Introduction to Nuclear Fusion

43
I ntroduction to Nuclear Fusion Prof. Dr. Yong-Su Na

Transcript of Introduction to Nuclear Fusion

Page 1: Introduction to Nuclear Fusion

Introduction to Nuclear Fusion

Prof. Dr. Yong-Su Na

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Force balance in a tokamak

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▽p:plasma load

on a magnetic flux surface

JpxBΦ:strength

of BΦ

JΦxBp:strength

of Bp

ψψµψ

ddFF

ddpR −−=∆ 2

0*

Tokamak Equilibrium• The Grad-Shafranov Equation

What kind of forces does a plasma have regarding equilibrium?

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• Basic Forces Acting on Tokamak Plasmas- Radial force balance

- Toroidal force balance: Tire tube force

)(~ IIIRNET pApAeF −−

Tokamak Equilibrium

J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note

X Jt

Bp

Jt x Bp

Magnetic pressure,Tension force

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- Toroidal force balance: 1/R force

- Toroidal force balance: Hoop force

022 2/)(~ µIIIIIIRNET ABABeF −

IIIIII

IIIIII

III

ABABAABB

22

,

>

<>=φφ

0

222

22

µπ BaFNET =

IIIIII

IIIIII

III

ABABAABB

22

,

>

<>=φφ

netpressure

Tokamak Equilibrium• Basic Forces Acting on Tokamak Plasmas

J.P. Freidberg, “Ideal Magneto-Hydro-Dynamics”, lecture note

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vpv BIRBILF 02π==

vp BI

×

vBBB >> θφ

• Basic Forces Acting on Tokamak Plasmas

Tokamak Equilibrium

- External coils required to provide the force balance

How about vertical movement?

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Plasma transport in a Tokamak

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Time

Temperature

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Energy Confinement Time

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τE

• τE is a measure of how fast the plasma loses its energy.• The loss rate is smallest, τE largest

if the fusion plasma is big and well insulated.

1/e

Time

Temperature

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Energy Confinement Time

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nD∇−=Γ : Fick’s law

: Fourier’s law

Thermal diffusivity

χτ

ττκχ

222

)( aaxDn E

E

≈→≈∆

≈≈≡

• Transport Coefficients

Tokamak Transport

: diffusion coefficient (m2/s)( )τ2

2xD ∆=

Tq ∇−= κ

2BkTn

D ∑⊥⊥ =

η- Particle transport in fully ionised plasmaswith magnetic field

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• Classical Transport

- Classical thermal conductivity (expectation): χi ~ 40χe

- Typical numbers expected: ~10-4 m2/s- Experimentally found: ~1 m2/s, χi ~ χe

Bohm diffusion (1946):eBkTD e

161

=⊥

David Bohm(1917-1992)

Tokamak Transport

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Bohm diffusion:eBkTD e

161

=⊥

F. F. Chen, “Introduction to Plasma Physics and Controlled Fusion” (2006)

τE in various types of discharges in the Model C

Stellarator

• Classical TransportTokamak Transport

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• Neoclassical Transport

- Major changes arise from toroidal effects characterised by inverse aspect ratio, ε = a/R0

Tokamak Transport

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• Particle Trapping

- Trapped fraction:

for a typical tokamak, ε ~ 1/3 → ftrap ~ 70%

- Particle trapping by magnetic mirrorstrapped particles with banana orbitsuntrapped particles with circular orbits

εε

εε

+=

+−

−=−=−=12

111111

max

min

BB

Rf

mtrap

[ ]

θεφθθθθ

θεθθ

φθε

θθε

φθ

θθ

φθ

cos1ˆ),(ˆ),(),(

cos1)0(),(

01)cos1(1)(1cos11

0

0

0

0

+=+=

+=

=⇒

=

∂∂

++∂∂

+∂∂

+⇒

=⋅∇

BrBrBrB

BrB

BR

Br

rBrr

B

r

Tokamak Transport

θHW. Derive this.

