Tokamak Fusion Basics and the MHD Equations Stephen C. Jardin - - PowerPoint PPT Presentation

MHD Simulations for Fusion Applications

Lecture 1

Tokamak Fusion Basics and the MHD Equations

Stephen C. Jardin Princeton Plasma Physics Laboratory

CEMRACS ‘10 Marseille, France July 19, 2010

1

Fusion Powers the Sun and Stars Can we harness Fusion power on earth?

The Case for Fusion Energy

• Worldwide demand for energy continues to increase

– Due to population increases and economic development – Most population growth and energy demand is in urban areas

• Implies need for large, centralized power generation
• Worldwide oil and gas production is near or past peak

– Need for alternative source: coal, fission, fusion

• Increasing evidence that release of greenhouse gases is causing global

climate change . . . “Global warming”

– Historical data and 100+ year detailed climate projections – This makes nuclear (fission or fusion) preferable to fossil (coal)

• Fusion has some advantages over fission that could become critical:

– Inherent safety (no China syndrome) – No weapons proliferation considerations (security) – Greatly reduced waste disposal problems (no Yucca Mt.)

Controlled Fusion uses isotopes of Hydrogen in a High Temperature Ionized Gas (Plasma)

Deuterium Tritium Helium nuclei (α-particle) … sustains reaction Neutron Deuterium exists in nature (0.015% abundant in Hydrogen) α T Tritium has a 12 year half life: produced via 6Li + n T + 4He Lithium is naturally abundant Lithium proton neutron

key

Need ~ 5 atmosphere @ 10 keV

Controlled Fusion Basics

Create a mixture of D and T (plasma), heat it to high temperature, and the D and T will fuse to produce energy. PDT = nDnT <σv>(Uα+Un) at 10 keV, <σv> ~ T2 PDT ~ (plasma pressure)2

Operating point ~ 10 keV

Note: 1 keV = 10,000,000 deg(K)

Toroidal Magnetic Confinement

Charged particles have helical orbits in a magnetic field; they describe circular orbits perpendicular to the field and free-stream in the direction

• f the field.

TOKAMAK creates toroidal magnetic

fields to confine particles in the 3rd

• dimension. Includes an induced

toroidal plasma current to heat and confine the plasma

“TOKAMAK”: Russian abbreviation

for “toroidal chamber”

• 500 MW fusion output
• Cost: $ 5-10 B
• Originally to begin operation in 2015 (now 2028 full power)

ITER is now under construction

scale

International Thermonuclear Experimental Reactor:

• European Union
• Japan
• United States
• Russia
• Korea
• China
• India
• World’s largest tokamak
• all super-conducting coils

Tore Supra ITER

ITER has a site… Cadarache, France

June 28, 2005

Ministerial Level Meeting Moscow, Russia

Progress in Magnetic Fusion Research and Next Step to ITER

Years

Megawatts

10 1,000 100 10 1,000 100 10 100 1,000

Kilowatts Watts Milliwatts

1,000 100 10

Fusion Power

1975 1985 1995 2005

Data from Tokamak Experiments Worldwide

2015

TFTR (U.S.) JET (EU)

2025

ITER (Multilateral)

Start of ITER Operations Operation with full power test

1 2 3 4 5 6 7 8 9 10 Power Gain

TFTR/JET ITER

50 100 150 200 250 300 350 400 450 500 Power (MW) Plasma Duration (Seconds)

Power (MW) Duration (Seconds) Power Gain (Output/Input)

A Big Next Step to ITER

Plasma Parameters

Simulations are needed in 4 areas

• How to heat the plasma to thermonuclear

temperatures ( ~ 100,000,000o C)

• How to reduce the background turbulence
• How to eliminate device-scale instabilities
• How to optimize the operation of the whole

device

10-10 10-2 104 100

SEC.

CURRENT DIFFUSION

10-8 10-6 10-4 102 ωLH

• 1

Ωci

• 1

τA Ωce

• 1

ISLAND GROWTH ENERGY CONFINEMENT SAWTOOTH CRASH TURBULENCE ELECTRON TRANSIT

(a) RF codes (b) Micro- turbulence codes (c) Extended- MHD codes (d) Transport Codes

These 4 areas address different timescales and are normally studied using different codes

Extended MHD Codes solve 3D fluid equations for device-scale stability

10-10 10-2 104 100

SEC.

