ffs) = Cs =

Css′

Force field F = es

ms [E + v × B] (note divvF = 0)

E, B given by Maxwell equations: ∂B ∂t + curl E = 0 Υ = es

fs(x, v, t)dv3

− 1 c2 ∂E ∂t + curl B = µ0J J = es

fsv(x, v, t)dv3

ε0 div E = Υ div B = 0

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Kinetic models

Self-consistent and closed model extremely heavy from computational point of view

6 D model : 3 space dimensions, 3 velocity dimensions covers huge range of time and space scales

some simplifications possible : Ampere law, Quasi-Neutral assumption, Electrostatic assumption, gyrokinetic theory Used for some very specific tasks

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Kinetic models : one example of applications

Anomalous transport in tokamaks

Radial diffusion of mass and tem- perature exceed by order of magni- tude “laminar” values due to micro-turbulence and micro- instabilities : Estimate the value of turbulent transport coefficients by direct sim- ulation Development of gyrokinetic codes (e.g GYSELA code of CEA)

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Outline

1

Fusion Plasmas

2

Kinetic Models

3

Fluid models

4

The MHD limit

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From kinetic to fluid

Idea of fluid models : instead of computing the whole distribution function fs compute only a small number of its moments, i.e : Fluid variables are moments of the distribution function fs

fluid density

ns(x, t) =

fs(x, v, t)dv 3
fluid velocity

ns(x, t)us(x, t) =

fs(x, v, t)vdv 3

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From kinetic to fluid

Pressure tensor

Ps(x, t) = ms

fs(x, v, t)(v − us) ⊗ (v − us)dv 3
scalar pressure = 1/3 trace of pressure tensor

ps(x, t) = ms 3

fs(x, v, t)|(v − us)|2dv 3
temperature

Ts(x, t) = ps(x, t) ns(x, t)

Energy flux

Qs(x, t) = ms 2

fs(x, v, t)|v|2vdv 3

More complex fluid models, e.g 14 Moment closure by D. Levermore

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From kinetic to fluid

Mass conservation

First velocity moment of the Boltzmann (Vlasov) equation

[∂fs

∂t + divx(vfs) + divv(Ffs) = Cs] gives

fsdv 3 def

= ns

vfsdv 3 def

= nsus

divv(Ffs) = 0
Csdv 3 = 0

∂ns ∂t + divx(nsus) = 0

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Fluid models

proceeding in the same way for the other moments :

msv× Boltzmann and integrating, gives momentum (velocity) equation

1 2ms|v|2× Boltzmann and integrating, gives energy (temperature) equation

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Fluid models

s ∈ ion, electron, neutral particle

∂ns ∂t + div(nsus) = 0 ms(∂nsus ∂t + div(nus ⊗ us)) + ∇ps + divxΠs − nses(E + us × B) =

s′=s

Rss′ ∂ ∂t (ms ns 2 |us|2 + 3 2ps) + divQ − esus.E = Qs

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The 2-fluid model

Much simpler than kinetic model (3D) Clear mathematical structure (1 compressible Navier-Stokes system per species coupled by force terms) But needs closure assumptions Questionable in tokamaks very large disparity in length and time scales still costly e.g for a ion-electron model 10 dof per mesh point not really used but good starting point for subsequent approximations

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From 2-fluid to one Fluid models

Normalization units

time τobs velocity u∗ lenght L∗ = u∗τobs ion gyro-period τci = mi/ZieB electron gyro-period τce = me/eB plasma period τpe = (ε0me/nee2)1/2 τce/τpe ∼ 1 collision τcoll ∝ (Λ/lnΛ)τpe Non dimensional parameters εi = τci/τobs Mi = u∗/νTi, Λ = 4π 3 nλ3

D,

µ = me/mi Relevant asymptotic regimes εi → 0 Λ → ∞ µ = me/mi → 0 etc Give a huge set of one fluid models

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Outline

1

Fusion Plasmas

2

Kinetic Models

3

Fluid models

4

The MHD limit

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From 2-fluid to one Fluid models

The MHD scalings

MHD : violent instabilities affecting the whole plasma region : x∗ = e.ga minor radius fast event : speed of MHD waves : Alfen velocity vA = B √µ0ρ u∗ = O(νTi) ∼ O(vA)

MHD asymptotic limit

εi = τci τ → 0, Mi = O(1)

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THE MHD MODELS

Ideal MHD

Zero order model in εi (εi = 0)

∂ ∂t ρ + div(ρu) = 0 ∂ ∂t ρu + div(ρu ⊗ u) + ∇p − (J × B) = 0 E + u × B = 0 need the current J → Ampere’s law J = curl B need the magnetic field B → Faraday’s law ∂B ∂t + curl E = 0 ⇒ autonomous system for a One Fluid model

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THE MHD MODELS

Ideal MHD - summary

∂ ∂t ρ + div(ρu) = 0 ∂ ∂t ρu + div(ρu ⊗ u) + ∇p + 1 µ0 (B × curl B) = 0 ∂B ∂t − curl (u × B) = 0

+ energy equation

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One example of hydrodynamic instabilities in magnetized plasmas

2 scalar variables : ion density n; ion parallel velocity u|| u = u||b + u⊥ ∂ ∂t ρ + div(ρu) = 0 ∂ ∂t ρu|| + divρu||u + ∇||p = ρu.Db Dt with (this is the ideal Ohms law solved for the ⊥ velocity) u⊥ = E × B B2 B given, E computed by the adiabatic assumption

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Hydrodynamic instabilities

In the core plasma (white region) the plasma rotate counterclockwise In the edge region, the plasma touches the limiter Due to electric charging of the wall (Bohm’s BC) the plasma enters the wall → intense shear near the wall Development of hydrodynamical in- stabilities (Kelvin-Helmholtz like) ? momentum plot at t=0

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Hydrodynamic instabilities

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Hydrodynamic instabilities

Oscillation of the plasma boundary

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Conclusions

large number of unsolved phenomena requiring the development

f

numerical models numerical methods parallel algorithms

short overview of numerical simulations for fusion plasmas See you next year in Cemracs2014

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