Discuss the particle motion

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• Particle Trapping

trapped particles passing particles

Tokamak Transport 2, 21

BBrv LBD

∇×±= ⊥∇

Bv

200

20

2||

, RqBmv

RDBRv ×

=

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• Particle Trapping

Tokamak Transport

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Toroidal direction

Ion gyro-motion

Poloidaldirection

Projection of poloidallytrapped ion trajectory

R

B

http://tfy.tkk.fi/fusion/research/

• Neoclassical Bootstrap current

Tokamak Transport

Tim Hender, “Neoclassical Tearing Modes in Tokamaks”, KPS/DPP, Daejun, Korea, 24 April 2009

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- More & faster particles on orbits nearer the core (green .vs. blue) lead to a net “banana current”.

- This is transferred to a helical bootstrap current via collisions.

Currents due to neighbouring bananas largely cancel

orbits tighter where field

stronger

pJ BS ∇∝• Neoclassical Bootstrap current

Tokamak Transport

http://tfy.tkk.fi/fusion/research/Tim Hender, “Neoclassical Tearing Modes in Tokamaks”, KPS/DPP, Daejun, Korea, 24 April 2009

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- Collisional excursion across flux surfacesuntrapped particles: 2rg (2rLi)

• Particle Trapping

Tokamak Transport

Magnetic field lines

Magnetic flux surfaces

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- Collisional excursion across flux surfacesuntrapped particles: 2rg (2rLi)

• Particle Trapping

Tokamak Transport

Magnetic field lines

Magnetic flux surfaces

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- Collisional excursion across flux surfacesuntrapped particles: 2rg (2rLi)trapped particles: Δrtrap >> 2rg – enhanced radial diffusion

across the confining magnetic field

- If the fraction of trapped particle is large, this leakage enhancement constitutes a substantial problem in tokamak confinement.

• Particle Trapping

Tokamak Transport

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- Collisional excursion across flux surfacesuntrapped particles: 2rg (2rLi)trapped particles: Δrtrap >> 2rg – enhanced radial diffusion

across the confining magnetic field

- If the fraction of trapped particle is large, this leakage enhancement constitutes a substantial problem in tokamak confinement.

• Particle Trapping

Tokamak Transport

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J. Wesson, Tokamaks (2004)

- May increase D, χ up to two orders of magnitude:- χi 'only' wrong by factor 3-5- D, χe still wrong by up to two orders of magnitude!

2BkTn

D ∑⊥⊥ =

η

Tokamak Transport• Neoclassical Transports

Bohm diffusioneBkTD e

161

=⊥

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- Normal (water) flow: Hydrodynamic equations can developnonlinear turbulent solutions (Reynolds, 1883)

ρ: density of the fluid (kg/m³)v: mean velocity of the object relative

to the fluid (m/s)L: a characteristic linear dimension,

(travelled length of the fluid; hydraulic diameter when dealingwith river systems) (m)

μ: dynamic viscosity of the fluid(Pa·s or N·s/m² or kg/(m·s))

ν: kinematic viscosity (μ/ρ) (m²/s)

νµρ vLvL

===forces viscousforces inertialRe

A vortex street around a cylinder. This occurs around cylinders, for any fluid, cylinder size and fluid speed, provided that there is a Reynolds number of between ~40 and 103

Tokamak Transport• Transport in fusion plasmas is ‘anomalous’

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- Normal (water) flow: Hydrodynamic equations can developnonlinear turbulent solutions (Reynolds, 1883)

- Transport mainly governed by turbulence: radial extent of turbulent eddy: 1 - 2 cmtypical lifetime of turbulent eddy: 0.5 - 1 ms

- Anomalous transport coefficients are of the order 1 m2/s

( )τ

2

~ xD ∆: diffusion coefficient (m2/s)

Tokamak Transport• Transport in fusion plasmas is ‘anomalous’

http://scidacreview.org/0801/html/fusion.html

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- often associated with non-Maxwellian velocity distributions: deviation from thermodynamic equilibrium (nonuniformity, anisotropy of distributions) → free energy source which can drive instabilities