CURRENT DIFFUSION

10-8 10-6 10-4 102 ωLH

• 1

Ωci

• 1

τA Ωce

• 1

ISLAND GROWTH ENERGY CONFINEMENT SAWTOOTH CRASH TURBULENCE ELECTRON TRANSIT

• Sawtooth cycle is one example
• f global phenomena that need

to be understood

• Can cause degradation of

confinement, or plasma termination if it couples with

• ther modes
• There are several codes in the

US and elsewhere that are being used to study this and related phenomena:

• NIMROD
• M3D

Quicktime Movie shows Poincare plot of magnetic field at one toroidal location

• Example of a recent 3D

calculation using M3D code

• “Internal Kink” mode in a

small tokamak (Sawtooth Oscillations)

• Good agreement between

M3D, NIMROD, and experimental results

• 500 wallclock hours and
• ver 200,000 CPU-hours

Excellent Agreement between NIMROD and M3D

Kinetic energy vs time in lowest toroidal harmonics M3D NIMROD M3D NIMROD Flux Surfaces during crash at 2 times

15

2-Fluid MHD Equations:

( )

2 2

1 ( ) continuity Maxwell ( ) momentum Ohm's law 3 3 electron energy 2 2 3 3 2 2

i e e e i i e e i GV

p ne n n t t nM p J Q t p p p t p p p t μ η μ η

Δ

∂ + ∇ • = ∂ ∂ = −∇× ∇ = = ∇× ∂ ∂ +

• ∇

+ ∇ = × − ∂ + × = + ∂ ⎛ ⎞ + ∇ = ∇ × − ∇ ⎜ ⎟ ∂ ⎝ ⎠ ∂ ⎛ ⎞ + + ∇ = ∇ + −∇ + −∇ − ∇ ⎜ ⎟ ∂ ⎝ ⎠ Π J V B E B J B V V V J V J B E B B V V q V V i i i i i i

2

ion energy

i

V Q μ

Δ

+ ∇ −∇ − V q i i

Resistive MH 2- Idea flui D d l MHD MHD

number density magnetic field current density electric field mass density

i

n nM ρ ≡ Β J E fluid velocity electron pressure ion pressure electron charge

e i e i

p p p p p e ≡ + V viscosity resistivity heat fluxes equipartition permeability Q μ μ η

Δ i e

q ,q

16

Ideal MHD Equations:

( ) continuity Maxwell ( ) momentum Ohm's law 3 3 energy 2 2

i

n n t t nM p t p p p t μ ∂ + ∇ • = ∂ ∂ = −∇× ∇ = = ∇× ∂ ∂ +

• ∇

+ ∇ = × ∂ + × = ∂ ⎛ ⎞ + ∇ = − ∇ ⎜ ⎟ ∂ ⎝ ⎠ V B E B J B V V V J B E V B V V i i i

Ideal MHD

number density magnetic field current density electric field mass density

i

n nM ρ ≡ Β J E fluid velocity electron pressure ion pressure

e i e i

p p p p p ≡ + V permeability μ

17

Ideal MHD Equations:

5/3

( ) continuity Maxwell ( ) momentum Ohm's law 3 3 energy 2 2 entropy t t p t p p p t s s p s t ρ ρ μ ρ ρ − ∂ + ∇ • = ∂ ∂ = −∇× ∇ = = ∇× ∂ ∂ +

• ∇

+ ∇ = × ∂ + × = ∂ ⎛ ⎞ + ∇ = − ∇ ⎜ ⎟ ∂ ⎝ ⎠ ∂ ≡ ⇒ + ∇ = ∂ V B E B J B V V V J B E V B V V V i i i i

Ideal MHD

number density magnetic field current density electric field mass density

i

n nM ρ ≡ Β J E fluid velocity electron pressure ion pressure

e i e i

p p p p p ≡ + V can be eliminated / is redundant is redundant permeability t ρ μ ∂ ∂ ∇ E,J B i

18

Ideal MHD Equations:

( ) ( ) ( )

3/5

1 ( ) 3 3 2 2 / t p t p p p t s s t p s ρ μ ρ ∂ = ∇× × ∂ ∂ +

• ∇

+ ∇ = ∇× × ∂ ∂ ⎛ ⎞ + ∇ = − ∇ ⎜ ⎟ ∂ ⎝ ⎠ ∂ + ∇ = ∂ = B V B V V V B B V V V i i i mass density magnetic field fluid velocity entropy density fluid pressure s p ρ Β V is redundant permeability μ ∇ B i

Quasi-linear Symmetric real characteristics Hyperbolic

Ideal MHD characteristics:

The characteristic curves are the surfaces along which the solution is

• propagated. In 1D, the characteristic curves would be lines in (x,t)

s s u t x ∂ ∂ + = ∂ ∂ Boundary data (normally IC and BC) can be given on any curve that each characteristic curve intersects only once: Cannot be tangent to characteristic curve To calculate characteristics in 3D, we suppose that the boundary conditions are given on a 3D surface and ask under what conditions this is insufficient to determine the solution away from this

• surface. If so, is a characteristic surface.

Perform a coordinate transformation: and look for power series solution away from the boundary surface ( , ) t φ φ = r

( )

( , ) , , , t φ χ σ τ → r φ φ =

( ) ( ) ( ) ( ) ( ) ( )

, , , , ,

φ φ φ φ

φ χ σ τ χ σ τ φ φ χ χ σ σ τ τ φ χ σ τ ∂ ∂ ∂ ∂ = + − + − + − + − ∂ ∂ ∂ ∂ v v v v v v These can all be calculated since they are surface derivatives within φ φ = If this cannot be constructed, then is a characteristic surface φ φ

20

Ideal MHD characteristics-2:

z A x A x S z A z S z A z A x A x S z S

u n V n V n c u n V u n c n V u n V u n V u n c n c u u − − ⎡ ⎤ ⎢ ⎥ − − ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ − − ⎢ ⎥ = ⎢ ⎥ − − ⎢ ⎥ − ⎢ ⎥ ⎢ ⎥ − ⎢ ⎥ − ⎢ ⎥ ⎣ ⎦ A

1 x y z x y z S

V V V B B B c p s ρ ρ ρ ρ μ ρ μ ρ μ

−

⎡ ⎤ ′ ⎢ ⎥ ′ ⎢ ⎥ ⎢ ⎥ ′ ⎢ ⎥ ⎢ ⎥ ′ ⎢ ⎥ = ⎢ ⎥ ′ ⎢ ⎥ ⎢ ⎥ ′ ⎢ ⎥ ′ ⎢ ⎥ ⎢ ⎥ ′ ⎢ ⎥ ⎣ ⎦ X

= ⋅⋅⋅ A X i

( ) ( ) ( )

Introduce a characteristic surface ( , ) ˆ spatial normal / characteristic speed: /

t

t u φ φ φ φ φ φ φ φ = = ∇ ∇ ≡ + ∇ ∇ ∂ ′ = ∂ r n Vi

( ) ( )

0,0, ˆ ˆ is in z direction propagation in (x,z) ,0,

x z

B n n = = B n B

5 3

/ /

A S

V B c p μ ρ ρ ≡ ≡

Ideal MHD All known quantities All terms containing derivatives involving φ

if det = 0 is characteristic surface φ → A

21

Ideal MHD wave speeds:

( ) ( ) ( ) ( )

2 2 2 2 1/2 2 2 2 2 2 2 2 2 2 1 1 2 2 1/2 2 2 2 2 2 2 2 2 2 1 1 2 2

entropy disturbance Alfven wave slow wave fas 4 t wave 4

A An s A S A S An S f A S A S An S

u u u u V u u V c V c V c u u V c V c V c = = = = ⎡ ⎤ = = + − + − ⎢ ⎥ ⎣ ⎦ ⎡ ⎤ = = + + + − ⎢ ⎥ ⎣ ⎦

( ) ( )