- kinetic approach required: limited MHD approach

• Two-stream or beam-plasma instability- Particle bunching → E perturbation → bunching↑ → unstable

• Drift (or Universal) instability- driven by ∇p (or ∇n) in magnetic field- excited by drift waves with a phase velocity of vDe with a very short wavelength- most unstable, dominant for anomalous transport

• Trapped particle modes- Preferably when the perturbation frequency < bounce frequency- drift instability enhanced by trapped particle effects- Trapped Electron Mode (TEM), Trapped Ion Mode (TIM)

Tokamak Transport• Microinstabilities

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NCanomalousNC

NCanomalousNC DDDDχχχχ >+=

>+=exp

exp

G. Tardini et al, NF 42 258 (2002)

Tokamak Transport• Profile consistency (or profile resilience or stiffness)

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How to reduce plasma transport?

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- 1982 IAEA FEC, F. Wagner et al. (ASDEX, Germany)- Transition to H-mode: state with reduced turbulence at the plasma edge- Formation of an edge transport barrier: steep pressure gradient at the edge

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• Suppression of Anomalous Transport: H-mode

Tokamak Transport

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χτ

2

aE ≈

Tokamak Transport• Suppression of Anomalous Transport: H-mode

- 1982 IAEA FEC, F. Wagner et al. (ASDEX, Germany)- Transition to H-mode: state with reduced turbulence at the plasma edge- Formation of an edge transport barrier: steep pressure gradient at the edge

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Tokamak Transport• Suppression of Anomalous Transport: H-mode

- 1982 IAEA FEC, F. Wagner et al. (ASDEX, Germany)- Transition to H-mode: state with reduced turbulence at the plasma edge- Formation of an edge transport barrier: steep pressure gradient at the edge

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Tokamak Transport• Suppression of Anomalous Transport: H-mode

- 1982 IAEA FEC, F. Wagner et al. (ASDEX, Germany)- Transition to H-mode: state with reduced turbulence at the plasma edge- Formation of an edge transport barrier: steep pressure gradient at the edge

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Pth = 2.84M-1Bt0.82n20

0.58R1.0a0.81

L-H transition threshold power

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• First H-mode Transition in KSTAR (November 8, 2010)

- B0 = 2.0 T, Heating = 1.5 MW (NBI : 1.3 MW, ECH : 0.2 MW) After Boronisation on November 7, 2010

Tokamak Transport

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• First H-mode Transition in KSTAR (November 8, 2010)Tokamak Transport

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• Suppression of Anomalous Transport: ITBs

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H-mode Reversed shear mode

Tokamak Transport

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• Suppression of Anomalous Transport

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Tokamak Transport

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Non-monotonic current profile

Turbulence suppression

High pressure gradients

Large bootstrap current

Non-inductive current drive

Improved confinement suitable for the steady-state operation

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Plasma Current

CS Coil Current(kA)

(MA)

t (s)

t (s)

Power Supply Limit

Inherent drawback of Tokamak!

Faraday‘s law

SdBdtdv

S

⋅−= ∫

Pulsed Operation

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Plasma Current

CS Coil Current(kA)

(MA)

t (s)

t (s)

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Steady-state operationby self-generated and externally driven current

d/dt ~ 0

Steady-State Operation

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- XGC1 simulation

Turbulence Simulations

Courtesy of S. H. Ku (PPPL)

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power heating appliedenergy stored

=≈

∂∂

−=

inin

E PW

tWP

WτIn steady conditions,

neglecting radiation loss, Ohmic heating replaced by

total input power

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ακαεαααααα κετ RMnPBIfitEth RMnPBCI=,

in engineering variables

• Since tokamak transport is anomalous, empirical scaling laws for energy confinement are necessary.

• To predict the performance of future devices, the energy confinement time is one of the most important parameter.

• Empirical scaling laws: regression analysis from available experimental database.

Energy Confinement Time

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78.058.097.119.041.069.015.093.0)2(98 0562.0 aRMnPBIy,IPBth,E κετ −=

τE in KSTAR and ITER?Why should ITER be large?

• H-mode confinement scaling

Energy Confinement Time