2 2 2 4 2 2 2 2 2

det

An A S An S

D u u V u V c u V c = ⎡ ⎤ = − − + + = ⎣ ⎦ A

( ) ( )

5 3 2 2 2

0,0, ˆ ,0, / /

x z A S An Z A

B n n V B c p V n V μ ρ ρ = = ≡ ≡ ≡ B n

2 2 2 2 2 2 2 2 2 2 2

Alfven wave slow wave fast wave

A An s z S A x S f

u u V u u u u n c V n c = = = = +

• In normal magnetically

confined plasmas, we take the low-β limit

2 2 S A

c V

22

Ideal MHD surface diagrams

2 2 2 2 2 2 2 2 2 2 2

Alfven wave slow wave fast wave

A An s z S A x S f

u u V u u u u n c V n c = = = = +

• (

) ( )

5 3 2 2 2

0,0, ˆ ,0, / /

x z A S An Z A

B n n V B c p V n V μ ρ ρ = = ≡ ≡ ≡ B n Reciprocal normal surface diagram Ray surface diagram

1 x y z x y z S

V V V B B B c p s ρ ρ ρ ρ μ ρ μ ρ μ

−

⎡ ⎤ ′ ⎢ ⎥ ′ ⎢ ⎥ ⎢ ⎥ ′ ⎢ ⎥ ⎢ ⎥ ′ ⎢ ⎥ = = ⎢ ⎥ ′ ⎢ ⎥ ⎢ ⎥ ′ ⎢ ⎥ ′ ⎢ ⎥ ⎢ ⎥ ′ ⎢ ⎥ ⎣ ⎦ X

1 1 1 1 1 1 1 / 1 1

S A

c V ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ± ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ± ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ± ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ± ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦

entropy Alfven fast nx = 1 nz = 0 fast nx = 0 nz = 1 slow nx = 0 nz = 1 Only the fast wave can propagate perpendicular to the background field, and does so by compressing and expanding the field The Alfven wave only propagates parallel to the magnetic field, and does so by bending the field. It is purely transverse (incompressible)

( ) ( )

5 3 2 2 2

0,0, ˆ ,0, / /

x z A S An Z A

B n n V B c p V n V μ ρ ρ = = ≡ ≡ ≡ B n

Ideal MHD eigenvectors

The slow wave does not perturb the magnetic field, only the pressure

Slow Wave Alfven Wave Fast Wave

Background magnetic field direction

propagation propagation propagation

• only propagates

parallel to B

• only compresses fluid

in parallel direction

• does not perturb

magnetic field

• only propagates

parallel to B

• incompressible
• only bends the

field, does not compress it

• can propagate

perpendicular to B

• only compresses

fluid in ⊥ direction

• compresses the

magnetic field

• This is the

troublesome wave!

Tokamaks have Magnetic Surfaces, or Flux Surfaces

φ Magnetic field is primarily into the screen, however it has a twist to it. After many transits, it forms 2D surfaces in 3D space. Because the particles are free to stream along the field, the temperatures and densities are nearly uniform on these surfaces. Only the Fast Wave can propagate across these surfaces, but it will have a very small amplitude compared to the other waves.

Tokamak schematic Tokamak cross section

• The field lines in a tokamak are dominantly in the toroidal direction.
• The magnetic field forms “flux surfaces”.
• Only the fast wave can propagate across these surfaces.
• Since the gradients across surfaces are large (requiring high resolution), the

time-scales associated with the fast wave are very short

• However, the amplitude will always be small because it compresses the field.

The presence of the fast wave makes explicit time integration not practical

Must deal with Fast Wave

Summary

• Nuclear fusion is a promising energy source that will be demonstrated in

the coming decades by way of the tokamak (ITER)

• Global dynamics of the plasma in the tokamak are described by a set of

fluid like equations called the MHD equations

• A subset of the full-MHD equations with the dissipative terms removed

are called the ideal-MHD equations

• These have wave solutions that illustrate that there are 3 fundamentally

different types of waves.

• Unstable plasma motions are always associated with the slow wave

and Alfven wave.

• The fast wave is a major source of trouble computationally because it is

the fastest and the only one that propagates across the surfaces

• Largely because of the fast wave, implicit methods are essential

28